This is gsl-ref.info, produced by makeinfo version 5.1 from gsl-ref.texi. Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 The GSL Team. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections and no cover texts. A copy of the license is included in the section entitled "GNU Free Documentation License". INFO-DIR-SECTION Software libraries START-INFO-DIR-ENTRY * gsl-ref: (gsl-ref). GNU Scientific Library - Reference END-INFO-DIR-ENTRY  File: gsl-ref.info, Node: Minimization References and Further Reading, Prev: Minimization Examples, Up: One dimensional Minimization 34.9 References and Further Reading =================================== Further information on Brent's algorithm is available in the following book, Richard Brent, 'Algorithms for minimization without derivatives', Prentice-Hall (1973), republished by Dover in paperback (2002), ISBN 0-486-41998-3.  File: gsl-ref.info, Node: Multidimensional Root-Finding, Next: Multidimensional Minimization, Prev: One dimensional Minimization, Up: Top 35 Multidimensional Root-Finding ******************************** This chapter describes functions for multidimensional root-finding (solving nonlinear systems with n equations in n unknowns). The library provides low level components for a variety of iterative solvers and convergence tests. These can be combined by the user to achieve the desired solution, with full access to the intermediate steps of the iteration. Each class of methods uses the same framework, so that you can switch between solvers at runtime without needing to recompile your program. Each instance of a solver keeps track of its own state, allowing the solvers to be used in multi-threaded programs. The solvers are based on the original Fortran library MINPACK. The header file 'gsl_multiroots.h' contains prototypes for the multidimensional root finding functions and related declarations. * Menu: * Overview of Multidimensional Root Finding:: * Initializing the Multidimensional Solver:: * Providing the multidimensional system of equations to solve:: * Iteration of the multidimensional solver:: * Search Stopping Parameters for the multidimensional solver:: * Algorithms using Derivatives:: * Algorithms without Derivatives:: * Example programs for Multidimensional Root finding:: * References and Further Reading for Multidimensional Root Finding::  File: gsl-ref.info, Node: Overview of Multidimensional Root Finding, Next: Initializing the Multidimensional Solver, Up: Multidimensional Root-Finding 35.1 Overview ============= The problem of multidimensional root finding requires the simultaneous solution of n equations, f_i, in n variables, x_i, f_i (x_1, ..., x_n) = 0 for i = 1 ... n. In general there are no bracketing methods available for n dimensional systems, and no way of knowing whether any solutions exist. All algorithms proceed from an initial guess using a variant of the Newton iteration, x -> x' = x - J^{-1} f(x) where x, f are vector quantities and J is the Jacobian matrix J_{ij} = d f_i / d x_j. Additional strategies can be used to enlarge the region of convergence. These include requiring a decrease in the norm |f| on each step proposed by Newton's method, or taking steepest-descent steps in the direction of the negative gradient of |f|. Several root-finding algorithms are available within a single framework. The user provides a high-level driver for the algorithms, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are, * initialize solver state, S, for algorithm T * update S using the iteration T * test S for convergence, and repeat iteration if necessary The evaluation of the Jacobian matrix can be problematic, either because programming the derivatives is intractable or because computation of the n^2 terms of the matrix becomes too expensive. For these reasons the algorithms provided by the library are divided into two classes according to whether the derivatives are available or not. The state for solvers with an analytic Jacobian matrix is held in a 'gsl_multiroot_fdfsolver' struct. The updating procedure requires both the function and its derivatives to be supplied by the user. The state for solvers which do not use an analytic Jacobian matrix is held in a 'gsl_multiroot_fsolver' struct. The updating procedure uses only function evaluations (not derivatives). The algorithms estimate the matrix J or J^{-1} by approximate methods.  File: gsl-ref.info, Node: Initializing the Multidimensional Solver, Next: Providing the multidimensional system of equations to solve, Prev: Overview of Multidimensional Root Finding, Up: Multidimensional Root-Finding 35.2 Initializing the Solver ============================ The following functions initialize a multidimensional solver, either with or without derivatives. The solver itself depends only on the dimension of the problem and the algorithm and can be reused for different problems. -- Function: gsl_multiroot_fsolver * gsl_multiroot_fsolver_alloc (const gsl_multiroot_fsolver_type * T, size_t N) This function returns a pointer to a newly allocated instance of a solver of type T for a system of N dimensions. For example, the following code creates an instance of a hybrid solver, to solve a 3-dimensional system of equations. const gsl_multiroot_fsolver_type * T = gsl_multiroot_fsolver_hybrid; gsl_multiroot_fsolver * s = gsl_multiroot_fsolver_alloc (T, 3); If there is insufficient memory to create the solver then the function returns a null pointer and the error handler is invoked with an error code of 'GSL_ENOMEM'. -- Function: gsl_multiroot_fdfsolver * gsl_multiroot_fdfsolver_alloc (const gsl_multiroot_fdfsolver_type * T, size_t N) This function returns a pointer to a newly allocated instance of a derivative solver of type T for a system of N dimensions. For example, the following code creates an instance of a Newton-Raphson solver, for a 2-dimensional system of equations. const gsl_multiroot_fdfsolver_type * T = gsl_multiroot_fdfsolver_newton; gsl_multiroot_fdfsolver * s = gsl_multiroot_fdfsolver_alloc (T, 2); If there is insufficient memory to create the solver then the function returns a null pointer and the error handler is invoked with an error code of 'GSL_ENOMEM'. -- Function: int gsl_multiroot_fsolver_set (gsl_multiroot_fsolver * S, gsl_multiroot_function * F, const gsl_vector * X) -- Function: int gsl_multiroot_fdfsolver_set (gsl_multiroot_fdfsolver * S, gsl_multiroot_function_fdf * FDF, const gsl_vector * X) These functions set, or reset, an existing solver S to use the function F or function and derivative FDF, and the initial guess X. Note that the initial position is copied from X, this argument is not modified by subsequent iterations. -- Function: void gsl_multiroot_fsolver_free (gsl_multiroot_fsolver * S) -- Function: void gsl_multiroot_fdfsolver_free (gsl_multiroot_fdfsolver * S) These functions free all the memory associated with the solver S. -- Function: const char * gsl_multiroot_fsolver_name (const gsl_multiroot_fsolver * S) -- Function: const char * gsl_multiroot_fdfsolver_name (const gsl_multiroot_fdfsolver * S) These functions return a pointer to the name of the solver. For example, printf ("s is a '%s' solver\n", gsl_multiroot_fdfsolver_name (s)); would print something like 's is a 'newton' solver'.  File: gsl-ref.info, Node: Providing the multidimensional system of equations to solve, Next: Iteration of the multidimensional solver, Prev: Initializing the Multidimensional Solver, Up: Multidimensional Root-Finding 35.3 Providing the function to solve ==================================== You must provide n functions of n variables for the root finders to operate on. In order to allow for general parameters the functions are defined by the following data types: -- Data Type: gsl_multiroot_function This data type defines a general system of functions with parameters. 'int (* f) (const gsl_vector * X, void * PARAMS, gsl_vector * F)' this function should store the vector result f(x,params) in F for argument X and parameters PARAMS, returning an appropriate error code if the function cannot be computed. 'size_t n' the dimension of the system, i.e. the number of components of the vectors X and F. 'void * params' a pointer to the parameters of the function. Here is an example using Powell's test function, f_1(x) = A x_0 x_1 - 1, f_2(x) = exp(-x_0) + exp(-x_1) - (1 + 1/A) with A = 10^4. The following code defines a 'gsl_multiroot_function' system 'F' which you could pass to a solver: struct powell_params { double A; }; int powell (gsl_vector * x, void * p, gsl_vector * f) { struct powell_params * params = (struct powell_params *)p; const double A = (params->A); const double x0 = gsl_vector_get(x,0); const double x1 = gsl_vector_get(x,1); gsl_vector_set (f, 0, A * x0 * x1 - 1); gsl_vector_set (f, 1, (exp(-x0) + exp(-x1) - (1.0 + 1.0/A))); return GSL_SUCCESS } gsl_multiroot_function F; struct powell_params params = { 10000.0 }; F.f = &powell; F.n = 2; F.params = ¶ms; -- Data Type: gsl_multiroot_function_fdf This data type defines a general system of functions with parameters and the corresponding Jacobian matrix of derivatives, 'int (* f) (const gsl_vector * X, void * PARAMS, gsl_vector * F)' this function should store the vector result f(x,params) in F for argument X and parameters PARAMS, returning an appropriate error code if the function cannot be computed. 'int (* df) (const gsl_vector * X, void * PARAMS, gsl_matrix * J)' this function should store the N-by-N matrix result J_ij = d f_i(x,params) / d x_j in J for argument X and parameters PARAMS, returning an appropriate error code if the function cannot be computed. 'int (* fdf) (const gsl_vector * X, void * PARAMS, gsl_vector * F, gsl_matrix * J)' This function should set the values of the F and J as above, for arguments X and parameters PARAMS. This function provides an optimization of the separate functions for f(x) and J(x)--it is always faster to compute the function and its derivative at the same time. 'size_t n' the dimension of the system, i.e. the number of components of the vectors X and F. 'void * params' a pointer to the parameters of the function. The example of Powell's test function defined above can be extended to include analytic derivatives using the following code, int powell_df (gsl_vector * x, void * p, gsl_matrix * J) { struct powell_params * params = (struct powell_params *)p; const double A = (params->A); const double x0 = gsl_vector_get(x,0); const double x1 = gsl_vector_get(x,1); gsl_matrix_set (J, 0, 0, A * x1); gsl_matrix_set (J, 0, 1, A * x0); gsl_matrix_set (J, 1, 0, -exp(-x0)); gsl_matrix_set (J, 1, 1, -exp(-x1)); return GSL_SUCCESS } int powell_fdf (gsl_vector * x, void * p, gsl_matrix * f, gsl_matrix * J) { struct powell_params * params = (struct powell_params *)p; const double A = (params->A); const double x0 = gsl_vector_get(x,0); const double x1 = gsl_vector_get(x,1); const double u0 = exp(-x0); const double u1 = exp(-x1); gsl_vector_set (f, 0, A * x0 * x1 - 1); gsl_vector_set (f, 1, u0 + u1 - (1 + 1/A)); gsl_matrix_set (J, 0, 0, A * x1); gsl_matrix_set (J, 0, 1, A * x0); gsl_matrix_set (J, 1, 0, -u0); gsl_matrix_set (J, 1, 1, -u1); return GSL_SUCCESS } gsl_multiroot_function_fdf FDF; FDF.f = &powell_f; FDF.df = &powell_df; FDF.fdf = &powell_fdf; FDF.n = 2; FDF.params = 0; Note that the function 'powell_fdf' is able to reuse existing terms from the function when calculating the Jacobian, thus saving time.  File: gsl-ref.info, Node: Iteration of the multidimensional solver, Next: Search Stopping Parameters for the multidimensional solver, Prev: Providing the multidimensional system of equations to solve, Up: Multidimensional Root-Finding 35.4 Iteration ============== The following functions drive the iteration of each algorithm. Each function performs one iteration to update the state of any solver of the corresponding type. The same functions work for all solvers so that different methods can be substituted at runtime without modifications to the code. -- Function: int gsl_multiroot_fsolver_iterate (gsl_multiroot_fsolver * S) -- Function: int gsl_multiroot_fdfsolver_iterate (gsl_multiroot_fdfsolver * S) These functions perform a single iteration of the solver S. If the iteration encounters an unexpected problem then an error code will be returned, 'GSL_EBADFUNC' the iteration encountered a singular point where the function or its derivative evaluated to 'Inf' or 'NaN'. 'GSL_ENOPROG' the iteration is not making any progress, preventing the algorithm from continuing. The solver maintains a current best estimate of the root 's->x' and its function value 's->f' at all times. This information can be accessed with the following auxiliary functions, -- Function: gsl_vector * gsl_multiroot_fsolver_root (const gsl_multiroot_fsolver * S) -- Function: gsl_vector * gsl_multiroot_fdfsolver_root (const gsl_multiroot_fdfsolver * S) These functions return the current estimate of the root for the solver S, given by 's->x'. -- Function: gsl_vector * gsl_multiroot_fsolver_f (const gsl_multiroot_fsolver * S) -- Function: gsl_vector * gsl_multiroot_fdfsolver_f (const gsl_multiroot_fdfsolver * S) These functions return the function value f(x) at the current estimate of the root for the solver S, given by 's->f'. -- Function: gsl_vector * gsl_multiroot_fsolver_dx (const gsl_multiroot_fsolver * S) -- Function: gsl_vector * gsl_multiroot_fdfsolver_dx (const gsl_multiroot_fdfsolver * S) These functions return the last step dx taken by the solver S, given by 's->dx'.  File: gsl-ref.info, Node: Search Stopping Parameters for the multidimensional solver, Next: Algorithms using Derivatives, Prev: Iteration of the multidimensional solver, Up: Multidimensional Root-Finding 35.5 Search Stopping Parameters =============================== A root finding procedure should stop when one of the following conditions is true: * A multidimensional root has been found to within the user-specified precision. * A user-specified maximum number of iterations has been reached. * An error has occurred. The handling of these conditions is under user control. The functions below allow the user to test the precision of the current result in several standard ways. -- Function: int gsl_multiroot_test_delta (const gsl_vector * DX, const gsl_vector * X, double EPSABS, double EPSREL) This function tests for the convergence of the sequence by comparing the last step DX with the absolute error EPSABS and relative error EPSREL to the current position X. The test returns 'GSL_SUCCESS' if the following condition is achieved, |dx_i| < epsabs + epsrel |x_i| for each component of X and returns 'GSL_CONTINUE' otherwise. -- Function: int gsl_multiroot_test_residual (const gsl_vector * F, double EPSABS) This function tests the residual value F against the absolute error bound EPSABS. The test returns 'GSL_SUCCESS' if the following condition is achieved, \sum_i |f_i| < epsabs and returns 'GSL_CONTINUE' otherwise. This criterion is suitable for situations where the precise location of the root, x, is unimportant provided a value can be found where the residual is small enough.  File: gsl-ref.info, Node: Algorithms using Derivatives, Next: Algorithms without Derivatives, Prev: Search Stopping Parameters for the multidimensional solver, Up: Multidimensional Root-Finding 35.6 Algorithms using Derivatives ================================= The root finding algorithms described in this section make use of both the function and its derivative. They require an initial guess for the location of the root, but there is no absolute guarantee of convergence--the function must be suitable for this technique and the initial guess must be sufficiently close to the root for it to work. When the conditions are satisfied then convergence is quadratic. -- Derivative Solver: gsl_multiroot_fdfsolver_hybridsj This is a modified version of Powell's Hybrid method as implemented in the HYBRJ algorithm in MINPACK. Minpack was written by Jorge J. More', Burton S. Garbow and Kenneth E. Hillstrom. The Hybrid algorithm retains the fast convergence of Newton's method but will also reduce the residual when Newton's method is unreliable. The algorithm uses a generalized trust region to keep each step under control. In order to be accepted a proposed new position x' must satisfy the condition |D (x' - x)| < \delta, where D is a diagonal scaling matrix and \delta is the size of the trust region. The components of D are computed internally, using the column norms of the Jacobian to estimate the sensitivity of the residual to each component of x. This improves the behavior of the algorithm for badly scaled functions. On each iteration the algorithm first determines the standard Newton step by solving the system J dx = - f. If this step falls inside the trust region it is used as a trial step in the next stage. If not, the algorithm uses the linear combination of the Newton and gradient directions which is predicted to minimize the norm of the function while staying inside the trust region, dx = - \alpha J^{-1} f(x) - \beta \nabla |f(x)|^2. This combination of Newton and gradient directions is referred to as a "dogleg step". The proposed step is now tested by evaluating the function at the resulting point, x'. If the step reduces the norm of the function sufficiently then it is accepted and size of the trust region is increased. If the proposed step fails to improve the solution then the size of the trust region is decreased and another trial step is computed. The speed of the algorithm is increased by computing the changes to the Jacobian approximately, using a rank-1 update. If two successive attempts fail to reduce the residual then the full Jacobian is recomputed. The algorithm also monitors the progress of the solution and returns an error if several steps fail to make any improvement, 'GSL_ENOPROG' the iteration is not making any progress, preventing the algorithm from continuing. 'GSL_ENOPROGJ' re-evaluations of the Jacobian indicate that the iteration is not making any progress, preventing the algorithm from continuing. -- Derivative Solver: gsl_multiroot_fdfsolver_hybridj This algorithm is an unscaled version of 'hybridsj'. The steps are controlled by a spherical trust region |x' - x| < \delta, instead of a generalized region. This can be useful if the generalized region estimated by 'hybridsj' is inappropriate. -- Derivative Solver: gsl_multiroot_fdfsolver_newton Newton's Method is the standard root-polishing algorithm. The algorithm begins with an initial guess for the location of the solution. On each iteration a linear approximation to the function F is used to estimate the step which will zero all the components of the residual. The iteration is defined by the following sequence, x -> x' = x - J^{-1} f(x) where the Jacobian matrix J is computed from the derivative functions provided by F. The step dx is obtained by solving the linear system, J dx = - f(x) using LU decomposition. If the Jacobian matrix is singular, an error code of 'GSL_EDOM' is returned. -- Derivative Solver: gsl_multiroot_fdfsolver_gnewton This is a modified version of Newton's method which attempts to improve global convergence by requiring every step to reduce the Euclidean norm of the residual, |f(x)|. If the Newton step leads to an increase in the norm then a reduced step of relative size, t = (\sqrt(1 + 6 r) - 1) / (3 r) is proposed, with r being the ratio of norms |f(x')|^2/|f(x)|^2. This procedure is repeated until a suitable step size is found.  File: gsl-ref.info, Node: Algorithms without Derivatives, Next: Example programs for Multidimensional Root finding, Prev: Algorithms using Derivatives, Up: Multidimensional Root-Finding 35.7 Algorithms without Derivatives =================================== The algorithms described in this section do not require any derivative information to be supplied by the user. Any derivatives needed are approximated by finite differences. Note that if the finite-differencing step size chosen by these routines is inappropriate, an explicit user-supplied numerical derivative can always be used with the algorithms described in the previous section. -- Solver: gsl_multiroot_fsolver_hybrids This is a version of the Hybrid algorithm which replaces calls to the Jacobian function by its finite difference approximation. The finite difference approximation is computed using 'gsl_multiroots_fdjac' with a relative step size of 'GSL_SQRT_DBL_EPSILON'. Note that this step size will not be suitable for all problems. -- Solver: gsl_multiroot_fsolver_hybrid This is a finite difference version of the Hybrid algorithm without internal scaling. -- Solver: gsl_multiroot_fsolver_dnewton The "discrete Newton algorithm" is the simplest method of solving a multidimensional system. It uses the Newton iteration x -> x - J^{-1} f(x) where the Jacobian matrix J is approximated by taking finite differences of the function F. The approximation scheme used by this implementation is, J_{ij} = (f_i(x + \delta_j) - f_i(x)) / \delta_j where \delta_j is a step of size \sqrt\epsilon |x_j| with \epsilon being the machine precision (\epsilon \approx 2.22 \times 10^-16). The order of convergence of Newton's algorithm is quadratic, but the finite differences require n^2 function evaluations on each iteration. The algorithm may become unstable if the finite differences are not a good approximation to the true derivatives. -- Solver: gsl_multiroot_fsolver_broyden The "Broyden algorithm" is a version of the discrete Newton algorithm which attempts to avoids the expensive update of the Jacobian matrix on each iteration. The changes to the Jacobian are also approximated, using a rank-1 update, J^{-1} \to J^{-1} - (J^{-1} df - dx) dx^T J^{-1} / dx^T J^{-1} df where the vectors dx and df are the changes in x and f. On the first iteration the inverse Jacobian is estimated using finite differences, as in the discrete Newton algorithm. This approximation gives a fast update but is unreliable if the changes are not small, and the estimate of the inverse Jacobian becomes worse as time passes. The algorithm has a tendency to become unstable unless it starts close to the root. The Jacobian is refreshed if this instability is detected (consult the source for details). This algorithm is included only for demonstration purposes, and is not recommended for serious use.  File: gsl-ref.info, Node: Example programs for Multidimensional Root finding, Next: References and Further Reading for Multidimensional Root Finding, Prev: Algorithms without Derivatives, Up: Multidimensional Root-Finding 35.8 Examples ============= The multidimensional solvers are used in a similar way to the one-dimensional root finding algorithms. This first example demonstrates the 'hybrids' scaled-hybrid algorithm, which does not require derivatives. The program solves the Rosenbrock system of equations, f_1 (x, y) = a (1 - x) f_2 (x, y) = b (y - x^2) with a = 1, b = 10. The solution of this system lies at (x,y) = (1,1) in a narrow valley. The first stage of the program is to define the system of equations, #include #include #include #include struct rparams { double a; double b; }; int rosenbrock_f (const gsl_vector * x, void *params, gsl_vector * f) { double a = ((struct rparams *) params)->a; double b = ((struct rparams *) params)->b; const double x0 = gsl_vector_get (x, 0); const double x1 = gsl_vector_get (x, 1); const double y0 = a * (1 - x0); const double y1 = b * (x1 - x0 * x0); gsl_vector_set (f, 0, y0); gsl_vector_set (f, 1, y1); return GSL_SUCCESS; } The main program begins by creating the function object 'f', with the arguments '(x,y)' and parameters '(a,b)'. The solver 's' is initialized to use this function, with the 'hybrids' method. int main (void) { const gsl_multiroot_fsolver_type *T; gsl_multiroot_fsolver *s; int status; size_t i, iter = 0; const size_t n = 2; struct rparams p = {1.0, 10.0}; gsl_multiroot_function f = {&rosenbrock_f, n, &p}; double x_init[2] = {-10.0, -5.0}; gsl_vector *x = gsl_vector_alloc (n); gsl_vector_set (x, 0, x_init[0]); gsl_vector_set (x, 1, x_init[1]); T = gsl_multiroot_fsolver_hybrids; s = gsl_multiroot_fsolver_alloc (T, 2); gsl_multiroot_fsolver_set (s, &f, x); print_state (iter, s); do { iter++; status = gsl_multiroot_fsolver_iterate (s); print_state (iter, s); if (status) /* check if solver is stuck */ break; status = gsl_multiroot_test_residual (s->f, 1e-7); } while (status == GSL_CONTINUE && iter < 1000); printf ("status = %s\n", gsl_strerror (status)); gsl_multiroot_fsolver_free (s); gsl_vector_free (x); return 0; } Note that it is important to check the return status of each solver step, in case the algorithm becomes stuck. If an error condition is detected, indicating that the algorithm cannot proceed, then the error can be reported to the user, a new starting point chosen or a different algorithm used. The intermediate state of the solution is displayed by the following function. The solver state contains the vector 's->x' which is the current position, and the vector 's->f' with corresponding function values. int print_state (size_t iter, gsl_multiroot_fsolver * s) { printf ("iter = %3u x = % .3f % .3f " "f(x) = % .3e % .3e\n", iter, gsl_vector_get (s->x, 0), gsl_vector_get (s->x, 1), gsl_vector_get (s->f, 0), gsl_vector_get (s->f, 1)); } Here are the results of running the program. The algorithm is started at (-10,-5) far from the solution. Since the solution is hidden in a narrow valley the earliest steps follow the gradient of the function downhill, in an attempt to reduce the large value of the residual. Once the root has been approximately located, on iteration 8, the Newton behavior takes over and convergence is very rapid. iter = 0 x = -10.000 -5.000 f(x) = 1.100e+01 -1.050e+03 iter = 1 x = -10.000 -5.000 f(x) = 1.100e+01 -1.050e+03 iter = 2 x = -3.976 24.827 f(x) = 4.976e+00 9.020e+01 iter = 3 x = -3.976 24.827 f(x) = 4.976e+00 9.020e+01 iter = 4 x = -3.976 24.827 f(x) = 4.976e+00 9.020e+01 iter = 5 x = -1.274 -5.680 f(x) = 2.274e+00 -7.302e+01 iter = 6 x = -1.274 -5.680 f(x) = 2.274e+00 -7.302e+01 iter = 7 x = 0.249 0.298 f(x) = 7.511e-01 2.359e+00 iter = 8 x = 0.249 0.298 f(x) = 7.511e-01 2.359e+00 iter = 9 x = 1.000 0.878 f(x) = 1.268e-10 -1.218e+00 iter = 10 x = 1.000 0.989 f(x) = 1.124e-11 -1.080e-01 iter = 11 x = 1.000 1.000 f(x) = 0.000e+00 0.000e+00 status = success Note that the algorithm does not update the location on every iteration. Some iterations are used to adjust the trust-region parameter, after trying a step which was found to be divergent, or to recompute the Jacobian, when poor convergence behavior is detected. The next example program adds derivative information, in order to accelerate the solution. There are two derivative functions 'rosenbrock_df' and 'rosenbrock_fdf'. The latter computes both the function and its derivative simultaneously. This allows the optimization of any common terms. For simplicity we substitute calls to the separate 'f' and 'df' functions at this point in the code below. int rosenbrock_df (const gsl_vector * x, void *params, gsl_matrix * J) { const double a = ((struct rparams *) params)->a; const double b = ((struct rparams *) params)->b; const double x0 = gsl_vector_get (x, 0); const double df00 = -a; const double df01 = 0; const double df10 = -2 * b * x0; const double df11 = b; gsl_matrix_set (J, 0, 0, df00); gsl_matrix_set (J, 0, 1, df01); gsl_matrix_set (J, 1, 0, df10); gsl_matrix_set (J, 1, 1, df11); return GSL_SUCCESS; } int rosenbrock_fdf (const gsl_vector * x, void *params, gsl_vector * f, gsl_matrix * J) { rosenbrock_f (x, params, f); rosenbrock_df (x, params, J); return GSL_SUCCESS; } The main program now makes calls to the corresponding 'fdfsolver' versions of the functions, int main (void) { const gsl_multiroot_fdfsolver_type *T; gsl_multiroot_fdfsolver *s; int status; size_t i, iter = 0; const size_t n = 2; struct rparams p = {1.0, 10.0}; gsl_multiroot_function_fdf f = {&rosenbrock_f, &rosenbrock_df, &rosenbrock_fdf, n, &p}; double x_init[2] = {-10.0, -5.0}; gsl_vector *x = gsl_vector_alloc (n); gsl_vector_set (x, 0, x_init[0]); gsl_vector_set (x, 1, x_init[1]); T = gsl_multiroot_fdfsolver_gnewton; s = gsl_multiroot_fdfsolver_alloc (T, n); gsl_multiroot_fdfsolver_set (s, &f, x); print_state (iter, s); do { iter++; status = gsl_multiroot_fdfsolver_iterate (s); print_state (iter, s); if (status) break; status = gsl_multiroot_test_residual (s->f, 1e-7); } while (status == GSL_CONTINUE && iter < 1000); printf ("status = %s\n", gsl_strerror (status)); gsl_multiroot_fdfsolver_free (s); gsl_vector_free (x); return 0; } The addition of derivative information to the 'hybrids' solver does not make any significant difference to its behavior, since it able to approximate the Jacobian numerically with sufficient accuracy. To illustrate the behavior of a different derivative solver we switch to 'gnewton'. This is a traditional Newton solver with the constraint that it scales back its step if the full step would lead "uphill". Here is the output for the 'gnewton' algorithm, iter = 0 x = -10.000 -5.000 f(x) = 1.100e+01 -1.050e+03 iter = 1 x = -4.231 -65.317 f(x) = 5.231e+00 -8.321e+02 iter = 2 x = 1.000 -26.358 f(x) = -8.882e-16 -2.736e+02 iter = 3 x = 1.000 1.000 f(x) = -2.220e-16 -4.441e-15 status = success The convergence is much more rapid, but takes a wide excursion out to the point (-4.23,-65.3). This could cause the algorithm to go astray in a realistic application. The hybrid algorithm follows the downhill path to the solution more reliably.  File: gsl-ref.info, Node: References and Further Reading for Multidimensional Root Finding, Prev: Example programs for Multidimensional Root finding, Up: Multidimensional Root-Finding 35.9 References and Further Reading =================================== The original version of the Hybrid method is described in the following articles by Powell, M.J.D. Powell, "A Hybrid Method for Nonlinear Equations" (Chap 6, p 87-114) and "A Fortran Subroutine for Solving systems of Nonlinear Algebraic Equations" (Chap 7, p 115-161), in 'Numerical Methods for Nonlinear Algebraic Equations', P. Rabinowitz, editor. Gordon and Breach, 1970. The following papers are also relevant to the algorithms described in this section, J.J. More', M.Y. Cosnard, "Numerical Solution of Nonlinear Equations", 'ACM Transactions on Mathematical Software', Vol 5, No 1, (1979), p 64-85 C.G. Broyden, "A Class of Methods for Solving Nonlinear Simultaneous Equations", 'Mathematics of Computation', Vol 19 (1965), p 577-593 J.J. More', B.S. Garbow, K.E. Hillstrom, "Testing Unconstrained Optimization Software", ACM Transactions on Mathematical Software, Vol 7, No 1 (1981), p 17-41  File: gsl-ref.info, Node: Multidimensional Minimization, Next: Least-Squares Fitting, Prev: Multidimensional Root-Finding, Up: Top 36 Multidimensional Minimization ******************************** This chapter describes routines for finding minima of arbitrary multidimensional functions. The library provides low level components for a variety of iterative minimizers and convergence tests. These can be combined by the user to achieve the desired solution, while providing full access to the intermediate steps of the algorithms. Each class of methods uses the same framework, so that you can switch between minimizers at runtime without needing to recompile your program. Each instance of a minimizer keeps track of its own state, allowing the minimizers to be used in multi-threaded programs. The minimization algorithms can be used to maximize a function by inverting its sign. The header file 'gsl_multimin.h' contains prototypes for the minimization functions and related declarations. * Menu: * Multimin Overview:: * Multimin Caveats:: * Initializing the Multidimensional Minimizer:: * Providing a function to minimize:: * Multimin Iteration:: * Multimin Stopping Criteria:: * Multimin Algorithms with Derivatives:: * Multimin Algorithms without Derivatives:: * Multimin Examples:: * Multimin References and Further Reading::  File: gsl-ref.info, Node: Multimin Overview, Next: Multimin Caveats, Up: Multidimensional Minimization 36.1 Overview ============= The problem of multidimensional minimization requires finding a point x such that the scalar function, f(x_1, ..., x_n) takes a value which is lower than at any neighboring point. For smooth functions the gradient g = \nabla f vanishes at the minimum. In general there are no bracketing methods available for the minimization of n-dimensional functions. The algorithms proceed from an initial guess using a search algorithm which attempts to move in a downhill direction. Algorithms making use of the gradient of the function perform a one-dimensional line minimisation along this direction until the lowest point is found to a suitable tolerance. The search direction is then updated with local information from the function and its derivatives, and the whole process repeated until the true n-dimensional minimum is found. Algorithms which do not require the gradient of the function use different strategies. For example, the Nelder-Mead Simplex algorithm maintains n+1 trial parameter vectors as the vertices of a n-dimensional simplex. On each iteration it tries to improve the worst vertex of the simplex by geometrical transformations. The iterations are continued until the overall size of the simplex has decreased sufficiently. Both types of algorithms use a standard framework. The user provides a high-level driver for the algorithms, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are, * initialize minimizer state, S, for algorithm T * update S using the iteration T * test S for convergence, and repeat iteration if necessary Each iteration step consists either of an improvement to the line-minimisation in the current direction or an update to the search direction itself. The state for the minimizers is held in a 'gsl_multimin_fdfminimizer' struct or a 'gsl_multimin_fminimizer' struct.  File: gsl-ref.info, Node: Multimin Caveats, Next: Initializing the Multidimensional Minimizer, Prev: Multimin Overview, Up: Multidimensional Minimization 36.2 Caveats ============ Note that the minimization algorithms can only search for one local minimum at a time. When there are several local minima in the search area, the first minimum to be found will be returned; however it is difficult to predict which of the minima this will be. In most cases, no error will be reported if you try to find a local minimum in an area where there is more than one. It is also important to note that the minimization algorithms find local minima; there is no way to determine whether a minimum is a global minimum of the function in question.  File: gsl-ref.info, Node: Initializing the Multidimensional Minimizer, Next: Providing a function to minimize, Prev: Multimin Caveats, Up: Multidimensional Minimization 36.3 Initializing the Multidimensional Minimizer ================================================ The following function initializes a multidimensional minimizer. The minimizer itself depends only on the dimension of the problem and the algorithm and can be reused for different problems. -- Function: gsl_multimin_fdfminimizer * gsl_multimin_fdfminimizer_alloc (const gsl_multimin_fdfminimizer_type * T, size_t N) -- Function: gsl_multimin_fminimizer * gsl_multimin_fminimizer_alloc (const gsl_multimin_fminimizer_type * T, size_t N) This function returns a pointer to a newly allocated instance of a minimizer of type T for an N-dimension function. If there is insufficient memory to create the minimizer then the function returns a null pointer and the error handler is invoked with an error code of 'GSL_ENOMEM'. -- Function: int gsl_multimin_fdfminimizer_set (gsl_multimin_fdfminimizer * S, gsl_multimin_function_fdf * FDF, const gsl_vector * X, double STEP_SIZE, double TOL) -- Function: int gsl_multimin_fminimizer_set (gsl_multimin_fminimizer * S, gsl_multimin_function * F, const gsl_vector * X, const gsl_vector * STEP_SIZE) The function 'gsl_multimin_fdfminimizer_set' initializes the minimizer S to minimize the function FDF starting from the initial point X. The size of the first trial step is given by STEP_SIZE. The accuracy of the line minimization is specified by TOL. The precise meaning of this parameter depends on the method used. Typically the line minimization is considered successful if the gradient of the function g is orthogonal to the current search direction p to a relative accuracy of TOL, where dot(p,g) < tol |p| |g|. A TOL value of 0.1 is suitable for most purposes, since line minimization only needs to be carried out approximately. Note that setting TOL to zero will force the use of "exact" line-searches, which are extremely expensive. The function 'gsl_multimin_fminimizer_set' initializes the minimizer S to minimize the function F, starting from the initial point X. The size of the initial trial steps is given in vector STEP_SIZE. The precise meaning of this parameter depends on the method used. -- Function: void gsl_multimin_fdfminimizer_free (gsl_multimin_fdfminimizer * S) -- Function: void gsl_multimin_fminimizer_free (gsl_multimin_fminimizer * S) This function frees all the memory associated with the minimizer S. -- Function: const char * gsl_multimin_fdfminimizer_name (const gsl_multimin_fdfminimizer * S) -- Function: const char * gsl_multimin_fminimizer_name (const gsl_multimin_fminimizer * S) This function returns a pointer to the name of the minimizer. For example, printf ("s is a '%s' minimizer\n", gsl_multimin_fdfminimizer_name (s)); would print something like 's is a 'conjugate_pr' minimizer'.  File: gsl-ref.info, Node: Providing a function to minimize, Next: Multimin Iteration, Prev: Initializing the Multidimensional Minimizer, Up: Multidimensional Minimization 36.4 Providing a function to minimize ===================================== You must provide a parametric function of n variables for the minimizers to operate on. You may also need to provide a routine which calculates the gradient of the function and a third routine which calculates both the function value and the gradient together. In order to allow for general parameters the functions are defined by the following data types: -- Data Type: gsl_multimin_function_fdf This data type defines a general function of n variables with parameters and the corresponding gradient vector of derivatives, 'double (* f) (const gsl_vector * X, void * PARAMS)' this function should return the result f(x,params) for argument X and parameters PARAMS. If the function cannot be computed, an error value of 'GSL_NAN' should be returned. 'void (* df) (const gsl_vector * X, void * PARAMS, gsl_vector * G)' this function should store the N-dimensional gradient g_i = d f(x,params) / d x_i in the vector G for argument X and parameters PARAMS, returning an appropriate error code if the function cannot be computed. 'void (* fdf) (const gsl_vector * X, void * PARAMS, double * f, gsl_vector * G)' This function should set the values of the F and G as above, for arguments X and parameters PARAMS. This function provides an optimization of the separate functions for f(x) and g(x)--it is always faster to compute the function and its derivative at the same time. 'size_t n' the dimension of the system, i.e. the number of components of the vectors X. 'void * params' a pointer to the parameters of the function. -- Data Type: gsl_multimin_function This data type defines a general function of n variables with parameters, 'double (* f) (const gsl_vector * X, void * PARAMS)' this function should return the result f(x,params) for argument X and parameters PARAMS. If the function cannot be computed, an error value of 'GSL_NAN' should be returned. 'size_t n' the dimension of the system, i.e. the number of components of the vectors X. 'void * params' a pointer to the parameters of the function. The following example function defines a simple two-dimensional paraboloid with five parameters, /* Paraboloid centered on (p[0],p[1]), with scale factors (p[2],p[3]) and minimum p[4] */ double my_f (const gsl_vector *v, void *params) { double x, y; double *p = (double *)params; x = gsl_vector_get(v, 0); y = gsl_vector_get(v, 1); return p[2] * (x - p[0]) * (x - p[0]) + p[3] * (y - p[1]) * (y - p[1]) + p[4]; } /* The gradient of f, df = (df/dx, df/dy). */ void my_df (const gsl_vector *v, void *params, gsl_vector *df) { double x, y; double *p = (double *)params; x = gsl_vector_get(v, 0); y = gsl_vector_get(v, 1); gsl_vector_set(df, 0, 2.0 * p[2] * (x - p[0])); gsl_vector_set(df, 1, 2.0 * p[3] * (y - p[1])); } /* Compute both f and df together. */ void my_fdf (const gsl_vector *x, void *params, double *f, gsl_vector *df) { *f = my_f(x, params); my_df(x, params, df); } The function can be initialized using the following code, gsl_multimin_function_fdf my_func; /* Paraboloid center at (1,2), scale factors (10, 20), minimum value 30 */ double p[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 }; my_func.n = 2; /* number of function components */ my_func.f = &my_f; my_func.df = &my_df; my_func.fdf = &my_fdf; my_func.params = (void *)p;  File: gsl-ref.info, Node: Multimin Iteration, Next: Multimin Stopping Criteria, Prev: Providing a function to minimize, Up: Multidimensional Minimization 36.5 Iteration ============== The following function drives the iteration of each algorithm. The function performs one iteration to update the state of the minimizer. The same function works for all minimizers so that different methods can be substituted at runtime without modifications to the code. -- Function: int gsl_multimin_fdfminimizer_iterate (gsl_multimin_fdfminimizer * S) -- Function: int gsl_multimin_fminimizer_iterate (gsl_multimin_fminimizer * S) These functions perform a single iteration of the minimizer S. If the iteration encounters an unexpected problem then an error code will be returned. The error code 'GSL_ENOPROG' signifies that the minimizer is unable to improve on its current estimate, either due to numerical difficulty or because a genuine local minimum has been reached. The minimizer maintains a current best estimate of the minimum at all times. This information can be accessed with the following auxiliary functions, -- Function: gsl_vector * gsl_multimin_fdfminimizer_x (const gsl_multimin_fdfminimizer * S) -- Function: gsl_vector * gsl_multimin_fminimizer_x (const gsl_multimin_fminimizer * S) -- Function: double gsl_multimin_fdfminimizer_minimum (const gsl_multimin_fdfminimizer * S) -- Function: double gsl_multimin_fminimizer_minimum (const gsl_multimin_fminimizer * S) -- Function: gsl_vector * gsl_multimin_fdfminimizer_gradient (const gsl_multimin_fdfminimizer * S) -- Function: double gsl_multimin_fminimizer_size (const gsl_multimin_fminimizer * S) These functions return the current best estimate of the location of the minimum, the value of the function at that point, its gradient, and minimizer specific characteristic size for the minimizer S. -- Function: int gsl_multimin_fdfminimizer_restart (gsl_multimin_fdfminimizer * S) This function resets the minimizer S to use the current point as a new starting point.  File: gsl-ref.info, Node: Multimin Stopping Criteria, Next: Multimin Algorithms with Derivatives, Prev: Multimin Iteration, Up: Multidimensional Minimization 36.6 Stopping Criteria ====================== A minimization procedure should stop when one of the following conditions is true: * A minimum has been found to within the user-specified precision. * A user-specified maximum number of iterations has been reached. * An error has occurred. The handling of these conditions is under user control. The functions below allow the user to test the precision of the current result. -- Function: int gsl_multimin_test_gradient (const gsl_vector * G, double EPSABS) This function tests the norm of the gradient G against the absolute tolerance EPSABS. The gradient of a multidimensional function goes to zero at a minimum. The test returns 'GSL_SUCCESS' if the following condition is achieved, |g| < epsabs and returns 'GSL_CONTINUE' otherwise. A suitable choice of EPSABS can be made from the desired accuracy in the function for small variations in x. The relationship between these quantities is given by \delta f = g \delta x. -- Function: int gsl_multimin_test_size (const double SIZE, double EPSABS) This function tests the minimizer specific characteristic size (if applicable to the used minimizer) against absolute tolerance EPSABS. The test returns 'GSL_SUCCESS' if the size is smaller than tolerance, otherwise 'GSL_CONTINUE' is returned.  File: gsl-ref.info, Node: Multimin Algorithms with Derivatives, Next: Multimin Algorithms without Derivatives, Prev: Multimin Stopping Criteria, Up: Multidimensional Minimization 36.7 Algorithms with Derivatives ================================ There are several minimization methods available. The best choice of algorithm depends on the problem. The algorithms described in this section use the value of the function and its gradient at each evaluation point. -- Minimizer: gsl_multimin_fdfminimizer_conjugate_fr This is the Fletcher-Reeves conjugate gradient algorithm. The conjugate gradient algorithm proceeds as a succession of line minimizations. The sequence of search directions is used to build up an approximation to the curvature of the function in the neighborhood of the minimum. An initial search direction P is chosen using the gradient, and line minimization is carried out in that direction. The accuracy of the line minimization is specified by the parameter TOL. The minimum along this line occurs when the function gradient G and the search direction P are orthogonal. The line minimization terminates when dot(p,g) < tol |p| |g|. The search direction is updated using the Fletcher-Reeves formula p' = g' - \beta g where \beta=-|g'|^2/|g|^2, and the line minimization is then repeated for the new search direction. -- Minimizer: gsl_multimin_fdfminimizer_conjugate_pr This is the Polak-Ribiere conjugate gradient algorithm. It is similar to the Fletcher-Reeves method, differing only in the choice of the coefficient \beta. Both methods work well when the evaluation point is close enough to the minimum of the objective function that it is well approximated by a quadratic hypersurface. -- Minimizer: gsl_multimin_fdfminimizer_vector_bfgs2 -- Minimizer: gsl_multimin_fdfminimizer_vector_bfgs These methods use the vector Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. This is a quasi-Newton method which builds up an approximation to the second derivatives of the function f using the difference between successive gradient vectors. By combining the first and second derivatives the algorithm is able to take Newton-type steps towards the function minimum, assuming quadratic behavior in that region. The 'bfgs2' version of this minimizer is the most efficient version available, and is a faithful implementation of the line minimization scheme described in Fletcher's 'Practical Methods of Optimization', Algorithms 2.6.2 and 2.6.4. It supersedes the original 'bfgs' routine and requires substantially fewer function and gradient evaluations. The user-supplied tolerance TOL corresponds to the parameter \sigma used by Fletcher. A value of 0.1 is recommended for typical use (larger values correspond to less accurate line searches). -- Minimizer: gsl_multimin_fdfminimizer_steepest_descent The steepest descent algorithm follows the downhill gradient of the function at each step. When a downhill step is successful the step-size is increased by a factor of two. If the downhill step leads to a higher function value then the algorithm backtracks and the step size is decreased using the parameter TOL. A suitable value of TOL for most applications is 0.1. The steepest descent method is inefficient and is included only for demonstration purposes.  File: gsl-ref.info, Node: Multimin Algorithms without Derivatives, Next: Multimin Examples, Prev: Multimin Algorithms with Derivatives, Up: Multidimensional Minimization 36.8 Algorithms without Derivatives =================================== The algorithms described in this section use only the value of the function at each evaluation point. -- Minimizer: gsl_multimin_fminimizer_nmsimplex2 -- Minimizer: gsl_multimin_fminimizer_nmsimplex These methods use the Simplex algorithm of Nelder and Mead. Starting from the initial vector X = p_0, the algorithm constructs an additional n vectors p_i using the step size vector s = STEP_SIZE as follows: p_0 = (x_0, x_1, ... , x_n) p_1 = (x_0 + s_0, x_1, ... , x_n) p_2 = (x_0, x_1 + s_1, ... , x_n) ... = ... p_n = (x_0, x_1, ... , x_n + s_n) These vectors form the n+1 vertices of a simplex in n dimensions. On each iteration the algorithm uses simple geometrical transformations to update the vector corresponding to the highest function value. The geometric transformations are reflection, reflection followed by expansion, contraction and multiple contraction. Using these transformations the simplex moves through the space towards the minimum, where it contracts itself. After each iteration, the best vertex is returned. Note, that due to the nature of the algorithm not every step improves the current best parameter vector. Usually several iterations are required. The minimizer-specific characteristic size is calculated as the average distance from the geometrical center of the simplex to all its vertices. This size can be used as a stopping criteria, as the simplex contracts itself near the minimum. The size is returned by the function 'gsl_multimin_fminimizer_size'. The 'nmsimplex2' version of this minimiser is a new O(N) operations implementation of the earlier O(N^2) operations 'nmsimplex' minimiser. It uses the same underlying algorithm, but the simplex updates are computed more efficiently for high-dimensional problems. In addition, the size of simplex is calculated as the RMS distance of each vertex from the center rather than the mean distance, allowing a linear update of this quantity on each step. The memory usage is O(N^2) for both algorithms. -- Minimizer: gsl_multimin_fminimizer_nmsimplex2rand This method is a variant of 'nmsimplex2' which initialises the simplex around the starting point X using a randomly-oriented set of basis vectors instead of the fixed coordinate axes. The final dimensions of the simplex are scaled along the coordinate axes by the vector STEP_SIZE. The randomisation uses a simple deterministic generator so that repeated calls to 'gsl_multimin_fminimizer_set' for a given solver object will vary the orientation in a well-defined way.  File: gsl-ref.info, Node: Multimin Examples, Next: Multimin References and Further Reading, Prev: Multimin Algorithms without Derivatives, Up: Multidimensional Minimization 36.9 Examples ============= This example program finds the minimum of the paraboloid function defined earlier. The location of the minimum is offset from the origin in x and y, and the function value at the minimum is non-zero. The main program is given below, it requires the example function given earlier in this chapter. int main (void) { size_t iter = 0; int status; const gsl_multimin_fdfminimizer_type *T; gsl_multimin_fdfminimizer *s; /* Position of the minimum (1,2), scale factors 10,20, height 30. */ double par[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 }; gsl_vector *x; gsl_multimin_function_fdf my_func; my_func.n = 2; my_func.f = my_f; my_func.df = my_df; my_func.fdf = my_fdf; my_func.params = par; /* Starting point, x = (5,7) */ x = gsl_vector_alloc (2); gsl_vector_set (x, 0, 5.0); gsl_vector_set (x, 1, 7.0); T = gsl_multimin_fdfminimizer_conjugate_fr; s = gsl_multimin_fdfminimizer_alloc (T, 2); gsl_multimin_fdfminimizer_set (s, &my_func, x, 0.01, 1e-4); do { iter++; status = gsl_multimin_fdfminimizer_iterate (s); if (status) break; status = gsl_multimin_test_gradient (s->gradient, 1e-3); if (status == GSL_SUCCESS) printf ("Minimum found at:\n"); printf ("%5d %.5f %.5f %10.5f\n", iter, gsl_vector_get (s->x, 0), gsl_vector_get (s->x, 1), s->f); } while (status == GSL_CONTINUE && iter < 100); gsl_multimin_fdfminimizer_free (s); gsl_vector_free (x); return 0; } The initial step-size is chosen as 0.01, a conservative estimate in this case, and the line minimization parameter is set at 0.0001. The program terminates when the norm of the gradient has been reduced below 0.001. The output of the program is shown below, x y f 1 4.99629 6.99072 687.84780 2 4.98886 6.97215 683.55456 3 4.97400 6.93501 675.01278 4 4.94429 6.86073 658.10798 5 4.88487 6.71217 625.01340 6 4.76602 6.41506 561.68440 7 4.52833 5.82083 446.46694 8 4.05295 4.63238 261.79422 9 3.10219 2.25548 75.49762 10 2.85185 1.62963 67.03704 11 2.19088 1.76182 45.31640 12 0.86892 2.02622 30.18555 Minimum found at: 13 1.00000 2.00000 30.00000 Note that the algorithm gradually increases the step size as it successfully moves downhill, as can be seen by plotting the successive points. The conjugate gradient algorithm finds the minimum on its second direction because the function is purely quadratic. Additional iterations would be needed for a more complicated function. Here is another example using the Nelder-Mead Simplex algorithm to minimize the same example object function, as above. int main(void) { double par[5] = {1.0, 2.0, 10.0, 20.0, 30.0}; const gsl_multimin_fminimizer_type *T = gsl_multimin_fminimizer_nmsimplex2; gsl_multimin_fminimizer *s = NULL; gsl_vector *ss, *x; gsl_multimin_function minex_func; size_t iter = 0; int status; double size; /* Starting point */ x = gsl_vector_alloc (2); gsl_vector_set (x, 0, 5.0); gsl_vector_set (x, 1, 7.0); /* Set initial step sizes to 1 */ ss = gsl_vector_alloc (2); gsl_vector_set_all (ss, 1.0); /* Initialize method and iterate */ minex_func.n = 2; minex_func.f = my_f; minex_func.params = par; s = gsl_multimin_fminimizer_alloc (T, 2); gsl_multimin_fminimizer_set (s, &minex_func, x, ss); do { iter++; status = gsl_multimin_fminimizer_iterate(s); if (status) break; size = gsl_multimin_fminimizer_size (s); status = gsl_multimin_test_size (size, 1e-2); if (status == GSL_SUCCESS) { printf ("converged to minimum at\n"); } printf ("%5d %10.3e %10.3e f() = %7.3f size = %.3f\n", iter, gsl_vector_get (s->x, 0), gsl_vector_get (s->x, 1), s->fval, size); } while (status == GSL_CONTINUE && iter < 100); gsl_vector_free(x); gsl_vector_free(ss); gsl_multimin_fminimizer_free (s); return status; } The minimum search stops when the Simplex size drops to 0.01. The output is shown below. 1 6.500e+00 5.000e+00 f() = 512.500 size = 1.130 2 5.250e+00 4.000e+00 f() = 290.625 size = 1.409 3 5.250e+00 4.000e+00 f() = 290.625 size = 1.409 4 5.500e+00 1.000e+00 f() = 252.500 size = 1.409 5 2.625e+00 3.500e+00 f() = 101.406 size = 1.847 6 2.625e+00 3.500e+00 f() = 101.406 size = 1.847 7 0.000e+00 3.000e+00 f() = 60.000 size = 1.847 8 2.094e+00 1.875e+00 f() = 42.275 size = 1.321 9 2.578e-01 1.906e+00 f() = 35.684 size = 1.069 10 5.879e-01 2.445e+00 f() = 35.664 size = 0.841 11 1.258e+00 2.025e+00 f() = 30.680 size = 0.476 12 1.258e+00 2.025e+00 f() = 30.680 size = 0.367 13 1.093e+00 1.849e+00 f() = 30.539 size = 0.300 14 8.830e-01 2.004e+00 f() = 30.137 size = 0.172 15 8.830e-01 2.004e+00 f() = 30.137 size = 0.126 16 9.582e-01 2.060e+00 f() = 30.090 size = 0.106 17 1.022e+00 2.004e+00 f() = 30.005 size = 0.063 18 1.022e+00 2.004e+00 f() = 30.005 size = 0.043 19 1.022e+00 2.004e+00 f() = 30.005 size = 0.043 20 1.022e+00 2.004e+00 f() = 30.005 size = 0.027 21 1.022e+00 2.004e+00 f() = 30.005 size = 0.022 22 9.920e-01 1.997e+00 f() = 30.001 size = 0.016 23 9.920e-01 1.997e+00 f() = 30.001 size = 0.013 converged to minimum at 24 9.920e-01 1.997e+00 f() = 30.001 size = 0.008 The simplex size first increases, while the simplex moves towards the minimum. After a while the size begins to decrease as the simplex contracts around the minimum.  File: gsl-ref.info, Node: Multimin References and Further Reading, Prev: Multimin Examples, Up: Multidimensional Minimization 36.10 References and Further Reading ==================================== The conjugate gradient and BFGS methods are described in detail in the following book, R. Fletcher, 'Practical Methods of Optimization (Second Edition)' Wiley (1987), ISBN 0471915475. A brief description of multidimensional minimization algorithms and more recent references can be found in, C.W. Ueberhuber, 'Numerical Computation (Volume 2)', Chapter 14, Section 4.4 "Minimization Methods", p. 325-335, Springer (1997), ISBN 3-540-62057-5. The simplex algorithm is described in the following paper, J.A. Nelder and R. Mead, 'A simplex method for function minimization', Computer Journal vol. 7 (1965), 308-313.  File: gsl-ref.info, Node: Least-Squares Fitting, Next: Nonlinear Least-Squares Fitting, Prev: Multidimensional Minimization, Up: Top 37 Least-Squares Fitting ************************ This chapter describes routines for performing least squares fits to experimental data using linear combinations of functions. The data may be weighted or unweighted, i.e. with known or unknown errors. For weighted data the functions compute the best fit parameters and their associated covariance matrix. For unweighted data the covariance matrix is estimated from the scatter of the points, giving a variance-covariance matrix. The functions are divided into separate versions for simple one- or two-parameter regression and multiple-parameter fits. The functions are declared in the header file 'gsl_fit.h'. * Menu: * Fitting Overview:: * Linear regression:: * Linear fitting without a constant term:: * Multi-parameter fitting:: * Robust linear regression:: * Troubleshooting:: * Fitting Examples:: * Fitting References and Further Reading::  File: gsl-ref.info, Node: Fitting Overview, Next: Linear regression, Up: Least-Squares Fitting 37.1 Overview ============= Least-squares fits are found by minimizing \chi^2 (chi-squared), the weighted sum of squared residuals over n experimental datapoints (x_i, y_i) for the model Y(c,x), \chi^2 = \sum_i w_i (y_i - Y(c, x_i))^2 The p parameters of the model are c = {c_0, c_1, ...}. The weight factors w_i are given by w_i = 1/\sigma_i^2, where \sigma_i is the experimental error on the data-point y_i. The errors are assumed to be Gaussian and uncorrelated. For unweighted data the chi-squared sum is computed without any weight factors. The fitting routines return the best-fit parameters c and their p \times p covariance matrix. The covariance matrix measures the statistical errors on the best-fit parameters resulting from the errors on the data, \sigma_i, and is defined as C_{ab} = <\delta c_a \delta c_b> where < > denotes an average over the Gaussian error distributions of the underlying datapoints. The covariance matrix is calculated by error propagation from the data errors \sigma_i. The change in a fitted parameter \delta c_a caused by a small change in the data \delta y_i is given by \delta c_a = \sum_i (dc_a/dy_i) \delta y_i allowing the covariance matrix to be written in terms of the errors on the data, C_{ab} = \sum_{i,j} (dc_a/dy_i) (dc_b/dy_j) <\delta y_i \delta y_j> For uncorrelated data the fluctuations of the underlying datapoints satisfy <\delta y_i \delta y_j> = \sigma_i^2 \delta_{ij}, giving a corresponding parameter covariance matrix of C_{ab} = \sum_i (1/w_i) (dc_a/dy_i) (dc_b/dy_i) When computing the covariance matrix for unweighted data, i.e. data with unknown errors, the weight factors w_i in this sum are replaced by the single estimate w = 1/\sigma^2, where \sigma^2 is the computed variance of the residuals about the best-fit model, \sigma^2 = \sum (y_i - Y(c,x_i))^2 / (n-p). This is referred to as the "variance-covariance matrix". The standard deviations of the best-fit parameters are given by the square root of the corresponding diagonal elements of the covariance matrix, \sigma_{c_a} = \sqrt{C_{aa}}. The correlation coefficient of the fit parameters c_a and c_b is given by \rho_{ab} = C_{ab} / \sqrt{C_{aa} C_{bb}}.  File: gsl-ref.info, Node: Linear regression, Next: Linear fitting without a constant term, Prev: Fitting Overview, Up: Least-Squares Fitting 37.2 Linear regression ====================== The functions described in this section can be used to perform least-squares fits to a straight line model, Y(c,x) = c_0 + c_1 x. -- Function: int gsl_fit_linear (const double * X, const size_t XSTRIDE, const double * Y, const size_t YSTRIDE, size_t N, double * C0, double * C1, double * COV00, double * COV01, double * COV11, double * SUMSQ) This function computes the best-fit linear regression coefficients (C0,C1) of the model Y = c_0 + c_1 X for the dataset (X, Y), two vectors of length N with strides XSTRIDE and YSTRIDE. The errors on Y are assumed unknown so the variance-covariance matrix for the parameters (C0, C1) is estimated from the scatter of the points around the best-fit line and returned via the parameters (COV00, COV01, COV11). The sum of squares of the residuals from the best-fit line is returned in SUMSQ. Note: the correlation coefficient of the data can be computed using 'gsl_stats_correlation' (*note Correlation::), it does not depend on the fit. -- Function: int gsl_fit_wlinear (const double * X, const size_t XSTRIDE, const double * W, const size_t WSTRIDE, const double * Y, const size_t YSTRIDE, size_t N, double * C0, double * C1, double * COV00, double * COV01, double * COV11, double * CHISQ) This function computes the best-fit linear regression coefficients (C0,C1) of the model Y = c_0 + c_1 X for the weighted dataset (X, Y), two vectors of length N with strides XSTRIDE and YSTRIDE. The vector W, of length N and stride WSTRIDE, specifies the weight of each datapoint. The weight is the reciprocal of the variance for each datapoint in Y. The covariance matrix for the parameters (C0, C1) is computed using the weights and returned via the parameters (COV00, COV01, COV11). The weighted sum of squares of the residuals from the best-fit line, \chi^2, is returned in CHISQ. -- Function: int gsl_fit_linear_est (double X, double C0, double C1, double COV00, double COV01, double COV11, double * Y, double * Y_ERR) This function uses the best-fit linear regression coefficients C0, C1 and their covariance COV00, COV01, COV11 to compute the fitted function Y and its standard deviation Y_ERR for the model Y = c_0 + c_1 X at the point X.  File: gsl-ref.info, Node: Linear fitting without a constant term, Next: Multi-parameter fitting, Prev: Linear regression, Up: Least-Squares Fitting 37.3 Linear fitting without a constant term =========================================== The functions described in this section can be used to perform least-squares fits to a straight line model without a constant term, Y = c_1 X. -- Function: int gsl_fit_mul (const double * X, const size_t XSTRIDE, const double * Y, const size_t YSTRIDE, size_t N, double * C1, double * COV11, double * SUMSQ) This function computes the best-fit linear regression coefficient C1 of the model Y = c_1 X for the datasets (X, Y), two vectors of length N with strides XSTRIDE and YSTRIDE. The errors on Y are assumed unknown so the variance of the parameter C1 is estimated from the scatter of the points around the best-fit line and returned via the parameter COV11. The sum of squares of the residuals from the best-fit line is returned in SUMSQ. -- Function: int gsl_fit_wmul (const double * X, const size_t XSTRIDE, const double * W, const size_t WSTRIDE, const double * Y, const size_t YSTRIDE, size_t N, double * C1, double * COV11, double * SUMSQ) This function computes the best-fit linear regression coefficient C1 of the model Y = c_1 X for the weighted datasets (X, Y), two vectors of length N with strides XSTRIDE and YSTRIDE. The vector W, of length N and stride WSTRIDE, specifies the weight of each datapoint. The weight is the reciprocal of the variance for each datapoint in Y. The variance of the parameter C1 is computed using the weights and returned via the parameter COV11. The weighted sum of squares of the residuals from the best-fit line, \chi^2, is returned in CHISQ. -- Function: int gsl_fit_mul_est (double X, double C1, double COV11, double * Y, double * Y_ERR) This function uses the best-fit linear regression coefficient C1 and its covariance COV11 to compute the fitted function Y and its standard deviation Y_ERR for the model Y = c_1 X at the point X.  File: gsl-ref.info, Node: Multi-parameter fitting, Next: Robust linear regression, Prev: Linear fitting without a constant term, Up: Least-Squares Fitting 37.4 Multi-parameter fitting ============================ The functions described in this section perform least-squares fits to a general linear model, y = X c where y is a vector of n observations, X is an n by p matrix of predictor variables, and the elements of the vector c are the p unknown best-fit parameters which are to be estimated. The chi-squared value is given by \chi^2 = \sum_i w_i (y_i - \sum_j X_{ij} c_j)^2. This formulation can be used for fits to any number of functions and/or variables by preparing the n-by-p matrix X appropriately. For example, to fit to a p-th order polynomial in X, use the following matrix, X_{ij} = x_i^j where the index i runs over the observations and the index j runs from 0 to p-1. To fit to a set of p sinusoidal functions with fixed frequencies \omega_1, \omega_2, ..., \omega_p, use, X_{ij} = sin(\omega_j x_i) To fit to p independent variables x_1, x_2, ..., x_p, use, X_{ij} = x_j(i) where x_j(i) is the i-th value of the predictor variable x_j. The functions described in this section are declared in the header file 'gsl_multifit.h'. The solution of the general linear least-squares system requires an additional working space for intermediate results, such as the singular value decomposition of the matrix X. -- Function: gsl_multifit_linear_workspace * gsl_multifit_linear_alloc (size_t N, size_t P) This function allocates a workspace for fitting a model to N observations using P parameters. -- Function: void gsl_multifit_linear_free (gsl_multifit_linear_workspace * WORK) This function frees the memory associated with the workspace W. -- Function: int gsl_multifit_linear (const gsl_matrix * X, const gsl_vector * Y, gsl_vector * C, gsl_matrix * COV, double * CHISQ, gsl_multifit_linear_workspace * WORK) This function computes the best-fit parameters C of the model y = X c for the observations Y and the matrix of predictor variables X, using the preallocated workspace provided in WORK. The p-by-p variance-covariance matrix of the model parameters COV is set to \sigma^2 (X^T X)^{-1}, where \sigma is the standard deviation of the fit residuals. The sum of squares of the residuals from the best-fit, \chi^2, is returned in CHISQ. If the coefficient of determination is desired, it can be computed from the expression R^2 = 1 - \chi^2 / TSS, where the total sum of squares (TSS) of the observations Y may be computed from 'gsl_stats_tss'. The best-fit is found by singular value decomposition of the matrix X using the modified Golub-Reinsch SVD algorithm, with column scaling to improve the accuracy of the singular values. Any components which have zero singular value (to machine precision) are discarded from the fit. -- Function: int gsl_multifit_wlinear (const gsl_matrix * X, const gsl_vector * W, const gsl_vector * Y, gsl_vector * C, gsl_matrix * COV, double * CHISQ, gsl_multifit_linear_workspace * WORK) This function computes the best-fit parameters C of the weighted model y = X c for the observations Y with weights W and the matrix of predictor variables X, using the preallocated workspace provided in WORK. The p-by-p covariance matrix of the model parameters COV is computed as (X^T W X)^{-1}. The weighted sum of squares of the residuals from the best-fit, \chi^2, is returned in CHISQ. If the coefficient of determination is desired, it can be computed from the expression R^2 = 1 - \chi^2 / WTSS, where the weighted total sum of squares (WTSS) of the observations Y may be computed from 'gsl_stats_wtss'. -- Function: int gsl_multifit_linear_svd (const gsl_matrix * X, const gsl_vector * Y, double TOL, size_t * RANK, gsl_vector * C, gsl_matrix * COV, double * CHISQ, gsl_multifit_linear_workspace * WORK) -- Function: int gsl_multifit_wlinear_svd (const gsl_matrix * X, const gsl_vector * W, const gsl_vector * Y, double TOL, size_t * RANK, gsl_vector * C, gsl_matrix * COV, double * CHISQ, gsl_multifit_linear_workspace * WORK) In these functions components of the fit are discarded if the ratio of singular values s_i/s_0 falls below the user-specified tolerance TOL, and the effective rank is returned in RANK. -- Function: int gsl_multifit_linear_usvd (const gsl_matrix * X, const gsl_vector * Y, double TOL, size_t * RANK, gsl_vector * C, gsl_matrix * COV, double * CHISQ, gsl_multifit_linear_workspace * WORK) -- Function: int gsl_multifit_wlinear_usvd (const gsl_matrix * X, const gsl_vector * W, const gsl_vector * Y, double TOL, size_t * RANK, gsl_vector * C, gsl_matrix * COV, double * CHISQ, gsl_multifit_linear_workspace * WORK) These functions compute the fit using an SVD without column scaling. -- Function: int gsl_multifit_linear_est (const gsl_vector * X, const gsl_vector * C, const gsl_matrix * COV, double * Y, double * Y_ERR) This function uses the best-fit multilinear regression coefficients C and their covariance matrix COV to compute the fitted function value Y and its standard deviation Y_ERR for the model y = x.c at the point X. -- Function: int gsl_multifit_linear_residuals (const gsl_matrix * X, const gsl_vector * Y, const gsl_vector * C, gsl_vector * R) This function computes the vector of residuals r = y - X c for the observations Y, coefficients C and matrix of predictor variables X.  File: gsl-ref.info, Node: Robust linear regression, Next: Troubleshooting, Prev: Multi-parameter fitting, Up: Least-Squares Fitting 37.5 Robust linear regression ============================= Ordinary least squares (OLS) models are often heavily influenced by the presence of outliers. Outliers are data points which do not follow the general trend of the other observations, although there is strictly no precise definition of an outlier. Robust linear regression refers to regression algorithms which are robust to outliers. The most common type of robust regression is M-estimation. The general M-estimator minimizes the objective function \sum_i \rho(e_i) = \sum_i \rho (y_i - Y(c, x_i)) where e_i = y_i - Y(c, x_i) is the residual of the ith data point, and \rho(e_i) is a function which should have the following properties: \rho(e) \ge 0 \rho(0) = 0 \rho(-e) = \rho(e) \rho(e_1) > \rho(e_2) for |e_1| > |e_2| The special case of ordinary least squares is given by \rho(e_i) = e_i^2. Letting \psi = \rho' be the derivative of \rho, differentiating the objective function with respect to the coefficients c and setting the partial derivatives to zero produces the system of equations \sum_i \psi(e_i) X_i = 0 where X_i is a vector containing row i of the design matrix X. Next, we define a weight function w(e) = \psi(e)/e, and let w_i = w(e_i): \sum_i w_i e_i X_i = 0 This system of equations is equivalent to solving a weighted ordinary least squares problem, minimizing \chi^2 = \sum_i w_i e_i^2. The weights however, depend on the residuals e_i, which depend on the coefficients c, which depend on the weights. Therefore, an iterative solution is used, called Iteratively Reweighted Least Squares (IRLS). 1. Compute initial estimates of the coefficients c^{(0)} using ordinary least squares 2. For iteration k, form the residuals e_i^{(k)} = (y_i - X_i c^{(k-1)})/(t \sigma^{(k)} \sqrt{1 - h_i}), where t is a tuning constant depending on the choice of \psi, and h_i are the statistical leverages (diagonal elements of the matrix X (X^T X)^{-1} X^T). Including t and h_i in the residual calculation has been shown to improve the convergence of the method. The residual standard deviation is approximated as \sigma^{(k)} = MAD / 0.6745, where MAD is the Median-Absolute-Deviation of the n-p largest residuals from the previous iteration. 3. Compute new weights w_i^{(k)} = \psi(e_i^{(k)})/e_i^{(k)}. 4. Compute new coefficients c^{(k)} by solving the weighted least squares problem with weights w_i^{(k)}. 5. Steps 2 through 4 are iterated until the coefficients converge or until some maximum iteration limit is reached. Coefficients are tested for convergence using the critera: |c_i^(k) - c_i^(k-1)| \le \epsilon \times max(|c_i^(k)|, |c_i^(k-1)|) for all 0 \le i < p where \epsilon is a small tolerance factor. The key to this method lies in selecting the function \psi(e_i) to assign smaller weights to large residuals, and larger weights to smaller residuals. As the iteration proceeds, outliers are assigned smaller and smaller weights, eventually having very little or no effect on the fitted model. -- Function: gsl_multifit_robust_workspace * gsl_multifit_robust_alloc (const gsl_multifit_robust_type * T, const size_t N, const size_t P) This function allocates a workspace for fitting a model to N observations using P parameters. The type T specifies the function \psi and can be selected from the following choices. -- Robust type: gsl_multifit_robust_default This specifies the 'gsl_multifit_robust_bisquare' type (see below) and is a good general purpose choice for robust regression. -- Robust type: gsl_multifit_robust_bisquare This is Tukey's biweight (bisquare) function and is a good general purpose choice for robust regression. The weight function is given by w(e) = (1 - e^2)^2 and the default tuning constant is t = 4.685. -- Robust type: gsl_multifit_robust_cauchy This is Cauchy's function, also known as the Lorentzian function. This function does not guarantee a unique solution, meaning different choices of the coefficient vector C could minimize the objective function. Therefore this option should be used with care. The weight function is given by w(e) = 1 / (1 + e^2) and the default tuning constant is t = 2.385. -- Robust type: gsl_multifit_robust_fair This is the fair \rho function, which guarantees a unique solution and has continuous derivatives to three orders. The weight function is given by w(e) = 1 / (1 + |e|) and the default tuning constant is t = 1.400. -- Robust type: gsl_multifit_robust_huber This specifies Huber's \rho function, which is a parabola in the vicinity of zero and increases linearly for a given threshold |e| > t. This function is also considered an excellent general purpose robust estimator, however, occasional difficulties can be encountered due to the discontinuous first derivative of the \psi function. The weight function is given by w(e) = 1/max(1,|e|) and the default tuning constant is t = 1.345. -- Robust type: gsl_multifit_robust_ols This specifies the ordinary least squares solution, which can be useful for quickly checking the difference between the various robust and OLS solutions. The weight function is given by w(e) = 1 and the default tuning constant is t = 1. -- Robust type: gsl_multifit_robust_welsch This specifies the Welsch function which can perform well in cases where the residuals have an exponential distribution. The weight function is given by w(e) = \exp(-e^2) and the default tuning constant is t = 2.985. -- Function: void gsl_multifit_robust_free (gsl_multifit_robust_workspace * W) This function frees the memory associated with the workspace W. -- Function: const char * gsl_multifit_robust_name (const gsl_multifit_robust_workspace * W) This function returns the name of the robust type T specified to 'gsl_multifit_robust_alloc'. -- Function: int gsl_multifit_robust_tune (const double TUNE, gsl_multifit_robust_workspace * W) This function sets the tuning constant t used to adjust the residuals at each iteration to TUNE. Decreasing the tuning constant increases the downweight assigned to large residuals, while increasing the tuning constant decreases the downweight assigned to large residuals. -- Function: int gsl_multifit_robust (const gsl_matrix * X, const gsl_vector * Y, gsl_vector * C, gsl_matrix * COV, gsl_multifit_robust_workspace * W) This function computes the best-fit parameters C of the model y = X c for the observations Y and the matrix of predictor variables X, attemping to reduce the influence of outliers using the algorithm outlined above. The p-by-p variance-covariance matrix of the model parameters COV is estimated as \sigma^2 (X^T X)^{-1}, where \sigma is an approximation of the residual standard deviation using the theory of robust regression. Special care must be taken when estimating \sigma and other statistics such as R^2, and so these are computed internally and are available by calling the function 'gsl_multifit_robust_statistics'. -- Function: int gsl_multifit_robust_est (const gsl_vector * X, const gsl_vector * C, const gsl_matrix * COV, double * Y, double * Y_ERR) This function uses the best-fit robust regression coefficients C and their covariance matrix COV to compute the fitted function value Y and its standard deviation Y_ERR for the model y = x.c at the point X. -- Function: gsl_multifit_robust_stats gsl_multifit_robust_statistics (const gsl_multifit_robust_workspace * W) This function returns a structure containing relevant statistics from a robust regression. The function 'gsl_multifit_robust' must be called first to perform the regression and calculate these statistics. The returned 'gsl_multifit_robust_stats' structure contains the following fields. double 'sigma_ols' This contains the standard deviation of the residuals as computed from ordinary least squares (OLS). double 'sigma_mad' This contains an estimate of the standard deviation of the final residuals using the Median-Absolute-Deviation statistic double 'sigma_rob' This contains an estimate of the standard deviation of the final residuals from the theory of robust regression (see Street et al, 1988). double 'sigma' This contains an estimate of the standard deviation of the final residuals by attemping to reconcile 'sigma_rob' and 'sigma_ols' in a reasonable way. double 'Rsq' This contains the R^2 coefficient of determination statistic using the estimate 'sigma'. double 'adj_Rsq' This contains the adjusted R^2 coefficient of determination statistic using the estimate 'sigma'. double 'rmse' This contains the root mean squared error of the final residuals double 'sse' This contains the residual sum of squares taking into account the robust covariance matrix. size_t 'dof' This contains the number of degrees of freedom n - p size_t 'numit' Upon successful convergence, this contains the number of iterations performed gsl_vector * 'weights' This contains the final weight vector of length N gsl_vector * 'r' This contains the final residual vector of length N, r = y - X c  File: gsl-ref.info, Node: Troubleshooting, Next: Fitting Examples, Prev: Robust linear regression, Up: Least-Squares Fitting 37.6 Troubleshooting ==================== When using models based on polynomials, care should be taken when constructing the design matrix X. If the x values are large, then the matrix X could be ill-conditioned since its columns are powers of x, leading to unstable least-squares solutions. In this case it can often help to center and scale the x values using the mean and standard deviation: x' = (x - mu)/sigma and then construct the X matrix using the transformed values x'.  File: gsl-ref.info, Node: Fitting Examples, Next: Fitting References and Further Reading, Prev: Troubleshooting, Up: Least-Squares Fitting 37.7 Examples ============= The following program computes a least squares straight-line fit to a simple dataset, and outputs the best-fit line and its associated one standard-deviation error bars. #include #include int main (void) { int i, n = 4; double x[4] = { 1970, 1980, 1990, 2000 }; double y[4] = { 12, 11, 14, 13 }; double w[4] = { 0.1, 0.2, 0.3, 0.4 }; double c0, c1, cov00, cov01, cov11, chisq; gsl_fit_wlinear (x, 1, w, 1, y, 1, n, &c0, &c1, &cov00, &cov01, &cov11, &chisq); printf ("# best fit: Y = %g + %g X\n", c0, c1); printf ("# covariance matrix:\n"); printf ("# [ %g, %g\n# %g, %g]\n", cov00, cov01, cov01, cov11); printf ("# chisq = %g\n", chisq); for (i = 0; i < n; i++) printf ("data: %g %g %g\n", x[i], y[i], 1/sqrt(w[i])); printf ("\n"); for (i = -30; i < 130; i++) { double xf = x[0] + (i/100.0) * (x[n-1] - x[0]); double yf, yf_err; gsl_fit_linear_est (xf, c0, c1, cov00, cov01, cov11, &yf, &yf_err); printf ("fit: %g %g\n", xf, yf); printf ("hi : %g %g\n", xf, yf + yf_err); printf ("lo : %g %g\n", xf, yf - yf_err); } return 0; } The following commands extract the data from the output of the program and display it using the GNU plotutils 'graph' utility, $ ./demo > tmp $ more tmp # best fit: Y = -106.6 + 0.06 X # covariance matrix: # [ 39602, -19.9 # -19.9, 0.01] # chisq = 0.8 $ for n in data fit hi lo ; do grep "^$n" tmp | cut -d: -f2 > $n ; done $ graph -T X -X x -Y y -y 0 20 -m 0 -S 2 -Ie data -S 0 -I a -m 1 fit -m 2 hi -m 2 lo The next program performs a quadratic fit y = c_0 + c_1 x + c_2 x^2 to a weighted dataset using the generalised linear fitting function 'gsl_multifit_wlinear'. The model matrix X for a quadratic fit is given by, X = [ 1 , x_0 , x_0^2 ; 1 , x_1 , x_1^2 ; 1 , x_2 , x_2^2 ; ... , ... , ... ] where the column of ones corresponds to the constant term c_0. The two remaining columns corresponds to the terms c_1 x and c_2 x^2. The program reads N lines of data in the format (X, Y, ERR) where ERR is the error (standard deviation) in the value Y. #include #include int main (int argc, char **argv) { int i, n; double xi, yi, ei, chisq; gsl_matrix *X, *cov; gsl_vector *y, *w, *c; if (argc != 2) { fprintf (stderr,"usage: fit n < data\n"); exit (-1); } n = atoi (argv[1]); X = gsl_matrix_alloc (n, 3); y = gsl_vector_alloc (n); w = gsl_vector_alloc (n); c = gsl_vector_alloc (3); cov = gsl_matrix_alloc (3, 3); for (i = 0; i < n; i++) { int count = fscanf (stdin, "%lg %lg %lg", &xi, &yi, &ei); if (count != 3) { fprintf (stderr, "error reading file\n"); exit (-1); } printf ("%g %g +/- %g\n", xi, yi, ei); gsl_matrix_set (X, i, 0, 1.0); gsl_matrix_set (X, i, 1, xi); gsl_matrix_set (X, i, 2, xi*xi); gsl_vector_set (y, i, yi); gsl_vector_set (w, i, 1.0/(ei*ei)); } { gsl_multifit_linear_workspace * work = gsl_multifit_linear_alloc (n, 3); gsl_multifit_wlinear (X, w, y, c, cov, &chisq, work); gsl_multifit_linear_free (work); } #define C(i) (gsl_vector_get(c,(i))) #define COV(i,j) (gsl_matrix_get(cov,(i),(j))) { printf ("# best fit: Y = %g + %g X + %g X^2\n", C(0), C(1), C(2)); printf ("# covariance matrix:\n"); printf ("[ %+.5e, %+.5e, %+.5e \n", COV(0,0), COV(0,1), COV(0,2)); printf (" %+.5e, %+.5e, %+.5e \n", COV(1,0), COV(1,1), COV(1,2)); printf (" %+.5e, %+.5e, %+.5e ]\n", COV(2,0), COV(2,1), COV(2,2)); printf ("# chisq = %g\n", chisq); } gsl_matrix_free (X); gsl_vector_free (y); gsl_vector_free (w); gsl_vector_free (c); gsl_matrix_free (cov); return 0; } A suitable set of data for fitting can be generated using the following program. It outputs a set of points with gaussian errors from the curve y = e^x in the region 0 < x < 2. #include #include #include int main (void) { double x; const gsl_rng_type * T; gsl_rng * r; gsl_rng_env_setup (); T = gsl_rng_default; r = gsl_rng_alloc (T); for (x = 0.1; x < 2; x+= 0.1) { double y0 = exp (x); double sigma = 0.1 * y0; double dy = gsl_ran_gaussian (r, sigma); printf ("%g %g %g\n", x, y0 + dy, sigma); } gsl_rng_free(r); return 0; } The data can be prepared by running the resulting executable program, $ GSL_RNG_TYPE=mt19937_1999 ./generate > exp.dat $ more exp.dat 0.1 0.97935 0.110517 0.2 1.3359 0.12214 0.3 1.52573 0.134986 0.4 1.60318 0.149182 0.5 1.81731 0.164872 0.6 1.92475 0.182212 .... To fit the data use the previous program, with the number of data points given as the first argument. In this case there are 19 data points. $ ./fit 19 < exp.dat 0.1 0.97935 +/- 0.110517 0.2 1.3359 +/- 0.12214 ... # best fit: Y = 1.02318 + 0.956201 X + 0.876796 X^2 # covariance matrix: [ +1.25612e-02, -3.64387e-02, +1.94389e-02 -3.64387e-02, +1.42339e-01, -8.48761e-02 +1.94389e-02, -8.48761e-02, +5.60243e-02 ] # chisq = 23.0987 The parameters of the quadratic fit match the coefficients of the expansion of e^x, taking into account the errors on the parameters and the O(x^3) difference between the exponential and quadratic functions for the larger values of x. The errors on the parameters are given by the square-root of the corresponding diagonal elements of the covariance matrix. The chi-squared per degree of freedom is 1.4, indicating a reasonable fit to the data. The next program demonstrates the advantage of robust least squares on a dataset with outliers. The program generates linear (x,y) data pairs on the line y = 1.45 x + 3.88, adds some random noise, and inserts 3 outliers into the dataset. Both the robust and ordinary least squares (OLS) coefficients are computed for comparison. #include #include #include int dofit(const gsl_multifit_robust_type *T, const gsl_matrix *X, const gsl_vector *y, gsl_vector *c, gsl_matrix *cov) { int s; gsl_multifit_robust_workspace * work = gsl_multifit_robust_alloc (T, X->size1, X->size2); s = gsl_multifit_robust (X, y, c, cov, work); gsl_multifit_robust_free (work); return s; } int main (int argc, char **argv) { int i; size_t n; const size_t p = 2; /* linear fit */ gsl_matrix *X, *cov; gsl_vector *x, *y, *c, *c_ols; const double a = 1.45; /* slope */ const double b = 3.88; /* intercept */ gsl_rng *r; if (argc != 2) { fprintf (stderr,"usage: robfit n\n"); exit (-1); } n = atoi (argv[1]); X = gsl_matrix_alloc (n, p); x = gsl_vector_alloc (n); y = gsl_vector_alloc (n); c = gsl_vector_alloc (p); c_ols = gsl_vector_alloc (p); cov = gsl_matrix_alloc (p, p); r = gsl_rng_alloc(gsl_rng_default); /* generate linear dataset */ for (i = 0; i < n - 3; i++) { double dx = 10.0 / (n - 1.0); double ei = gsl_rng_uniform(r); double xi = -5.0 + i * dx; double yi = a * xi + b; gsl_vector_set (x, i, xi); gsl_vector_set (y, i, yi + ei); } /* add a few outliers */ gsl_vector_set(x, n - 3, 4.7); gsl_vector_set(y, n - 3, -8.3); gsl_vector_set(x, n - 2, 3.5); gsl_vector_set(y, n - 2, -6.7); gsl_vector_set(x, n - 1, 4.1); gsl_vector_set(y, n - 1, -6.0); /* construct design matrix X for linear fit */ for (i = 0; i < n; ++i) { double xi = gsl_vector_get(x, i); gsl_matrix_set (X, i, 0, 1.0); gsl_matrix_set (X, i, 1, xi); } /* perform robust and OLS fit */ dofit(gsl_multifit_robust_ols, X, y, c_ols, cov); dofit(gsl_multifit_robust_bisquare, X, y, c, cov); /* output data and model */ for (i = 0; i < n; ++i) { double xi = gsl_vector_get(x, i); double yi = gsl_vector_get(y, i); gsl_vector_view v = gsl_matrix_row(X, i); double y_ols, y_rob, y_err; gsl_multifit_robust_est(&v.vector, c, cov, &y_rob, &y_err); gsl_multifit_robust_est(&v.vector, c_ols, cov, &y_ols, &y_err); printf("%g %g %g %g\n", xi, yi, y_rob, y_ols); } #define C(i) (gsl_vector_get(c,(i))) #define COV(i,j) (gsl_matrix_get(cov,(i),(j))) { printf ("# best fit: Y = %g + %g X\n", C(0), C(1)); printf ("# covariance matrix:\n"); printf ("# [ %+.5e, %+.5e\n", COV(0,0), COV(0,1)); printf ("# %+.5e, %+.5e\n", COV(1,0), COV(1,1)); } gsl_matrix_free (X); gsl_vector_free (x); gsl_vector_free (y); gsl_vector_free (c); gsl_vector_free (c_ols); gsl_matrix_free (cov); gsl_rng_free(r); return 0; } The output from the program is shown in the following plot.  File: gsl-ref.info, Node: Fitting References and Further Reading, Prev: Fitting Examples, Up: Least-Squares Fitting 37.8 References and Further Reading =================================== A summary of formulas and techniques for least squares fitting can be found in the "Statistics" chapter of the Annual Review of Particle Physics prepared by the Particle Data Group, 'Review of Particle Properties', R.M. Barnett et al., Physical Review D54, 1 (1996) The Review of Particle Physics is available online at the website given above. The tests used to prepare these routines are based on the NIST Statistical Reference Datasets. The datasets and their documentation are available from NIST at the following website, . The GSL implementation of robust linear regression closely follows the publications DuMouchel, W. and F. O'Brien (1989), "Integrating a robust option into a multiple regression computing environment," Computer Science and Statistics: Proceedings of the 21st Symposium on the Interface, American Statistical Association Street, J.O., R.J. Carroll, and D. Ruppert (1988), "A note on computing robust regression estimates via iteratively reweighted least squares," The American Statistician, v. 42, pp. 152-154.  File: gsl-ref.info, Node: Nonlinear Least-Squares Fitting, Next: Basis Splines, Prev: Least-Squares Fitting, Up: Top 38 Nonlinear Least-Squares Fitting ********************************** This chapter describes functions for multidimensional nonlinear least-squares fitting. The library provides low level components for a variety of iterative solvers and convergence tests. These can be combined by the user to achieve the desired solution, with full access to the intermediate steps of the iteration. Each class of methods uses the same framework, so that you can switch between solvers at runtime without needing to recompile your program. Each instance of a solver keeps track of its own state, allowing the solvers to be used in multi-threaded programs. The header file 'gsl_multifit_nlin.h' contains prototypes for the multidimensional nonlinear fitting functions and related declarations. * Menu: * Overview of Nonlinear Least-Squares Fitting:: * Initializing the Nonlinear Least-Squares Solver:: * Providing the Function to be Minimized:: * Finite Difference Jacobian:: * Iteration of the Minimization Algorithm:: * Search Stopping Parameters for Minimization Algorithms:: * High Level Driver:: * Minimization Algorithms using Derivatives:: * Minimization Algorithms without Derivatives:: * Computing the covariance matrix of best fit parameters:: * Example programs for Nonlinear Least-Squares Fitting:: * References and Further Reading for Nonlinear Least-Squares Fitting::  File: gsl-ref.info, Node: Overview of Nonlinear Least-Squares Fitting, Next: Initializing the Nonlinear Least-Squares Solver, Up: Nonlinear Least-Squares Fitting 38.1 Overview ============= The problem of multidimensional nonlinear least-squares fitting requires the minimization of the squared residuals of n functions, f_i, in p parameters, x_i, \Phi(x) = (1/2) || F(x) ||^2 = (1/2) \sum_{i=1}^{n} f_i(x_1, ..., x_p)^2 All algorithms proceed from an initial guess using the linearization, \psi(p) = || F(x+p) || ~=~ || F(x) + J p || where x is the initial point, p is the proposed step and J is the Jacobian matrix J_{ij} = d f_i / d x_j. Additional strategies are used to enlarge the region of convergence. These include requiring a decrease in the norm ||F|| on each step or using a trust region to avoid steps which fall outside the linear regime. To perform a weighted least-squares fit of a nonlinear model Y(x,t) to data (t_i, y_i) with independent Gaussian errors \sigma_i, use function components of the following form, f_i = (Y(x, t_i) - y_i) / \sigma_i Note that the model parameters are denoted by x in this chapter since the non-linear least-squares algorithms are described geometrically (i.e. finding the minimum of a surface). The independent variable of any data to be fitted is denoted by t. With the definition above the Jacobian is J_{ij} =(1 / \sigma_i) d Y_i / d x_j, where Y_i = Y(x,t_i).  File: gsl-ref.info, Node: Initializing the Nonlinear Least-Squares Solver, Next: Providing the Function to be Minimized, Prev: Overview of Nonlinear Least-Squares Fitting, Up: Nonlinear Least-Squares Fitting 38.2 Initializing the Solver ============================ -- Function: gsl_multifit_fsolver * gsl_multifit_fsolver_alloc (const gsl_multifit_fsolver_type * T, size_t N, size_t P) This function returns a pointer to a newly allocated instance of a solver of type T for N observations and P parameters. The number of observations N must be greater than or equal to parameters P. If there is insufficient memory to create the solver then the function returns a null pointer and the error handler is invoked with an error code of 'GSL_ENOMEM'. -- Function: gsl_multifit_fdfsolver * gsl_multifit_fdfsolver_alloc (const gsl_multifit_fdfsolver_type * T, size_t N, size_t P) This function returns a pointer to a newly allocated instance of a derivative solver of type T for N observations and P parameters. For example, the following code creates an instance of a Levenberg-Marquardt solver for 100 data points and 3 parameters, const gsl_multifit_fdfsolver_type * T = gsl_multifit_fdfsolver_lmder; gsl_multifit_fdfsolver * s = gsl_multifit_fdfsolver_alloc (T, 100, 3); The number of observations N must be greater than or equal to parameters P. If there is insufficient memory to create the solver then the function returns a null pointer and the error handler is invoked with an error code of 'GSL_ENOMEM'. -- Function: int gsl_multifit_fsolver_set (gsl_multifit_fsolver * S, gsl_multifit_function * F, const gsl_vector * X) This function initializes, or reinitializes, an existing solver S to use the function F and the initial guess X. -- Function: int gsl_multifit_fdfsolver_set (gsl_multifit_fdfsolver * S, gsl_multifit_function_fdf * FDF, const gsl_vector * X) This function initializes, or reinitializes, an existing solver S to use the function and derivative FDF and the initial guess X. -- Function: void gsl_multifit_fsolver_free (gsl_multifit_fsolver * S) -- Function: void gsl_multifit_fdfsolver_free (gsl_multifit_fdfsolver * S) These functions free all the memory associated with the solver S. -- Function: const char * gsl_multifit_fsolver_name (const gsl_multifit_fsolver * S) -- Function: const char * gsl_multifit_fdfsolver_name (const gsl_multifit_fdfsolver * S) These functions return a pointer to the name of the solver. For example, printf ("s is a '%s' solver\n", gsl_multifit_fdfsolver_name (s)); would print something like 's is a 'lmder' solver'.  File: gsl-ref.info, Node: Providing the Function to be Minimized, Next: Finite Difference Jacobian, Prev: Initializing the Nonlinear Least-Squares Solver, Up: Nonlinear Least-Squares Fitting 38.3 Providing the Function to be Minimized =========================================== You must provide n functions of p variables for the minimization algorithms to operate on. In order to allow for arbitrary parameters the functions are defined by the following data types: -- Data Type: gsl_multifit_function This data type defines a general system of functions with arbitrary parameters. 'int (* f) (const gsl_vector * X, void * PARAMS, gsl_vector * F)' this function should store the vector result f(x,params) in F for argument X and arbitrary parameters PARAMS, returning an appropriate error code if the function cannot be computed. 'size_t n' the number of functions, i.e. the number of components of the vector F. 'size_t p' the number of independent variables, i.e. the number of components of the vector X. 'void * params' a pointer to the arbitrary parameters of the function. -- Data Type: gsl_multifit_function_fdf This data type defines a general system of functions with arbitrary parameters and the corresponding Jacobian matrix of derivatives, 'int (* f) (const gsl_vector * X, void * PARAMS, gsl_vector * F)' this function should store the vector result f(x,params) in F for argument X and arbitrary parameters PARAMS, returning an appropriate error code if the function cannot be computed. 'int (* df) (const gsl_vector * X, void * PARAMS, gsl_matrix * J)' this function should store the N-by-P matrix result J_ij = d f_i(x,params) / d x_j in J for argument X and arbitrary parameters PARAMS, returning an appropriate error code if the function cannot be computed. If an analytic Jacobian is unavailable, or too expensive to compute, this function pointer may be set to NULL, in which case the Jacobian will be internally computed using finite difference approximations of the function F. 'int (* fdf) (const gsl_vector * X, void * PARAMS, gsl_vector * F, gsl_matrix * J)' This function should set the values of the F and J as above, for arguments X and arbitrary parameters PARAMS. This function provides an optimization of the separate functions for f(x) and J(x)--it is always faster to compute the function and its derivative at the same time. If an analytic Jacobian is unavailable, or too expensive to compute, this function pointer may be set to NULL, in which case the Jacobian will be internally computed using finite difference approximations of the function F. 'size_t n' the number of functions, i.e. the number of components of the vector F. 'size_t p' the number of independent variables, i.e. the number of components of the vector X. 'void * params' a pointer to the arbitrary parameters of the function. Note that when fitting a non-linear model against experimental data, the data is passed to the functions above using the PARAMS argument and the trial best-fit parameters through the X argument.  File: gsl-ref.info, Node: Finite Difference Jacobian, Next: Iteration of the Minimization Algorithm, Prev: Providing the Function to be Minimized, Up: Nonlinear Least-Squares Fitting 38.4 Finite Difference Jacobian =============================== For the algorithms which require a Jacobian matrix of derivatives of the fit functions, there are times when an analytic Jacobian may be unavailable or too expensive to compute. Therefore GSL supports approximating the Jacobian numerically using finite differences of the fit functions. This is typically done by setting the relevant function pointers of the 'gsl_multifit_function_fdf' data type to NULL, however the following functions allow the user to access the approximate Jacobian directly if needed. -- Function: int gsl_multifit_fdfsolver_dif_df (const gsl_vector * X, gsl_multifit_function_fdf * FDF, const gsl_vector * F, gsl_matrix * J) This function takes as input the current position X with the function values computed at the current position F, along with FDF which specifies the fit function and parameters and approximates the N-by-P Jacobian J using forward finite differences: J_ij = d f_i(x,params) / d x_j = (f_i(x^*,params) - f_i(x,params)) / d x_j. where x^* has the jth element perturbed by \Delta x_j and \Delta x_j = \epsilon |x_j|, where \epsilon is the square root of the machine precision 'GSL_DBL_EPSILON'. -- Function: int gsl_multifit_fdfsolver_dif_fdf (const gsl_vector * X, gsl_multifit_function_fdf * FDF, gsl_vector * F, gsl_matrix * J) This function computes the vector of function values F and the approximate Jacobian J at the position vector X using the system described in FDF. See 'gsl_multifit_fdfsolver_dif_df' for a description of how the Jacobian is computed.  File: gsl-ref.info, Node: Iteration of the Minimization Algorithm, Next: Search Stopping Parameters for Minimization Algorithms, Prev: Finite Difference Jacobian, Up: Nonlinear Least-Squares Fitting 38.5 Iteration ============== The following functions drive the iteration of each algorithm. Each function performs one iteration to update the state of any solver of the corresponding type. The same functions work for all solvers so that different methods can be substituted at runtime without modifications to the code. -- Function: int gsl_multifit_fsolver_iterate (gsl_multifit_fsolver * S) -- Function: int gsl_multifit_fdfsolver_iterate (gsl_multifit_fdfsolver * S) These functions perform a single iteration of the solver S. If the iteration encounters an unexpected problem then an error code will be returned. The solver maintains a current estimate of the best-fit parameters at all times. The solver struct S contains the following entries, which can be used to track the progress of the solution: 'gsl_vector * x' The current position. 'gsl_vector * f' The function value at the current position. 'gsl_vector * dx' The difference between the current position and the previous position, i.e. the last step, taken as a vector. 'gsl_matrix * J' The Jacobian matrix at the current position (for the 'gsl_multifit_fdfsolver' struct only) The best-fit information also can be accessed with the following auxiliary functions, -- Function: gsl_vector * gsl_multifit_fsolver_position (const gsl_multifit_fsolver * S) -- Function: gsl_vector * gsl_multifit_fdfsolver_position (const gsl_multifit_fdfsolver * S) These functions return the current position (i.e. best-fit parameters) 's->x' of the solver S.  File: gsl-ref.info, Node: Search Stopping Parameters for Minimization Algorithms, Next: High Level Driver, Prev: Iteration of the Minimization Algorithm, Up: Nonlinear Least-Squares Fitting 38.6 Search Stopping Parameters =============================== A minimization procedure should stop when one of the following conditions is true: * A minimum has been found to within the user-specified precision. * A user-specified maximum number of iterations has been reached. * An error has occurred. The handling of these conditions is under user control. The functions below allow the user to test the current estimate of the best-fit parameters in several standard ways. -- Function: int gsl_multifit_test_delta (const gsl_vector * DX, const gsl_vector * X, double EPSABS, double EPSREL) This function tests for the convergence of the sequence by comparing the last step DX with the absolute error EPSABS and relative error EPSREL to the current position X. The test returns 'GSL_SUCCESS' if the following condition is achieved, |dx_i| < epsabs + epsrel |x_i| for each component of X and returns 'GSL_CONTINUE' otherwise. -- Function: int gsl_multifit_test_gradient (const gsl_vector * G, double EPSABS) This function tests the residual gradient G against the absolute error bound EPSABS. Mathematically, the gradient should be exactly zero at the minimum. The test returns 'GSL_SUCCESS' if the following condition is achieved, \sum_i |g_i| < epsabs and returns 'GSL_CONTINUE' otherwise. This criterion is suitable for situations where the precise location of the minimum, x, is unimportant provided a value can be found where the gradient is small enough. -- Function: int gsl_multifit_gradient (const gsl_matrix * J, const gsl_vector * F, gsl_vector * G) This function computes the gradient G of \Phi(x) = (1/2) ||F(x)||^2 from the Jacobian matrix J and the function values F, using the formula g = J^T f.  File: gsl-ref.info, Node: High Level Driver, Next: Minimization Algorithms using Derivatives, Prev: Search Stopping Parameters for Minimization Algorithms, Up: Nonlinear Least-Squares Fitting 38.7 High Level Driver ====================== These routines provide a high level wrapper that combine the iteration and convergence testing for easy use. -- Function: int gsl_multifit_fsolver_driver (gsl_multifit_fsolver * S, const size_t MAXITER, const double EPSABS, const double EPSREL) -- Function: int gsl_multifit_fdfsolver_driver (gsl_multifit_fdfsolver * S, const size_t MAXITER, const double EPSABS, const double EPSREL) These functions iterate the solver S for a maximum of MAXITER iterations. After each iteration, the system is tested for convergence using 'gsl_multifit_test_delta' with the error tolerances EPSABS and EPSREL.  File: gsl-ref.info, Node: Minimization Algorithms using Derivatives, Next: Minimization Algorithms without Derivatives, Prev: High Level Driver, Up: Nonlinear Least-Squares Fitting 38.8 Minimization Algorithms using Derivatives ============================================== The minimization algorithms described in this section make use of both the function and its derivative. They require an initial guess for the location of the minimum. There is no absolute guarantee of convergence--the function must be suitable for this technique and the initial guess must be sufficiently close to the minimum for it to work. -- Derivative Solver: gsl_multifit_fdfsolver_lmsder This is a robust and efficient version of the Levenberg-Marquardt algorithm as implemented in the scaled LMDER routine in MINPACK. Minpack was written by Jorge J. More', Burton S. Garbow and Kenneth E. Hillstrom. The algorithm uses a generalized trust region to keep each step under control. In order to be accepted a proposed new position x' must satisfy the condition |D (x' - x)| < \delta, where D is a diagonal scaling matrix and \delta is the size of the trust region. The components of D are computed internally, using the column norms of the Jacobian to estimate the sensitivity of the residual to each component of x. This improves the behavior of the algorithm for badly scaled functions. On each iteration the algorithm attempts to minimize the linear system |F + J p| subject to the constraint |D p| < \Delta. The solution to this constrained linear system is found using the Levenberg-Marquardt method. The proposed step is now tested by evaluating the function at the resulting point, x'. If the step reduces the norm of the function sufficiently, and follows the predicted behavior of the function within the trust region, then it is accepted and the size of the trust region is increased. If the proposed step fails to improve the solution, or differs significantly from the expected behavior within the trust region, then the size of the trust region is decreased and another trial step is computed. The algorithm also monitors the progress of the solution and returns an error if the changes in the solution are smaller than the machine precision. The possible error codes are, 'GSL_ETOLF' the decrease in the function falls below machine precision 'GSL_ETOLX' the change in the position vector falls below machine precision 'GSL_ETOLG' the norm of the gradient, relative to the norm of the function, falls below machine precision 'GSL_ENOPROG' the routine has made 10 or more attempts to find a suitable trial step without success (but subsequent calls can be made to continue the search).(1) These error codes indicate that further iterations will be unlikely to change the solution from its current value. -- Derivative Solver: gsl_multifit_fdfsolver_lmder This is an unscaled version of the LMDER algorithm. The elements of the diagonal scaling matrix D are set to 1. This algorithm may be useful in circumstances where the scaled version of LMDER converges too slowly, or the function is already scaled appropriately. ---------- Footnotes ---------- (1) The return code 'GSL_CONTINUE' was used for this case in versions prior to 1.14.  File: gsl-ref.info, Node: Minimization Algorithms without Derivatives, Next: Computing the covariance matrix of best fit parameters, Prev: Minimization Algorithms using Derivatives, Up: Nonlinear Least-Squares Fitting 38.9 Minimization Algorithms without Derivatives ================================================ There are no algorithms implemented in this section at the moment.  File: gsl-ref.info, Node: Computing the covariance matrix of best fit parameters, Next: Example programs for Nonlinear Least-Squares Fitting, Prev: Minimization Algorithms without Derivatives, Up: Nonlinear Least-Squares Fitting 38.10 Computing the covariance matrix of best fit parameters ============================================================ -- Function: int gsl_multifit_covar (const gsl_matrix * J, double EPSREL, gsl_matrix * COVAR) This function uses the Jacobian matrix J to compute the covariance matrix of the best-fit parameters, COVAR. The parameter EPSREL is used to remove linear-dependent columns when J is rank deficient. The covariance matrix is given by, covar = (J^T J)^{-1} and is computed by QR decomposition of J with column-pivoting. Any columns of R which satisfy |R_{kk}| <= epsrel |R_{11}| are considered linearly-dependent and are excluded from the covariance matrix (the corresponding rows and columns of the covariance matrix are set to zero). If the minimisation uses the weighted least-squares function f_i = (Y(x, t_i) - y_i) / \sigma_i then the covariance matrix above gives the statistical error on the best-fit parameters resulting from the Gaussian errors \sigma_i on the underlying data y_i. This can be verified from the relation \delta f = J \delta c and the fact that the fluctuations in f from the data y_i are normalised by \sigma_i and so satisfy <\delta f \delta f^T> = I. For an unweighted least-squares function f_i = (Y(x, t_i) - y_i) the covariance matrix above should be multiplied by the variance of the residuals about the best-fit \sigma^2 = \sum (y_i - Y(x,t_i))^2 / (n-p) to give the variance-covariance matrix \sigma^2 C. This estimates the statistical error on the best-fit parameters from the scatter of the underlying data. For more information about covariance matrices see *note Fitting Overview::.  File: gsl-ref.info, Node: Example programs for Nonlinear Least-Squares Fitting, Next: References and Further Reading for Nonlinear Least-Squares Fitting, Prev: Computing the covariance matrix of best fit parameters, Up: Nonlinear Least-Squares Fitting 38.11 Examples ============== The following example program fits a weighted exponential model with background to experimental data, Y = A \exp(-\lambda t) + b. The first part of the program sets up the functions 'expb_f' and 'expb_df' to calculate the model and its Jacobian. The appropriate fitting function is given by, f_i = ((A \exp(-\lambda t_i) + b) - y_i)/\sigma_i where we have chosen t_i = i. The Jacobian matrix J is the derivative of these functions with respect to the three parameters (A, \lambda, b). It is given by, J_{ij} = d f_i / d x_j where x_0 = A, x_1 = \lambda and x_2 = b. /* expfit.c -- model functions for exponential + background */ struct data { size_t n; double * y; double * sigma; }; int expb_f (const gsl_vector * x, void *data, gsl_vector * f) { size_t n = ((struct data *)data)->n; double *y = ((struct data *)data)->y; double *sigma = ((struct data *) data)->sigma; double A = gsl_vector_get (x, 0); double lambda = gsl_vector_get (x, 1); double b = gsl_vector_get (x, 2); size_t i; for (i = 0; i < n; i++) { /* Model Yi = A * exp(-lambda * i) + b */ double t = i; double Yi = A * exp (-lambda * t) + b; gsl_vector_set (f, i, (Yi - y[i])/sigma[i]); } return GSL_SUCCESS; } int expb_df (const gsl_vector * x, void *data, gsl_matrix * J) { size_t n = ((struct data *)data)->n; double *sigma = ((struct data *) data)->sigma; double A = gsl_vector_get (x, 0); double lambda = gsl_vector_get (x, 1); size_t i; for (i = 0; i < n; i++) { /* Jacobian matrix J(i,j) = dfi / dxj, */ /* where fi = (Yi - yi)/sigma[i], */ /* Yi = A * exp(-lambda * i) + b */ /* and the xj are the parameters (A,lambda,b) */ double t = i; double s = sigma[i]; double e = exp(-lambda * t); gsl_matrix_set (J, i, 0, e/s); gsl_matrix_set (J, i, 1, -t * A * e/s); gsl_matrix_set (J, i, 2, 1/s); } return GSL_SUCCESS; } int expb_fdf (const gsl_vector * x, void *data, gsl_vector * f, gsl_matrix * J) { expb_f (x, data, f); expb_df (x, data, J); return GSL_SUCCESS; } The main part of the program sets up a Levenberg-Marquardt solver and some simulated random data. The data uses the known parameters (1.0,5.0,0.1) combined with Gaussian noise (standard deviation = 0.1) over a range of 40 timesteps. The initial guess for the parameters is chosen as (0.0, 1.0, 0.0). #include #include #include #include #include #include #include #include "expfit.c" #define N 40 void print_state (size_t iter, gsl_multifit_fdfsolver * s); int main (void) { const gsl_multifit_fdfsolver_type *T; gsl_multifit_fdfsolver *s; int status; unsigned int i, iter = 0; const size_t n = N; const size_t p = 3; gsl_matrix *covar = gsl_matrix_alloc (p, p); double y[N], sigma[N]; struct data d = { n, y, sigma}; gsl_multifit_function_fdf f; double x_init[3] = { 1.0, 0.0, 0.0 }; gsl_vector_view x = gsl_vector_view_array (x_init, p); const gsl_rng_type * type; gsl_rng * r; gsl_rng_env_setup(); type = gsl_rng_default; r = gsl_rng_alloc (type); f.f = &expb_f; f.df = &expb_df; f.fdf = &expb_fdf; f.n = n; f.p = p; f.params = &d; /* This is the data to be fitted */ for (i = 0; i < n; i++) { double t = i; y[i] = 1.0 + 5 * exp (-0.1 * t) + gsl_ran_gaussian (r, 0.1); sigma[i] = 0.1; printf ("data: %u %g %g\n", i, y[i], sigma[i]); }; T = gsl_multifit_fdfsolver_lmsder; s = gsl_multifit_fdfsolver_alloc (T, n, p); gsl_multifit_fdfsolver_set (s, &f, &x.vector); print_state (iter, s); do { iter++; status = gsl_multifit_fdfsolver_iterate (s); printf ("status = %s\n", gsl_strerror (status)); print_state (iter, s); if (status) break; status = gsl_multifit_test_delta (s->dx, s->x, 1e-4, 1e-4); } while (status == GSL_CONTINUE && iter < 500); gsl_multifit_covar (s->J, 0.0, covar); #define FIT(i) gsl_vector_get(s->x, i) #define ERR(i) sqrt(gsl_matrix_get(covar,i,i)) { double chi = gsl_blas_dnrm2(s->f); double dof = n - p; double c = GSL_MAX_DBL(1, chi / sqrt(dof)); printf("chisq/dof = %g\n", pow(chi, 2.0) / dof); printf ("A = %.5f +/- %.5f\n", FIT(0), c*ERR(0)); printf ("lambda = %.5f +/- %.5f\n", FIT(1), c*ERR(1)); printf ("b = %.5f +/- %.5f\n", FIT(2), c*ERR(2)); } printf ("status = %s\n", gsl_strerror (status)); gsl_multifit_fdfsolver_free (s); gsl_matrix_free (covar); gsl_rng_free (r); return 0; } void print_state (size_t iter, gsl_multifit_fdfsolver * s) { printf ("iter: %3u x = % 15.8f % 15.8f % 15.8f " "|f(x)| = %g\n", iter, gsl_vector_get (s->x, 0), gsl_vector_get (s->x, 1), gsl_vector_get (s->x, 2), gsl_blas_dnrm2 (s->f)); } The iteration terminates when the change in x is smaller than 0.0001, as both an absolute and relative change. Here are the results of running the program: iter: 0 x=1.00000000 0.00000000 0.00000000 |f(x)|=117.349 status=success iter: 1 x=1.64659312 0.01814772 0.64659312 |f(x)|=76.4578 status=success iter: 2 x=2.85876037 0.08092095 1.44796363 |f(x)|=37.6838 status=success iter: 3 x=4.94899512 0.11942928 1.09457665 |f(x)|=9.58079 status=success iter: 4 x=5.02175572 0.10287787 1.03388354 |f(x)|=5.63049 status=success iter: 5 x=5.04520433 0.10405523 1.01941607 |f(x)|=5.44398 status=success iter: 6 x=5.04535782 0.10404906 1.01924871 |f(x)|=5.44397 chisq/dof = 0.800996 A = 5.04536 +/- 0.06028 lambda = 0.10405 +/- 0.00316 b = 1.01925 +/- 0.03782 status = success The approximate values of the parameters are found correctly, and the chi-squared value indicates a good fit (the chi-squared per degree of freedom is approximately 1). In this case the errors on the parameters can be estimated from the square roots of the diagonal elements of the covariance matrix. If the chi-squared value shows a poor fit (i.e. chi^2/dof >> 1) then the error estimates obtained from the covariance matrix will be too small. In the example program the error estimates are multiplied by \sqrt{\chi^2/dof} in this case, a common way of increasing the errors for a poor fit. Note that a poor fit will result from the use an inappropriate model, and the scaled error estimates may then be outside the range of validity for Gaussian errors.  File: gsl-ref.info, Node: References and Further Reading for Nonlinear Least-Squares Fitting, Prev: Example programs for Nonlinear Least-Squares Fitting, Up: Nonlinear Least-Squares Fitting 38.12 References and Further Reading ==================================== The MINPACK algorithm is described in the following article, J.J. More', 'The Levenberg-Marquardt Algorithm: Implementation and Theory', Lecture Notes in Mathematics, v630 (1978), ed G. Watson. The following paper is also relevant to the algorithms described in this section, J.J. More', B.S. Garbow, K.E. Hillstrom, "Testing Unconstrained Optimization Software", ACM Transactions on Mathematical Software, Vol 7, No 1 (1981), p 17-41.  File: gsl-ref.info, Node: Basis Splines, Next: Physical Constants, Prev: Nonlinear Least-Squares Fitting, Up: Top 39 Basis Splines **************** This chapter describes functions for the computation of smoothing basis splines (B-splines). A smoothing spline differs from an interpolating spline in that the resulting curve is not required to pass through each datapoint. *Note Interpolation::, for information about interpolating splines. The header file 'gsl_bspline.h' contains the prototypes for the bspline functions and related declarations. * Menu: * Overview of B-splines:: * Initializing the B-splines solver:: * Constructing the knots vector:: * Evaluation of B-spline basis functions:: * Evaluation of B-spline basis function derivatives:: * Working with the Greville abscissae:: * Example programs for B-splines:: * References and Further Reading::  File: gsl-ref.info, Node: Overview of B-splines, Next: Initializing the B-splines solver, Up: Basis Splines 39.1 Overview ============= B-splines are commonly used as basis functions to fit smoothing curves to large data sets. To do this, the abscissa axis is broken up into some number of intervals, where the endpoints of each interval are called "breakpoints". These breakpoints are then converted to "knots" by imposing various continuity and smoothness conditions at each interface. Given a nondecreasing knot vector t = {t_0, t_1, ..., t_{n+k-1}}, the n basis splines of order k are defined by B_(i,1)(x) = (1, t_i <= x < t_(i+1) (0, else B_(i,k)(x) = [(x - t_i)/(t_(i+k-1) - t_i)] B_(i,k-1)(x) + [(t_(i+k) - x)/(t_(i+k) - t_(i+1))] B_(i+1,k-1)(x) for i = 0, ..., n-1. The common case of cubic B-splines is given by k = 4. The above recurrence relation can be evaluated in a numerically stable way by the de Boor algorithm. If we define appropriate knots on an interval [a,b] then the B-spline basis functions form a complete set on that interval. Therefore we can expand a smoothing function as f(x) = \sum_i c_i B_(i,k)(x) given enough (x_j, f(x_j)) data pairs. The coefficients c_i can be readily obtained from a least-squares fit.  File: gsl-ref.info, Node: Initializing the B-splines solver, Next: Constructing the knots vector, Prev: Overview of B-splines, Up: Basis Splines 39.2 Initializing the B-splines solver ====================================== The computation of B-spline functions requires a preallocated workspace of type 'gsl_bspline_workspace'. If B-spline derivatives are also required, an additional 'gsl_bspline_deriv_workspace' is needed. -- Function: gsl_bspline_workspace * gsl_bspline_alloc (const size_t K, const size_t NBREAK) This function allocates a workspace for computing B-splines of order K. The number of breakpoints is given by NBREAK. This leads to n = nbreak + k - 2 basis functions. Cubic B-splines are specified by k = 4. The size of the workspace is O(5k + nbreak). -- Function: void gsl_bspline_free (gsl_bspline_workspace * W) This function frees the memory associated with the workspace W. -- Function: gsl_bspline_deriv_workspace * gsl_bspline_deriv_alloc (const size_t K) This function allocates a workspace for computing the derivatives of a B-spline basis function of order K. The size of the workspace is O(2k^2). -- Function: void gsl_bspline_deriv_free (gsl_bspline_deriv_workspace * W) This function frees the memory associated with the derivative workspace W.  File: gsl-ref.info, Node: Constructing the knots vector, Next: Evaluation of B-spline basis functions, Prev: Initializing the B-splines solver, Up: Basis Splines 39.3 Constructing the knots vector ================================== -- Function: int gsl_bspline_knots (const gsl_vector * BREAKPTS, gsl_bspline_workspace * W) This function computes the knots associated with the given breakpoints and stores them internally in 'w->knots'. -- Function: int gsl_bspline_knots_uniform (const double A, const double B, gsl_bspline_workspace * W) This function assumes uniformly spaced breakpoints on [a,b] and constructs the corresponding knot vector using the previously specified NBREAK parameter. The knots are stored in 'w->knots'.  File: gsl-ref.info, Node: Evaluation of B-spline basis functions, Next: Evaluation of B-spline basis function derivatives, Prev: Constructing the knots vector, Up: Basis Splines 39.4 Evaluation of B-splines ============================ -- Function: int gsl_bspline_eval (const double X, gsl_vector * B, gsl_bspline_workspace * W) This function evaluates all B-spline basis functions at the position X and stores them in the vector B, so that the i-th element is B_i(x). The vector B must be of length n = nbreak + k - 2. This value may also be obtained by calling 'gsl_bspline_ncoeffs'. Computing all the basis functions at once is more efficient than computing them individually, due to the nature of the defining recurrence relation. -- Function: int gsl_bspline_eval_nonzero (const double X, gsl_vector * BK, size_t * ISTART, size_t * IEND, gsl_bspline_workspace * W) This function evaluates all potentially nonzero B-spline basis functions at the position X and stores them in the vector BK, so that the i-th element is B_(istart+i)(x). The last element of BK is B_(iend)(x). The vector BK must be of length k. By returning only the nonzero basis functions, this function allows quantities involving linear combinations of the B_i(x) to be computed without unnecessary terms (such linear combinations occur, for example, when evaluating an interpolated function). -- Function: size_t gsl_bspline_ncoeffs (gsl_bspline_workspace * W) This function returns the number of B-spline coefficients given by n = nbreak + k - 2.  File: gsl-ref.info, Node: Evaluation of B-spline basis function derivatives, Next: Working with the Greville abscissae, Prev: Evaluation of B-spline basis functions, Up: Basis Splines 39.5 Evaluation of B-spline derivatives ======================================= -- Function: int gsl_bspline_deriv_eval (const double X, const size_t NDERIV, gsl_matrix * DB, gsl_bspline_workspace * W, gsl_bspline_deriv_workspace * DW) This function evaluates all B-spline basis function derivatives of orders 0 through nderiv (inclusive) at the position X and stores them in the matrix DB. The (i,j)-th element of DB is d^jB_i(x)/dx^j. The matrix DB must be of size n = nbreak + k - 2 by nderiv + 1. The value n may also be obtained by calling 'gsl_bspline_ncoeffs'. Note that function evaluations are included as the zeroth order derivatives in DB. Computing all the basis function derivatives at once is more efficient than computing them individually, due to the nature of the defining recurrence relation. -- Function: int gsl_bspline_deriv_eval_nonzero (const double X, const size_t NDERIV, gsl_matrix * DB, size_t * ISTART, size_t * IEND, gsl_bspline_workspace * W, gsl_bspline_deriv_workspace * DW) This function evaluates all potentially nonzero B-spline basis function derivatives of orders 0 through nderiv (inclusive) at the position X and stores them in the matrix DB. The (i,j)-th element of DB is d^j/dx^j B_(istart+i)(x). The last row of DB contains d^j/dx^j B_(iend)(x). The matrix DB must be of size k by at least nderiv + 1. Note that function evaluations are included as the zeroth order derivatives in DB. By returning only the nonzero basis functions, this function allows quantities involving linear combinations of the B_i(x) and their derivatives to be computed without unnecessary terms.  File: gsl-ref.info, Node: Working with the Greville abscissae, Next: Example programs for B-splines, Prev: Evaluation of B-spline basis function derivatives, Up: Basis Splines 39.6 Working with the Greville abscissae ======================================== The Greville abscissae are defined to be the mean location of k-1 consecutive knots in the knot vector for each basis spline function of order k. With the first and last knots in the 'gsl_bspline_workspace' knot vector excluded, there are 'gsl_bspline_ncoeffs' Greville abscissae for any given B-spline basis. These values are often used in B-spline collocation applications and may also be called Marsden-Schoenberg points. -- Function: double gsl_bspline_greville_abscissa (size_t I, gsl_bspline_workspace *W); Returns the location of the i-th Greville abscissa for the given B-spline basis. For the ill-defined case when k=1, the implementation chooses to return breakpoint interval midpoints.  File: gsl-ref.info, Node: Example programs for B-splines, Next: References and Further Reading, Prev: Working with the Greville abscissae, Up: Basis Splines 39.7 Examples ============= The following program computes a linear least squares fit to data using cubic B-spline basis functions with uniform breakpoints. The data is generated from the curve y(x) = \cos{(x)} \exp{(-x/10)} on the interval [0, 15] with Gaussian noise added. #include #include #include #include #include #include #include #include /* number of data points to fit */ #define N 200 /* number of fit coefficients */ #define NCOEFFS 12 /* nbreak = ncoeffs + 2 - k = ncoeffs - 2 since k = 4 */ #define NBREAK (NCOEFFS - 2) int main (void) { const size_t n = N; const size_t ncoeffs = NCOEFFS; const size_t nbreak = NBREAK; size_t i, j; gsl_bspline_workspace *bw; gsl_vector *B; double dy; gsl_rng *r; gsl_vector *c, *w; gsl_vector *x, *y; gsl_matrix *X, *cov; gsl_multifit_linear_workspace *mw; double chisq, Rsq, dof, tss; gsl_rng_env_setup(); r = gsl_rng_alloc(gsl_rng_default); /* allocate a cubic bspline workspace (k = 4) */ bw = gsl_bspline_alloc(4, nbreak); B = gsl_vector_alloc(ncoeffs); x = gsl_vector_alloc(n); y = gsl_vector_alloc(n); X = gsl_matrix_alloc(n, ncoeffs); c = gsl_vector_alloc(ncoeffs); w = gsl_vector_alloc(n); cov = gsl_matrix_alloc(ncoeffs, ncoeffs); mw = gsl_multifit_linear_alloc(n, ncoeffs); printf("#m=0,S=0\n"); /* this is the data to be fitted */ for (i = 0; i < n; ++i) { double sigma; double xi = (15.0 / (N - 1)) * i; double yi = cos(xi) * exp(-0.1 * xi); sigma = 0.1 * yi; dy = gsl_ran_gaussian(r, sigma); yi += dy; gsl_vector_set(x, i, xi); gsl_vector_set(y, i, yi); gsl_vector_set(w, i, 1.0 / (sigma * sigma)); printf("%f %f\n", xi, yi); } /* use uniform breakpoints on [0, 15] */ gsl_bspline_knots_uniform(0.0, 15.0, bw); /* construct the fit matrix X */ for (i = 0; i < n; ++i) { double xi = gsl_vector_get(x, i); /* compute B_j(xi) for all j */ gsl_bspline_eval(xi, B, bw); /* fill in row i of X */ for (j = 0; j < ncoeffs; ++j) { double Bj = gsl_vector_get(B, j); gsl_matrix_set(X, i, j, Bj); } } /* do the fit */ gsl_multifit_wlinear(X, w, y, c, cov, &chisq, mw); dof = n - ncoeffs; tss = gsl_stats_wtss(w->data, 1, y->data, 1, y->size); Rsq = 1.0 - chisq / tss; fprintf(stderr, "chisq/dof = %e, Rsq = %f\n", chisq / dof, Rsq); /* output the smoothed curve */ { double xi, yi, yerr; printf("#m=1,S=0\n"); for (xi = 0.0; xi < 15.0; xi += 0.1) { gsl_bspline_eval(xi, B, bw); gsl_multifit_linear_est(B, c, cov, &yi, &yerr); printf("%f %f\n", xi, yi); } } gsl_rng_free(r); gsl_bspline_free(bw); gsl_vector_free(B); gsl_vector_free(x); gsl_vector_free(y); gsl_matrix_free(X); gsl_vector_free(c); gsl_vector_free(w); gsl_matrix_free(cov); gsl_multifit_linear_free(mw); return 0; } /* main() */ The output can be plotted with GNU 'graph'. $ ./a.out > bspline.dat chisq/dof = 1.118217e+00, Rsq = 0.989771 $ graph -T ps -X x -Y y -x 0 15 -y -1 1.3 < bspline.dat > bspline.ps  File: gsl-ref.info, Node: References and Further Reading, Prev: Example programs for B-splines, Up: Basis Splines 39.8 References and Further Reading =================================== Further information on the algorithms described in this section can be found in the following book, C. de Boor, 'A Practical Guide to Splines' (1978), Springer-Verlag, ISBN 0-387-90356-9. Further information of Greville abscissae and B-spline collocation can be found in the following paper, Richard W. Johnson, Higher order B-spline collocation at the Greville abscissae. 'Applied Numerical Mathematics'. vol. 52, 2005, 63-75. A large collection of B-spline routines is available in the PPPACK library available at , which is also part of SLATEC.  File: gsl-ref.info, Node: Physical Constants, Next: IEEE floating-point arithmetic, Prev: Basis Splines, Up: Top 40 Physical Constants ********************* This chapter describes macros for the values of physical constants, such as the speed of light, c, and gravitational constant, G. The values are available in different unit systems, including the standard MKSA system (meters, kilograms, seconds, amperes) and the CGSM system (centimeters, grams, seconds, gauss), which is commonly used in Astronomy. The definitions of constants in the MKSA system are available in the file 'gsl_const_mksa.h'. The constants in the CGSM system are defined in 'gsl_const_cgsm.h'. Dimensionless constants, such as the fine structure constant, which are pure numbers are defined in 'gsl_const_num.h'. * Menu: * Fundamental Constants:: * Astronomy and Astrophysics:: * Atomic and Nuclear Physics:: * Measurement of Time:: * Imperial Units :: * Speed and Nautical Units:: * Printers Units:: * Volume Area and Length:: * Mass and Weight :: * Thermal Energy and Power:: * Pressure:: * Viscosity:: * Light and Illumination:: * Radioactivity:: * Force and Energy:: * Prefixes:: * Physical Constant Examples:: * Physical Constant References and Further Reading:: The full list of constants is described briefly below. Consult the header files themselves for the values of the constants used in the library.  File: gsl-ref.info, Node: Fundamental Constants, Next: Astronomy and Astrophysics, Up: Physical Constants 40.1 Fundamental Constants ========================== 'GSL_CONST_MKSA_SPEED_OF_LIGHT' The speed of light in vacuum, c. 'GSL_CONST_MKSA_VACUUM_PERMEABILITY' The permeability of free space, \mu_0. This constant is defined in the MKSA system only. 'GSL_CONST_MKSA_VACUUM_PERMITTIVITY' The permittivity of free space, \epsilon_0. This constant is defined in the MKSA system only. 'GSL_CONST_MKSA_PLANCKS_CONSTANT_H' Planck's constant, h. 'GSL_CONST_MKSA_PLANCKS_CONSTANT_HBAR' Planck's constant divided by 2\pi, \hbar. 'GSL_CONST_NUM_AVOGADRO' Avogadro's number, N_a. 'GSL_CONST_MKSA_FARADAY' The molar charge of 1 Faraday. 'GSL_CONST_MKSA_BOLTZMANN' The Boltzmann constant, k. 'GSL_CONST_MKSA_MOLAR_GAS' The molar gas constant, R_0. 'GSL_CONST_MKSA_STANDARD_GAS_VOLUME' The standard gas volume, V_0. 'GSL_CONST_MKSA_STEFAN_BOLTZMANN_CONSTANT' The Stefan-Boltzmann radiation constant, \sigma. 'GSL_CONST_MKSA_GAUSS' The magnetic field of 1 Gauss.  File: gsl-ref.info, Node: Astronomy and Astrophysics, Next: Atomic and Nuclear Physics, Prev: Fundamental Constants, Up: Physical Constants 40.2 Astronomy and Astrophysics =============================== 'GSL_CONST_MKSA_ASTRONOMICAL_UNIT' The length of 1 astronomical unit (mean earth-sun distance), au. 'GSL_CONST_MKSA_GRAVITATIONAL_CONSTANT' The gravitational constant, G. 'GSL_CONST_MKSA_LIGHT_YEAR' The distance of 1 light-year, ly. 'GSL_CONST_MKSA_PARSEC' The distance of 1 parsec, pc. 'GSL_CONST_MKSA_GRAV_ACCEL' The standard gravitational acceleration on Earth, g. 'GSL_CONST_MKSA_SOLAR_MASS' The mass of the Sun.  File: gsl-ref.info, Node: Atomic and Nuclear Physics, Next: Measurement of Time, Prev: Astronomy and Astrophysics, Up: Physical Constants 40.3 Atomic and Nuclear Physics =============================== 'GSL_CONST_MKSA_ELECTRON_CHARGE' The charge of the electron, e. 'GSL_CONST_MKSA_ELECTRON_VOLT' The energy of 1 electron volt, eV. 'GSL_CONST_MKSA_UNIFIED_ATOMIC_MASS' The unified atomic mass, amu. 'GSL_CONST_MKSA_MASS_ELECTRON' The mass of the electron, m_e. 'GSL_CONST_MKSA_MASS_MUON' The mass of the muon, m_\mu. 'GSL_CONST_MKSA_MASS_PROTON' The mass of the proton, m_p. 'GSL_CONST_MKSA_MASS_NEUTRON' The mass of the neutron, m_n. 'GSL_CONST_NUM_FINE_STRUCTURE' The electromagnetic fine structure constant \alpha. 'GSL_CONST_MKSA_RYDBERG' The Rydberg constant, Ry, in units of energy. This is related to the Rydberg inverse wavelength R_\infty by Ry = h c R_\infty. 'GSL_CONST_MKSA_BOHR_RADIUS' The Bohr radius, a_0. 'GSL_CONST_MKSA_ANGSTROM' The length of 1 angstrom. 'GSL_CONST_MKSA_BARN' The area of 1 barn. 'GSL_CONST_MKSA_BOHR_MAGNETON' The Bohr Magneton, \mu_B. 'GSL_CONST_MKSA_NUCLEAR_MAGNETON' The Nuclear Magneton, \mu_N. 'GSL_CONST_MKSA_ELECTRON_MAGNETIC_MOMENT' The absolute value of the magnetic moment of the electron, \mu_e. The physical magnetic moment of the electron is negative. 'GSL_CONST_MKSA_PROTON_MAGNETIC_MOMENT' The magnetic moment of the proton, \mu_p. 'GSL_CONST_MKSA_THOMSON_CROSS_SECTION' The Thomson cross section, \sigma_T. 'GSL_CONST_MKSA_DEBYE' The electric dipole moment of 1 Debye, D.  File: gsl-ref.info, Node: Measurement of Time, Next: Imperial Units, Prev: Atomic and Nuclear Physics, Up: Physical Constants 40.4 Measurement of Time ======================== 'GSL_CONST_MKSA_MINUTE' The number of seconds in 1 minute. 'GSL_CONST_MKSA_HOUR' The number of seconds in 1 hour. 'GSL_CONST_MKSA_DAY' The number of seconds in 1 day. 'GSL_CONST_MKSA_WEEK' The number of seconds in 1 week.  File: gsl-ref.info, Node: Imperial Units, Next: Speed and Nautical Units, Prev: Measurement of Time, Up: Physical Constants 40.5 Imperial Units =================== 'GSL_CONST_MKSA_INCH' The length of 1 inch. 'GSL_CONST_MKSA_FOOT' The length of 1 foot. 'GSL_CONST_MKSA_YARD' The length of 1 yard. 'GSL_CONST_MKSA_MILE' The length of 1 mile. 'GSL_CONST_MKSA_MIL' The length of 1 mil (1/1000th of an inch).  File: gsl-ref.info, Node: Speed and Nautical Units, Next: Printers Units, Prev: Imperial Units, Up: Physical Constants 40.6 Speed and Nautical Units ============================= 'GSL_CONST_MKSA_KILOMETERS_PER_HOUR' The speed of 1 kilometer per hour. 'GSL_CONST_MKSA_MILES_PER_HOUR' The speed of 1 mile per hour. 'GSL_CONST_MKSA_NAUTICAL_MILE' The length of 1 nautical mile. 'GSL_CONST_MKSA_FATHOM' The length of 1 fathom. 'GSL_CONST_MKSA_KNOT' The speed of 1 knot.  File: gsl-ref.info, Node: Printers Units, Next: Volume Area and Length, Prev: Speed and Nautical Units, Up: Physical Constants 40.7 Printers Units =================== 'GSL_CONST_MKSA_POINT' The length of 1 printer's point (1/72 inch). 'GSL_CONST_MKSA_TEXPOINT' The length of 1 TeX point (1/72.27 inch).  File: gsl-ref.info, Node: Volume Area and Length, Next: Mass and Weight, Prev: Printers Units, Up: Physical Constants 40.8 Volume, Area and Length ============================ 'GSL_CONST_MKSA_MICRON' The length of 1 micron. 'GSL_CONST_MKSA_HECTARE' The area of 1 hectare. 'GSL_CONST_MKSA_ACRE' The area of 1 acre. 'GSL_CONST_MKSA_LITER' The volume of 1 liter. 'GSL_CONST_MKSA_US_GALLON' The volume of 1 US gallon. 'GSL_CONST_MKSA_CANADIAN_GALLON' The volume of 1 Canadian gallon. 'GSL_CONST_MKSA_UK_GALLON' The volume of 1 UK gallon. 'GSL_CONST_MKSA_QUART' The volume of 1 quart. 'GSL_CONST_MKSA_PINT' The volume of 1 pint.  File: gsl-ref.info, Node: Mass and Weight, Next: Thermal Energy and Power, Prev: Volume Area and Length, Up: Physical Constants 40.9 Mass and Weight ==================== 'GSL_CONST_MKSA_POUND_MASS' The mass of 1 pound. 'GSL_CONST_MKSA_OUNCE_MASS' The mass of 1 ounce. 'GSL_CONST_MKSA_TON' The mass of 1 ton. 'GSL_CONST_MKSA_METRIC_TON' The mass of 1 metric ton (1000 kg). 'GSL_CONST_MKSA_UK_TON' The mass of 1 UK ton. 'GSL_CONST_MKSA_TROY_OUNCE' The mass of 1 troy ounce. 'GSL_CONST_MKSA_CARAT' The mass of 1 carat. 'GSL_CONST_MKSA_GRAM_FORCE' The force of 1 gram weight. 'GSL_CONST_MKSA_POUND_FORCE' The force of 1 pound weight. 'GSL_CONST_MKSA_KILOPOUND_FORCE' The force of 1 kilopound weight. 'GSL_CONST_MKSA_POUNDAL' The force of 1 poundal.  File: gsl-ref.info, Node: Thermal Energy and Power, Next: Pressure, Prev: Mass and Weight, Up: Physical Constants 40.10 Thermal Energy and Power ============================== 'GSL_CONST_MKSA_CALORIE' The energy of 1 calorie. 'GSL_CONST_MKSA_BTU' The energy of 1 British Thermal Unit, btu. 'GSL_CONST_MKSA_THERM' The energy of 1 Therm. 'GSL_CONST_MKSA_HORSEPOWER' The power of 1 horsepower.  File: gsl-ref.info, Node: Pressure, Next: Viscosity, Prev: Thermal Energy and Power, Up: Physical Constants 40.11 Pressure ============== 'GSL_CONST_MKSA_BAR' The pressure of 1 bar. 'GSL_CONST_MKSA_STD_ATMOSPHERE' The pressure of 1 standard atmosphere. 'GSL_CONST_MKSA_TORR' The pressure of 1 torr. 'GSL_CONST_MKSA_METER_OF_MERCURY' The pressure of 1 meter of mercury. 'GSL_CONST_MKSA_INCH_OF_MERCURY' The pressure of 1 inch of mercury. 'GSL_CONST_MKSA_INCH_OF_WATER' The pressure of 1 inch of water. 'GSL_CONST_MKSA_PSI' The pressure of 1 pound per square inch.  File: gsl-ref.info, Node: Viscosity, Next: Light and Illumination, Prev: Pressure, Up: Physical Constants 40.12 Viscosity =============== 'GSL_CONST_MKSA_POISE' The dynamic viscosity of 1 poise. 'GSL_CONST_MKSA_STOKES' The kinematic viscosity of 1 stokes.  File: gsl-ref.info, Node: Light and Illumination, Next: Radioactivity, Prev: Viscosity, Up: Physical Constants 40.13 Light and Illumination ============================ 'GSL_CONST_MKSA_STILB' The luminance of 1 stilb. 'GSL_CONST_MKSA_LUMEN' The luminous flux of 1 lumen. 'GSL_CONST_MKSA_LUX' The illuminance of 1 lux. 'GSL_CONST_MKSA_PHOT' The illuminance of 1 phot. 'GSL_CONST_MKSA_FOOTCANDLE' The illuminance of 1 footcandle. 'GSL_CONST_MKSA_LAMBERT' The luminance of 1 lambert. 'GSL_CONST_MKSA_FOOTLAMBERT' The luminance of 1 footlambert.  File: gsl-ref.info, Node: Radioactivity, Next: Force and Energy, Prev: Light and Illumination, Up: Physical Constants 40.14 Radioactivity =================== 'GSL_CONST_MKSA_CURIE' The activity of 1 curie. 'GSL_CONST_MKSA_ROENTGEN' The exposure of 1 roentgen. 'GSL_CONST_MKSA_RAD' The absorbed dose of 1 rad.  File: gsl-ref.info, Node: Force and Energy, Next: Prefixes, Prev: Radioactivity, Up: Physical Constants 40.15 Force and Energy ====================== 'GSL_CONST_MKSA_NEWTON' The SI unit of force, 1 Newton. 'GSL_CONST_MKSA_DYNE' The force of 1 Dyne = 10^-5 Newton. 'GSL_CONST_MKSA_JOULE' The SI unit of energy, 1 Joule. 'GSL_CONST_MKSA_ERG' The energy 1 erg = 10^-7 Joule.  File: gsl-ref.info, Node: Prefixes, Next: Physical Constant Examples, Prev: Force and Energy, Up: Physical Constants 40.16 Prefixes ============== These constants are dimensionless scaling factors. 'GSL_CONST_NUM_YOTTA' 10^24 'GSL_CONST_NUM_ZETTA' 10^21 'GSL_CONST_NUM_EXA' 10^18 'GSL_CONST_NUM_PETA' 10^15 'GSL_CONST_NUM_TERA' 10^12 'GSL_CONST_NUM_GIGA' 10^9 'GSL_CONST_NUM_MEGA' 10^6 'GSL_CONST_NUM_KILO' 10^3 'GSL_CONST_NUM_MILLI' 10^-3 'GSL_CONST_NUM_MICRO' 10^-6 'GSL_CONST_NUM_NANO' 10^-9 'GSL_CONST_NUM_PICO' 10^-12 'GSL_CONST_NUM_FEMTO' 10^-15 'GSL_CONST_NUM_ATTO' 10^-18 'GSL_CONST_NUM_ZEPTO' 10^-21 'GSL_CONST_NUM_YOCTO' 10^-24  File: gsl-ref.info, Node: Physical Constant Examples, Next: Physical Constant References and Further Reading, Prev: Prefixes, Up: Physical Constants 40.17 Examples ============== The following program demonstrates the use of the physical constants in a calculation. In this case, the goal is to calculate the range of light-travel times from Earth to Mars. The required data is the average distance of each planet from the Sun in astronomical units (the eccentricities and inclinations of the orbits will be neglected for the purposes of this calculation). The average radius of the orbit of Mars is 1.52 astronomical units, and for the orbit of Earth it is 1 astronomical unit (by definition). These values are combined with the MKSA values of the constants for the speed of light and the length of an astronomical unit to produce a result for the shortest and longest light-travel times in seconds. The figures are converted into minutes before being displayed. #include #include int main (void) { double c = GSL_CONST_MKSA_SPEED_OF_LIGHT; double au = GSL_CONST_MKSA_ASTRONOMICAL_UNIT; double minutes = GSL_CONST_MKSA_MINUTE; /* distance stored in meters */ double r_earth = 1.00 * au; double r_mars = 1.52 * au; double t_min, t_max; t_min = (r_mars - r_earth) / c; t_max = (r_mars + r_earth) / c; printf ("light travel time from Earth to Mars:\n"); printf ("minimum = %.1f minutes\n", t_min / minutes); printf ("maximum = %.1f minutes\n", t_max / minutes); return 0; } Here is the output from the program, light travel time from Earth to Mars: minimum = 4.3 minutes maximum = 21.0 minutes  File: gsl-ref.info, Node: Physical Constant References and Further Reading, Prev: Physical Constant Examples, Up: Physical Constants 40.18 References and Further Reading ==================================== The authoritative sources for physical constants are the 2006 CODATA recommended values, published in the article below. Further information on the values of physical constants is also available from the NIST website. P.J. Mohr, B.N. Taylor, D.B. Newell, "CODATA Recommended Values of the Fundamental Physical Constants: 2006", Reviews of Modern Physics, 80(2), pp. 633-730 (2008).  File: gsl-ref.info, Node: IEEE floating-point arithmetic, Next: Debugging Numerical Programs, Prev: Physical Constants, Up: Top 41 IEEE floating-point arithmetic ********************************* This chapter describes functions for examining the representation of floating point numbers and controlling the floating point environment of your program. The functions described in this chapter are declared in the header file 'gsl_ieee_utils.h'. * Menu: * Representation of floating point numbers:: * Setting up your IEEE environment:: * IEEE References and Further Reading::  File: gsl-ref.info, Node: Representation of floating point numbers, Next: Setting up your IEEE environment, Up: IEEE floating-point arithmetic 41.1 Representation of floating point numbers ============================================= The IEEE Standard for Binary Floating-Point Arithmetic defines binary formats for single and double precision numbers. Each number is composed of three parts: a "sign bit" (s), an "exponent" (E) and a "fraction" (f). The numerical value of the combination (s,E,f) is given by the following formula, (-1)^s (1.fffff...) 2^E The sign bit is either zero or one. The exponent ranges from a minimum value E_min to a maximum value E_max depending on the precision. The exponent is converted to an unsigned number e, known as the "biased exponent", for storage by adding a "bias" parameter, e = E + bias. The sequence fffff... represents the digits of the binary fraction f. The binary digits are stored in "normalized form", by adjusting the exponent to give a leading digit of 1. Since the leading digit is always 1 for normalized numbers it is assumed implicitly and does not have to be stored. Numbers smaller than 2^(E_min) are be stored in "denormalized form" with a leading zero, (-1)^s (0.fffff...) 2^(E_min) This allows gradual underflow down to 2^(E_min - p) for p bits of precision. A zero is encoded with the special exponent of 2^(E_min - 1) and infinities with the exponent of 2^(E_max + 1). The format for single precision numbers uses 32 bits divided in the following way, seeeeeeeefffffffffffffffffffffff s = sign bit, 1 bit e = exponent, 8 bits (E_min=-126, E_max=127, bias=127) f = fraction, 23 bits The format for double precision numbers uses 64 bits divided in the following way, seeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffffff s = sign bit, 1 bit e = exponent, 11 bits (E_min=-1022, E_max=1023, bias=1023) f = fraction, 52 bits It is often useful to be able to investigate the behavior of a calculation at the bit-level and the library provides functions for printing the IEEE representations in a human-readable form. -- Function: void gsl_ieee_fprintf_float (FILE * STREAM, const float * X) -- Function: void gsl_ieee_fprintf_double (FILE * STREAM, const double * X) These functions output a formatted version of the IEEE floating-point number pointed to by X to the stream STREAM. A pointer is used to pass the number indirectly, to avoid any undesired promotion from 'float' to 'double'. The output takes one of the following forms, 'NaN' the Not-a-Number symbol 'Inf, -Inf' positive or negative infinity '1.fffff...*2^E, -1.fffff...*2^E' a normalized floating point number '0.fffff...*2^E, -0.fffff...*2^E' a denormalized floating point number '0, -0' positive or negative zero The output can be used directly in GNU Emacs Calc mode by preceding it with '2#' to indicate binary. -- Function: void gsl_ieee_printf_float (const float * X) -- Function: void gsl_ieee_printf_double (const double * X) These functions output a formatted version of the IEEE floating-point number pointed to by X to the stream 'stdout'. The following program demonstrates the use of the functions by printing the single and double precision representations of the fraction 1/3. For comparison the representation of the value promoted from single to double precision is also printed. #include #include int main (void) { float f = 1.0/3.0; double d = 1.0/3.0; double fd = f; /* promote from float to double */ printf (" f="); gsl_ieee_printf_float(&f); printf ("\n"); printf ("fd="); gsl_ieee_printf_double(&fd); printf ("\n"); printf (" d="); gsl_ieee_printf_double(&d); printf ("\n"); return 0; } The binary representation of 1/3 is 0.01010101... . The output below shows that the IEEE format normalizes this fraction to give a leading digit of 1, f= 1.01010101010101010101011*2^-2 fd= 1.0101010101010101010101100000000000000000000000000000*2^-2 d= 1.0101010101010101010101010101010101010101010101010101*2^-2 The output also shows that a single-precision number is promoted to double-precision by adding zeros in the binary representation.  File: gsl-ref.info, Node: Setting up your IEEE environment, Next: IEEE References and Further Reading, Prev: Representation of floating point numbers, Up: IEEE floating-point arithmetic 41.2 Setting up your IEEE environment ===================================== The IEEE standard defines several "modes" for controlling the behavior of floating point operations. These modes specify the important properties of computer arithmetic: the direction used for rounding (e.g. whether numbers should be rounded up, down or to the nearest number), the rounding precision and how the program should handle arithmetic exceptions, such as division by zero. Many of these features can now be controlled via standard functions such as 'fpsetround', which should be used whenever they are available. Unfortunately in the past there has been no universal API for controlling their behavior--each system has had its own low-level way of accessing them. To help you write portable programs GSL allows you to specify modes in a platform-independent way using the environment variable 'GSL_IEEE_MODE'. The library then takes care of all the necessary machine-specific initializations for you when you call the function 'gsl_ieee_env_setup'. -- Function: void gsl_ieee_env_setup () This function reads the environment variable 'GSL_IEEE_MODE' and attempts to set up the corresponding specified IEEE modes. The environment variable should be a list of keywords, separated by commas, like this, 'GSL_IEEE_MODE' = "KEYWORD,KEYWORD,..." where KEYWORD is one of the following mode-names, 'single-precision' 'double-precision' 'extended-precision' 'round-to-nearest' 'round-down' 'round-up' 'round-to-zero' 'mask-all' 'mask-invalid' 'mask-denormalized' 'mask-division-by-zero' 'mask-overflow' 'mask-underflow' 'trap-inexact' 'trap-common' If 'GSL_IEEE_MODE' is empty or undefined then the function returns immediately and no attempt is made to change the system's IEEE mode. When the modes from 'GSL_IEEE_MODE' are turned on the function prints a short message showing the new settings to remind you that the results of the program will be affected. If the requested modes are not supported by the platform being used then the function calls the error handler and returns an error code of 'GSL_EUNSUP'. When options are specified using this method, the resulting mode is based on a default setting of the highest available precision (double precision or extended precision, depending on the platform) in round-to-nearest mode, with all exceptions enabled apart from the INEXACT exception. The INEXACT exception is generated whenever rounding occurs, so it must generally be disabled in typical scientific calculations. All other floating-point exceptions are enabled by default, including underflows and the use of denormalized numbers, for safety. They can be disabled with the individual 'mask-' settings or together using 'mask-all'. The following adjusted combination of modes is convenient for many purposes, GSL_IEEE_MODE="double-precision,"\ "mask-underflow,"\ "mask-denormalized" This choice ignores any errors relating to small numbers (either denormalized, or underflowing to zero) but traps overflows, division by zero and invalid operations. Note that on the x86 series of processors this function sets both the original x87 mode and the newer MXCSR mode, which controls SSE floating-point operations. The SSE floating-point units do not have a precision-control bit, and always work in double-precision. The single-precision and extended-precision keywords have no effect in this case. To demonstrate the effects of different rounding modes consider the following program which computes e, the base of natural logarithms, by summing a rapidly-decreasing series, e = 1 + 1/2! + 1/3! + 1/4! + ... = 2.71828182846... #include #include #include int main (void) { double x = 1, oldsum = 0, sum = 0; int i = 0; gsl_ieee_env_setup (); /* read GSL_IEEE_MODE */ do { i++; oldsum = sum; sum += x; x = x / i; printf ("i=%2d sum=%.18f error=%g\n", i, sum, sum - M_E); if (i > 30) break; } while (sum != oldsum); return 0; } Here are the results of running the program in 'round-to-nearest' mode. This is the IEEE default so it isn't really necessary to specify it here, $ GSL_IEEE_MODE="round-to-nearest" ./a.out i= 1 sum=1.000000000000000000 error=-1.71828 i= 2 sum=2.000000000000000000 error=-0.718282 .... i=18 sum=2.718281828459045535 error=4.44089e-16 i=19 sum=2.718281828459045535 error=4.44089e-16 After nineteen terms the sum converges to within 4 \times 10^-16 of the correct value. If we now change the rounding mode to 'round-down' the final result is less accurate, $ GSL_IEEE_MODE="round-down" ./a.out i= 1 sum=1.000000000000000000 error=-1.71828 .... i=19 sum=2.718281828459041094 error=-3.9968e-15 The result is about 4 \times 10^-15 below the correct value, an order of magnitude worse than the result obtained in the 'round-to-nearest' mode. If we change to rounding mode to 'round-up' then the final result is higher than the correct value (when we add each term to the sum the final result is always rounded up, which increases the sum by at least one tick until the added term underflows to zero). To avoid this problem we would need to use a safer converge criterion, such as 'while (fabs(sum - oldsum) > epsilon)', with a suitably chosen value of epsilon. Finally we can see the effect of computing the sum using single-precision rounding, in the default 'round-to-nearest' mode. In this case the program thinks it is still using double precision numbers but the CPU rounds the result of each floating point operation to single-precision accuracy. This simulates the effect of writing the program using single-precision 'float' variables instead of 'double' variables. The iteration stops after about half the number of iterations and the final result is much less accurate, $ GSL_IEEE_MODE="single-precision" ./a.out .... i=12 sum=2.718281984329223633 error=1.5587e-07 with an error of O(10^-7), which corresponds to single precision accuracy (about 1 part in 10^7). Continuing the iterations further does not decrease the error because all the subsequent results are rounded to the same value.  File: gsl-ref.info, Node: IEEE References and Further Reading, Prev: Setting up your IEEE environment, Up: IEEE floating-point arithmetic 41.3 References and Further Reading =================================== The reference for the IEEE standard is, ANSI/IEEE Std 754-1985, IEEE Standard for Binary Floating-Point Arithmetic. A more pedagogical introduction to the standard can be found in the following paper, David Goldberg: What Every Computer Scientist Should Know About Floating-Point Arithmetic. 'ACM Computing Surveys', Vol. 23, No. 1 (March 1991), pages 5-48. Corrigendum: 'ACM Computing Surveys', Vol. 23, No. 3 (September 1991), page 413. and see also the sections by B. A. Wichmann and Charles B. Dunham in Surveyor's Forum: "What Every Computer Scientist Should Know About Floating-Point Arithmetic". 'ACM Computing Surveys', Vol. 24, No. 3 (September 1992), page 319. A detailed textbook on IEEE arithmetic and its practical use is available from SIAM Press, Michael L. Overton, 'Numerical Computing with IEEE Floating Point Arithmetic', SIAM Press, ISBN 0898715717.  File: gsl-ref.info, Node: Debugging Numerical Programs, Next: Contributors to GSL, Prev: IEEE floating-point arithmetic, Up: Top Appendix A Debugging Numerical Programs *************************************** This chapter describes some tips and tricks for debugging numerical programs which use GSL. * Menu: * Using gdb:: * Examining floating point registers:: * Handling floating point exceptions:: * GCC warning options for numerical programs:: * Debugging References::  File: gsl-ref.info, Node: Using gdb, Next: Examining floating point registers, Up: Debugging Numerical Programs A.1 Using gdb ============= Any errors reported by the library are passed to the function 'gsl_error'. By running your programs under gdb and setting a breakpoint in this function you can automatically catch any library errors. You can add a breakpoint for every session by putting break gsl_error into your '.gdbinit' file in the directory where your program is started. If the breakpoint catches an error then you can use a backtrace ('bt') to see the call-tree, and the arguments which possibly caused the error. By moving up into the calling function you can investigate the values of variables at that point. Here is an example from the program 'fft/test_trap', which contains the following line, status = gsl_fft_complex_wavetable_alloc (0, &complex_wavetable); The function 'gsl_fft_complex_wavetable_alloc' takes the length of an FFT as its first argument. When this line is executed an error will be generated because the length of an FFT is not allowed to be zero. To debug this problem we start 'gdb', using the file '.gdbinit' to define a breakpoint in 'gsl_error', $ gdb test_trap GDB is free software and you are welcome to distribute copies of it under certain conditions; type "show copying" to see the conditions. There is absolutely no warranty for GDB; type "show warranty" for details. GDB 4.16 (i586-debian-linux), Copyright 1996 Free Software Foundation, Inc. Breakpoint 1 at 0x8050b1e: file error.c, line 14. When we run the program this breakpoint catches the error and shows the reason for it. (gdb) run Starting program: test_trap Breakpoint 1, gsl_error (reason=0x8052b0d "length n must be positive integer", file=0x8052b04 "c_init.c", line=108, gsl_errno=1) at error.c:14 14 if (gsl_error_handler) The first argument of 'gsl_error' is always a string describing the error. Now we can look at the backtrace to see what caused the problem, (gdb) bt #0 gsl_error (reason=0x8052b0d "length n must be positive integer", file=0x8052b04 "c_init.c", line=108, gsl_errno=1) at error.c:14 #1 0x8049376 in gsl_fft_complex_wavetable_alloc (n=0, wavetable=0xbffff778) at c_init.c:108 #2 0x8048a00 in main (argc=1, argv=0xbffff9bc) at test_trap.c:94 #3 0x80488be in ___crt_dummy__ () We can see that the error was generated in the function 'gsl_fft_complex_wavetable_alloc' when it was called with an argument of N=0. The original call came from line 94 in the file 'test_trap.c'. By moving up to the level of the original call we can find the line that caused the error, (gdb) up #1 0x8049376 in gsl_fft_complex_wavetable_alloc (n=0, wavetable=0xbffff778) at c_init.c:108 108 GSL_ERROR ("length n must be positive integer", GSL_EDOM); (gdb) up #2 0x8048a00 in main (argc=1, argv=0xbffff9bc) at test_trap.c:94 94 status = gsl_fft_complex_wavetable_alloc (0, &complex_wavetable); Thus we have found the line that caused the problem. From this point we could also print out the values of other variables such as 'complex_wavetable'.  File: gsl-ref.info, Node: Examining floating point registers, Next: Handling floating point exceptions, Prev: Using gdb, Up: Debugging Numerical Programs A.2 Examining floating point registers ====================================== The contents of floating point registers can be examined using the command 'info float' (on supported platforms). (gdb) info float st0: 0xc4018b895aa17a945000 Valid Normal -7.838871e+308 st1: 0x3ff9ea3f50e4d7275000 Valid Normal 0.0285946 st2: 0x3fe790c64ce27dad4800 Valid Normal 6.7415931e-08 st3: 0x3ffaa3ef0df6607d7800 Spec Normal 0.0400229 st4: 0x3c028000000000000000 Valid Normal 4.4501477e-308 st5: 0x3ffef5412c22219d9000 Zero Normal 0.9580257 st6: 0x3fff8000000000000000 Valid Normal 1 st7: 0xc4028b65a1f6d243c800 Valid Normal -1.566206e+309 fctrl: 0x0272 53 bit; NEAR; mask DENOR UNDER LOS; fstat: 0xb9ba flags 0001; top 7; excep DENOR OVERF UNDER LOS ftag: 0x3fff fip: 0x08048b5c fcs: 0x051a0023 fopoff: 0x08086820 fopsel: 0x002b Individual registers can be examined using the variables $REG, where REG is the register name. (gdb) p $st1 $1 = 0.02859464454261210347719  File: gsl-ref.info, Node: Handling floating point exceptions, Next: GCC warning options for numerical programs, Prev: Examining floating point registers, Up: Debugging Numerical Programs A.3 Handling floating point exceptions ====================================== It is possible to stop the program whenever a 'SIGFPE' floating point exception occurs. This can be useful for finding the cause of an unexpected infinity or 'NaN'. The current handler settings can be shown with the command 'info signal SIGFPE'. (gdb) info signal SIGFPE Signal Stop Print Pass to program Description SIGFPE Yes Yes Yes Arithmetic exception Unless the program uses a signal handler the default setting should be changed so that SIGFPE is not passed to the program, as this would cause it to exit. The command 'handle SIGFPE stop nopass' prevents this. (gdb) handle SIGFPE stop nopass Signal Stop Print Pass to program Description SIGFPE Yes Yes No Arithmetic exception Depending on the platform it may be necessary to instruct the kernel to generate signals for floating point exceptions. For programs using GSL this can be achieved using the 'GSL_IEEE_MODE' environment variable in conjunction with the function 'gsl_ieee_env_setup' as described in *note IEEE floating-point arithmetic::. (gdb) set env GSL_IEEE_MODE=double-precision  File: gsl-ref.info, Node: GCC warning options for numerical programs, Next: Debugging References, Prev: Handling floating point exceptions, Up: Debugging Numerical Programs A.4 GCC warning options for numerical programs ============================================== Writing reliable numerical programs in C requires great care. The following GCC warning options are recommended when compiling numerical programs: gcc -ansi -pedantic -Werror -Wall -W -Wmissing-prototypes -Wstrict-prototypes -Wconversion -Wshadow -Wpointer-arith -Wcast-qual -Wcast-align -Wwrite-strings -Wnested-externs -fshort-enums -fno-common -Dinline= -g -O2 For details of each option consult the manual 'Using and Porting GCC'. The following table gives a brief explanation of what types of errors these options catch. '-ansi -pedantic' Use ANSI C, and reject any non-ANSI extensions. These flags help in writing portable programs that will compile on other systems. '-Werror' Consider warnings to be errors, so that compilation stops. This prevents warnings from scrolling off the top of the screen and being lost. You won't be able to compile the program until it is completely warning-free. '-Wall' This turns on a set of warnings for common programming problems. You need '-Wall', but it is not enough on its own. '-O2' Turn on optimization. The warnings for uninitialized variables in '-Wall' rely on the optimizer to analyze the code. If there is no optimization then these warnings aren't generated. '-W' This turns on some extra warnings not included in '-Wall', such as missing return values and comparisons between signed and unsigned integers. '-Wmissing-prototypes -Wstrict-prototypes' Warn if there are any missing or inconsistent prototypes. Without prototypes it is harder to detect problems with incorrect arguments. '-Wconversion' The main use of this option is to warn about conversions from signed to unsigned integers. For example, 'unsigned int x = -1'. If you need to perform such a conversion you can use an explicit cast. '-Wshadow' This warns whenever a local variable shadows another local variable. If two variables have the same name then it is a potential source of confusion. '-Wpointer-arith -Wcast-qual -Wcast-align' These options warn if you try to do pointer arithmetic for types which don't have a size, such as 'void', if you remove a 'const' cast from a pointer, or if you cast a pointer to a type which has a different size, causing an invalid alignment. '-Wwrite-strings' This option gives string constants a 'const' qualifier so that it will be a compile-time error to attempt to overwrite them. '-fshort-enums' This option makes the type of 'enum' as short as possible. Normally this makes an 'enum' different from an 'int'. Consequently any attempts to assign a pointer-to-int to a pointer-to-enum will generate a cast-alignment warning. '-fno-common' This option prevents global variables being simultaneously defined in different object files (you get an error at link time). Such a variable should be defined in one file and referred to in other files with an 'extern' declaration. '-Wnested-externs' This warns if an 'extern' declaration is encountered within a function. '-Dinline=' The 'inline' keyword is not part of ANSI C. Thus if you want to use '-ansi' with a program which uses inline functions you can use this preprocessor definition to remove the 'inline' keywords. '-g' It always makes sense to put debugging symbols in the executable so that you can debug it using 'gdb'. The only effect of debugging symbols is to increase the size of the file, and you can use the 'strip' command to remove them later if necessary.  File: gsl-ref.info, Node: Debugging References, Prev: GCC warning options for numerical programs, Up: Debugging Numerical Programs A.5 References and Further Reading ================================== The following books are essential reading for anyone writing and debugging numerical programs with GCC and GDB. R.M. Stallman, 'Using and Porting GNU CC', Free Software Foundation, ISBN 1882114388 R.M. Stallman, R.H. Pesch, 'Debugging with GDB: The GNU Source-Level Debugger', Free Software Foundation, ISBN 1882114779 For a tutorial introduction to the GNU C Compiler and related programs, see B.J. Gough, 'An Introduction to GCC', Network Theory Ltd, ISBN 0954161793  File: gsl-ref.info, Node: Contributors to GSL, Next: Autoconf Macros, Prev: Debugging Numerical Programs, Up: Top Appendix B Contributors to GSL ****************************** (See the AUTHORS file in the distribution for up-to-date information.) *Mark Galassi* Conceived GSL (with James Theiler) and wrote the design document. Wrote the simulated annealing package and the relevant chapter in the manual. *James Theiler* Conceived GSL (with Mark Galassi). Wrote the random number generators and the relevant chapter in this manual. *Jim Davies* Wrote the statistical routines and the relevant chapter in this manual. *Brian Gough* FFTs, numerical integration, random number generators and distributions, root finding, minimization and fitting, polynomial solvers, complex numbers, physical constants, permutations, vector and matrix functions, histograms, statistics, ieee-utils, revised CBLAS Level 2 & 3, matrix decompositions, eigensystems, cumulative distribution functions, testing, documentation and releases. *Reid Priedhorsky* Wrote and documented the initial version of the root finding routines while at Los Alamos National Laboratory, Mathematical Modeling and Analysis Group. *Gerard Jungman* Special Functions, Series acceleration, ODEs, BLAS, Linear Algebra, Eigensystems, Hankel Transforms. *Mike Booth* Wrote the Monte Carlo library. *Jorma Olavi Ta"htinen* Wrote the initial complex arithmetic functions. *Thomas Walter* Wrote the initial heapsort routines and Cholesky decomposition. *Fabrice Rossi* Multidimensional minimization. *Carlo Perassi* Implementation of the random number generators in Knuth's 'Seminumerical Algorithms', 3rd Ed. *Szymon Jaroszewicz* Wrote the routines for generating combinations. *Nicolas Darnis* Wrote the cyclic functions and the initial functions for canonical permutations. *Jason H. Stover* Wrote the major cumulative distribution functions. *Ivo Alxneit* Wrote the routines for wavelet transforms. *Tuomo Keskitalo* Improved the implementation of the ODE solvers and wrote the ode-initval2 routines. *Lowell Johnson* Implementation of the Mathieu functions. *Patrick Alken* Implementation of nonsymmetric and generalized eigensystems, B-splines, and robust linear regression. *Rhys Ulerich* Wrote the multiset routines. *Pavel Holoborodko* Wrote the fixed order Gauss-Legendre quadrature routines. *Pedro Gonnet* Wrote the CQUAD integration routines. Thanks to Nigel Lowry for help in proofreading the manual. The non-symmetric eigensystems routines contain code based on the LAPACK linear algebra library. LAPACK is distributed under the following license: Copyright (c) 1992-2006 The University of Tennessee. All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. * Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer listed in this license in the documentation and/or other materials provided with the distribution. * Neither the name of the copyright holders nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.  File: gsl-ref.info, Node: Autoconf Macros, Next: GSL CBLAS Library, Prev: Contributors to GSL, Up: Top Appendix C Autoconf Macros ************************** For applications using 'autoconf' the standard macro 'AC_CHECK_LIB' can be used to link with GSL automatically from a 'configure' script. The library itself depends on the presence of a CBLAS and math library as well, so these must also be located before linking with the main 'libgsl' file. The following commands should be placed in the 'configure.ac' file to perform these tests, AC_CHECK_LIB([m],[cos]) AC_CHECK_LIB([gslcblas],[cblas_dgemm]) AC_CHECK_LIB([gsl],[gsl_blas_dgemm]) It is important to check for 'libm' and 'libgslcblas' before 'libgsl', otherwise the tests will fail. Assuming the libraries are found the output during the configure stage looks like this, checking for cos in -lm... yes checking for cblas_dgemm in -lgslcblas... yes checking for gsl_blas_dgemm in -lgsl... yes If the library is found then the tests will define the macros 'HAVE_LIBGSL', 'HAVE_LIBGSLCBLAS', 'HAVE_LIBM' and add the options '-lgsl -lgslcblas -lm' to the variable 'LIBS'. The tests above will find any version of the library. They are suitable for general use, where the versions of the functions are not important. An alternative macro is available in the file 'gsl.m4' to test for a specific version of the library. To use this macro simply add the following line to your 'configure.in' file instead of the tests above: AX_PATH_GSL(GSL_VERSION, [action-if-found], [action-if-not-found]) The argument 'GSL_VERSION' should be the two or three digit MAJOR.MINOR or MAJOR.MINOR.MICRO version number of the release you require. A suitable choice for 'action-if-not-found' is, AC_MSG_ERROR(could not find required version of GSL) Then you can add the variables 'GSL_LIBS' and 'GSL_CFLAGS' to your Makefile.am files to obtain the correct compiler flags. 'GSL_LIBS' is equal to the output of the 'gsl-config --libs' command and 'GSL_CFLAGS' is equal to 'gsl-config --cflags' command. For example, libfoo_la_LDFLAGS = -lfoo $(GSL_LIBS) -lgslcblas Note that the macro 'AX_PATH_GSL' needs to use the C compiler so it should appear in the 'configure.in' file before the macro 'AC_LANG_CPLUSPLUS' for programs that use C++. To test for 'inline' the following test should be placed in your 'configure.in' file, AC_C_INLINE if test "$ac_cv_c_inline" != no ; then AC_DEFINE(HAVE_INLINE,1) AC_SUBST(HAVE_INLINE) fi and the macro will then be defined in the compilation flags or by including the file 'config.h' before any library headers. The following autoconf test will check for 'extern inline', dnl Check for "extern inline", using a modified version dnl of the test for AC_C_INLINE from acspecific.mt dnl AC_CACHE_CHECK([for extern inline], ac_cv_c_extern_inline, [ac_cv_c_extern_inline=no AC_TRY_COMPILE([extern $ac_cv_c_inline double foo(double x); extern $ac_cv_c_inline double foo(double x) { return x+1.0; }; double foo (double x) { return x + 1.0; };], [ foo(1.0) ], [ac_cv_c_extern_inline="yes"]) ]) if test "$ac_cv_c_extern_inline" != no ; then AC_DEFINE(HAVE_INLINE,1) AC_SUBST(HAVE_INLINE) fi The substitution of portability functions can be made automatically if you use 'autoconf'. For example, to test whether the BSD function 'hypot' is available you can include the following line in the configure file 'configure.in' for your application, AC_CHECK_FUNCS(hypot) and place the following macro definitions in the file 'config.h.in', /* Substitute gsl_hypot for missing system hypot */ #ifndef HAVE_HYPOT #define hypot gsl_hypot #endif The application source files can then use the include command '#include ' to substitute 'gsl_hypot' for each occurrence of 'hypot' when 'hypot' is not available.  File: gsl-ref.info, Node: GSL CBLAS Library, Next: GNU General Public License, Prev: Autoconf Macros, Up: Top Appendix D GSL CBLAS Library **************************** The prototypes for the low-level CBLAS functions are declared in the file 'gsl_cblas.h'. For the definition of the functions consult the documentation available from Netlib (*note BLAS References and Further Reading::). * Menu: * Level 1 CBLAS Functions:: * Level 2 CBLAS Functions:: * Level 3 CBLAS Functions:: * GSL CBLAS Examples::  File: gsl-ref.info, Node: Level 1 CBLAS Functions, Next: Level 2 CBLAS Functions, Up: GSL CBLAS Library D.1 Level 1 =========== -- Function: float cblas_sdsdot (const int N, const float ALPHA, const float * X, const int INCX, const float * Y, const int INCY) -- Function: double cblas_dsdot (const int N, const float * X, const int INCX, const float * Y, const int INCY) -- Function: float cblas_sdot (const int N, const float * X, const int INCX, const float * Y, const int INCY) -- Function: double cblas_ddot (const int N, const double * X, const int INCX, const double * Y, const int INCY) -- Function: void cblas_cdotu_sub (const int N, const void * X, const int INCX, const void * Y, const int INCY, void * DOTU) -- Function: void cblas_cdotc_sub (const int N, const void * X, const int INCX, const void * Y, const int INCY, void * DOTC) -- Function: void cblas_zdotu_sub (const int N, const void * X, const int INCX, const void * Y, const int INCY, void * DOTU) -- Function: void cblas_zdotc_sub (const int N, const void * X, const int INCX, const void * Y, const int INCY, void * DOTC) -- Function: float cblas_snrm2 (const int N, const float * X, const int INCX) -- Function: float cblas_sasum (const int N, const float * X, const int INCX) -- Function: double cblas_dnrm2 (const int N, const double * X, const int INCX) -- Function: double cblas_dasum (const int N, const double * X, const int INCX) -- Function: float cblas_scnrm2 (const int N, const void * X, const int INCX) -- Function: float cblas_scasum (const int N, const void * X, const int INCX) -- Function: double cblas_dznrm2 (const int N, const void * X, const int INCX) -- Function: double cblas_dzasum (const int N, const void * X, const int INCX) -- Function: CBLAS_INDEX cblas_isamax (const int N, const float * X, const int INCX) -- Function: CBLAS_INDEX cblas_idamax (const int N, const double * X, const int INCX) -- Function: CBLAS_INDEX cblas_icamax (const int N, const void * X, const int INCX) -- Function: CBLAS_INDEX cblas_izamax (const int N, const void * X, const int INCX) -- Function: void cblas_sswap (const int N, float * X, const int INCX, float * Y, const int INCY) -- Function: void cblas_scopy (const int N, const float * X, const int INCX, float * Y, const int INCY) -- Function: void cblas_saxpy (const int N, const float ALPHA, const float * X, const int INCX, float * Y, const int INCY) -- Function: void cblas_dswap (const int N, double * X, const int INCX, double * Y, const int INCY) -- Function: void cblas_dcopy (const int N, const double * X, const int INCX, double * Y, const int INCY) -- Function: void cblas_daxpy (const int N, const double ALPHA, const double * X, const int INCX, double * Y, const int INCY) -- Function: void cblas_cswap (const int N, void * X, const int INCX, void * Y, const int INCY) -- Function: void cblas_ccopy (const int N, const void * X, const int INCX, void * Y, const int INCY) -- Function: void cblas_caxpy (const int N, const void * ALPHA, const void * X, const int INCX, void * Y, const int INCY) -- Function: void cblas_zswap (const int N, void * X, const int INCX, void * Y, const int INCY) -- Function: void cblas_zcopy (const int N, const void * X, const int INCX, void * Y, const int INCY) -- Function: void cblas_zaxpy (const int N, const void * ALPHA, const void * X, const int INCX, void * Y, const int INCY) -- Function: void cblas_srotg (float * A, float * B, float * C, float * S) -- Function: void cblas_srotmg (float * D1, float * D2, float * B1, const float B2, float * P) -- Function: void cblas_srot (const int N, float * X, const int INCX, float * Y, const int INCY, const float C, const float S) -- Function: void cblas_srotm (const int N, float * X, const int INCX, float * Y, const int INCY, const float * P) -- Function: void cblas_drotg (double * A, double * B, double * C, double * S) -- Function: void cblas_drotmg (double * D1, double * D2, double * B1, const double B2, double * P) -- Function: void cblas_drot (const int N, double * X, const int INCX, double * Y, const int INCY, const double C, const double S) -- Function: void cblas_drotm (const int N, double * X, const int INCX, double * Y, const int INCY, const double * P) -- Function: void cblas_sscal (const int N, const float ALPHA, float * X, const int INCX) -- Function: void cblas_dscal (const int N, const double ALPHA, double * X, const int INCX) -- Function: void cblas_cscal (const int N, const void * ALPHA, void * X, const int INCX) -- Function: void cblas_zscal (const int N, const void * ALPHA, void * X, const int INCX) -- Function: void cblas_csscal (const int N, const float ALPHA, void * X, const int INCX) -- Function: void cblas_zdscal (const int N, const double ALPHA, void * X, const int INCX)  File: gsl-ref.info, Node: Level 2 CBLAS Functions, Next: Level 3 CBLAS Functions, Prev: Level 1 CBLAS Functions, Up: GSL CBLAS Library D.2 Level 2 =========== -- Function: void cblas_sgemv (const enum CBLAS_ORDER ORDER, const enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const float ALPHA, const float * A, const int LDA, const float * X, const int INCX, const float BETA, float * Y, const int INCY) -- Function: void cblas_sgbmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const int KL, const int KU, const float ALPHA, const float * A, const int LDA, const float * X, const int INCX, const float BETA, float * Y, const int INCY) -- Function: void cblas_strmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const float * A, const int LDA, float * X, const int INCX) -- Function: void cblas_stbmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const int K, const float * A, const int LDA, float * X, const int INCX) -- Function: void cblas_stpmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const float * AP, float * X, const int INCX) -- Function: void cblas_strsv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const float * A, const int LDA, float * X, const int INCX) -- Function: void cblas_stbsv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const int K, const float * A, const int LDA, float * X, const int INCX) -- Function: void cblas_stpsv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const float * AP, float * X, const int INCX) -- Function: void cblas_dgemv (const enum CBLAS_ORDER ORDER, const enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const double ALPHA, const double * A, const int LDA, const double * X, const int INCX, const double BETA, double * Y, const int INCY) -- Function: void cblas_dgbmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const int KL, const int KU, const double ALPHA, const double * A, const int LDA, const double * X, const int INCX, const double BETA, double * Y, const int INCY) -- Function: void cblas_dtrmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const double * A, const int LDA, double * X, const int INCX) -- Function: void cblas_dtbmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const int K, const double * A, const int LDA, double * X, const int INCX) -- Function: void cblas_dtpmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const double * AP, double * X, const int INCX) -- Function: void cblas_dtrsv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const double * A, const int LDA, double * X, const int INCX) -- Function: void cblas_dtbsv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const int K, const double * A, const int LDA, double * X, const int INCX) -- Function: void cblas_dtpsv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const double * AP, double * X, const int INCX) -- Function: void cblas_cgemv (const enum CBLAS_ORDER ORDER, const enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const void * ALPHA, const void * A, const int LDA, const void * X, const int INCX, const void * BETA, void * Y, const int INCY) -- Function: void cblas_cgbmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const int KL, const int KU, const void * ALPHA, const void * A, const int LDA, const void * X, const int INCX, const void * BETA, void * Y, const int INCY) -- Function: void cblas_ctrmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const void * A, const int LDA, void * X, const int INCX) -- Function: void cblas_ctbmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const int K, const void * A, const int LDA, void * X, const int INCX) -- Function: void cblas_ctpmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const void * AP, void * X, const int INCX) -- Function: void cblas_ctrsv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const void * A, const int LDA, void * X, const int INCX) -- Function: void cblas_ctbsv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const int K, const void * A, const int LDA, void * X, const int INCX) -- Function: void cblas_ctpsv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const void * AP, void * X, const int INCX) -- Function: void cblas_zgemv (const enum CBLAS_ORDER ORDER, const enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const void * ALPHA, const void * A, const int LDA, const void * X, const int INCX, const void * BETA, void * Y, const int INCY) -- Function: void cblas_zgbmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const int KL, const int KU, const void * ALPHA, const void * A, const int LDA, const void * X, const int INCX, const void * BETA, void * Y, const int INCY) -- Function: void cblas_ztrmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const void * A, const int LDA, void * X, const int INCX) -- Function: void cblas_ztbmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const int K, const void * A, const int LDA, void * X, const int INCX) -- Function: void cblas_ztpmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const void * AP, void * X, const int INCX) -- Function: void cblas_ztrsv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const void * A, const int LDA, void * X, const int INCX) -- Function: void cblas_ztbsv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const int K, const void * A, const int LDA, void * X, const int INCX) -- Function: void cblas_ztpsv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int N, const void * AP, void * X, const int INCX) -- Function: void cblas_ssymv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const float ALPHA, const float * A, const int LDA, const float * X, const int INCX, const float BETA, float * Y, const int INCY) -- Function: void cblas_ssbmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const int K, const float ALPHA, const float * A, const int LDA, const float * X, const int INCX, const float BETA, float * Y, const int INCY) -- Function: void cblas_sspmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const float ALPHA, const float * AP, const float * X, const int INCX, const float BETA, float * Y, const int INCY) -- Function: void cblas_sger (const enum CBLAS_ORDER ORDER, const int M, const int N, const float ALPHA, const float * X, const int INCX, const float * Y, const int INCY, float * A, const int LDA) -- Function: void cblas_ssyr (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const float ALPHA, const float * X, const int INCX, float * A, const int LDA) -- Function: void cblas_sspr (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const float ALPHA, const float * X, const int INCX, float * AP) -- Function: void cblas_ssyr2 (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const float ALPHA, const float * X, const int INCX, const float * Y, const int INCY, float * A, const int LDA) -- Function: void cblas_sspr2 (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const float ALPHA, const float * X, const int INCX, const float * Y, const int INCY, float * A) -- Function: void cblas_dsymv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const double ALPHA, const double * A, const int LDA, const double * X, const int INCX, const double BETA, double * Y, const int INCY) -- Function: void cblas_dsbmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const int K, const double ALPHA, const double * A, const int LDA, const double * X, const int INCX, const double BETA, double * Y, const int INCY) -- Function: void cblas_dspmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const double ALPHA, const double * AP, const double * X, const int INCX, const double BETA, double * Y, const int INCY) -- Function: void cblas_dger (const enum CBLAS_ORDER ORDER, const int M, const int N, const double ALPHA, const double * X, const int INCX, const double * Y, const int INCY, double * A, const int LDA) -- Function: void cblas_dsyr (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const double ALPHA, const double * X, const int INCX, double * A, const int LDA) -- Function: void cblas_dspr (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const double ALPHA, const double * X, const int INCX, double * AP) -- Function: void cblas_dsyr2 (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const double ALPHA, const double * X, const int INCX, const double * Y, const int INCY, double * A, const int LDA) -- Function: void cblas_dspr2 (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const double ALPHA, const double * X, const int INCX, const double * Y, const int INCY, double * A) -- Function: void cblas_chemv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const void * A, const int LDA, const void * X, const int INCX, const void * BETA, void * Y, const int INCY) -- Function: void cblas_chbmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const int K, const void * ALPHA, const void * A, const int LDA, const void * X, const int INCX, const void * BETA, void * Y, const int INCY) -- Function: void cblas_chpmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const void * AP, const void * X, const int INCX, const void * BETA, void * Y, const int INCY) -- Function: void cblas_cgeru (const enum CBLAS_ORDER ORDER, const int M, const int N, const void * ALPHA, const void * X, const int INCX, const void * Y, const int INCY, void * A, const int LDA) -- Function: void cblas_cgerc (const enum CBLAS_ORDER ORDER, const int M, const int N, const void * ALPHA, const void * X, const int INCX, const void * Y, const int INCY, void * A, const int LDA) -- Function: void cblas_cher (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const float ALPHA, const void * X, const int INCX, void * A, const int LDA) -- Function: void cblas_chpr (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const float ALPHA, const void * X, const int INCX, void * A) -- Function: void cblas_cher2 (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const void * X, const int INCX, const void * Y, const int INCY, void * A, const int LDA) -- Function: void cblas_chpr2 (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const void * X, const int INCX, const void * Y, const int INCY, void * AP) -- Function: void cblas_zhemv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const void * A, const int LDA, const void * X, const int INCX, const void * BETA, void * Y, const int INCY) -- Function: void cblas_zhbmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const int K, const void * ALPHA, const void * A, const int LDA, const void * X, const int INCX, const void * BETA, void * Y, const int INCY) -- Function: void cblas_zhpmv (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const void * AP, const void * X, const int INCX, const void * BETA, void * Y, const int INCY) -- Function: void cblas_zgeru (const enum CBLAS_ORDER ORDER, const int M, const int N, const void * ALPHA, const void * X, const int INCX, const void * Y, const int INCY, void * A, const int LDA) -- Function: void cblas_zgerc (const enum CBLAS_ORDER ORDER, const int M, const int N, const void * ALPHA, const void * X, const int INCX, const void * Y, const int INCY, void * A, const int LDA) -- Function: void cblas_zher (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const double ALPHA, const void * X, const int INCX, void * A, const int LDA) -- Function: void cblas_zhpr (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const double ALPHA, const void * X, const int INCX, void * A) -- Function: void cblas_zher2 (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const void * X, const int INCX, const void * Y, const int INCY, void * A, const int LDA) -- Function: void cblas_zhpr2 (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const void * X, const int INCX, const void * Y, const int INCY, void * AP)  File: gsl-ref.info, Node: Level 3 CBLAS Functions, Next: GSL CBLAS Examples, Prev: Level 2 CBLAS Functions, Up: GSL CBLAS Library D.3 Level 3 =========== -- Function: void cblas_sgemm (const enum CBLAS_ORDER ORDER, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_TRANSPOSE TRANSB, const int M, const int N, const int K, const float ALPHA, const float * A, const int LDA, const float * B, const int LDB, const float BETA, float * C, const int LDC) -- Function: void cblas_ssymm (const enum CBLAS_ORDER ORDER, const enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int M, const int N, const float ALPHA, const float * A, const int LDA, const float * B, const int LDB, const float BETA, float * C, const int LDC) -- Function: void cblas_ssyrk (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const int N, const int K, const float ALPHA, const float * A, const int LDA, const float BETA, float * C, const int LDC) -- Function: void cblas_ssyr2k (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const int N, const int K, const float ALPHA, const float * A, const int LDA, const float * B, const int LDB, const float BETA, float * C, const int LDC) -- Function: void cblas_strmm (const enum CBLAS_ORDER ORDER, const enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int M, const int N, const float ALPHA, const float * A, const int LDA, float * B, const int LDB) -- Function: void cblas_strsm (const enum CBLAS_ORDER ORDER, const enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int M, const int N, const float ALPHA, const float * A, const int LDA, float * B, const int LDB) -- Function: void cblas_dgemm (const enum CBLAS_ORDER ORDER, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_TRANSPOSE TRANSB, const int M, const int N, const int K, const double ALPHA, const double * A, const int LDA, const double * B, const int LDB, const double BETA, double * C, const int LDC) -- Function: void cblas_dsymm (const enum CBLAS_ORDER ORDER, const enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int M, const int N, const double ALPHA, const double * A, const int LDA, const double * B, const int LDB, const double BETA, double * C, const int LDC) -- Function: void cblas_dsyrk (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const int N, const int K, const double ALPHA, const double * A, const int LDA, const double BETA, double * C, const int LDC) -- Function: void cblas_dsyr2k (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const int N, const int K, const double ALPHA, const double * A, const int LDA, const double * B, const int LDB, const double BETA, double * C, const int LDC) -- Function: void cblas_dtrmm (const enum CBLAS_ORDER ORDER, const enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int M, const int N, const double ALPHA, const double * A, const int LDA, double * B, const int LDB) -- Function: void cblas_dtrsm (const enum CBLAS_ORDER ORDER, const enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int M, const int N, const double ALPHA, const double * A, const int LDA, double * B, const int LDB) -- Function: void cblas_cgemm (const enum CBLAS_ORDER ORDER, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_TRANSPOSE TRANSB, const int M, const int N, const int K, const void * ALPHA, const void * A, const int LDA, const void * B, const int LDB, const void * BETA, void * C, const int LDC) -- Function: void cblas_csymm (const enum CBLAS_ORDER ORDER, const enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int M, const int N, const void * ALPHA, const void * A, const int LDA, const void * B, const int LDB, const void * BETA, void * C, const int LDC) -- Function: void cblas_csyrk (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const int N, const int K, const void * ALPHA, const void * A, const int LDA, const void * BETA, void * C, const int LDC) -- Function: void cblas_csyr2k (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const int N, const int K, const void * ALPHA, const void * A, const int LDA, const void * B, const int LDB, const void * BETA, void * C, const int LDC) -- Function: void cblas_ctrmm (const enum CBLAS_ORDER ORDER, const enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int M, const int N, const void * ALPHA, const void * A, const int LDA, void * B, const int LDB) -- Function: void cblas_ctrsm (const enum CBLAS_ORDER ORDER, const enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int M, const int N, const void * ALPHA, const void * A, const int LDA, void * B, const int LDB) -- Function: void cblas_zgemm (const enum CBLAS_ORDER ORDER, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_TRANSPOSE TRANSB, const int M, const int N, const int K, const void * ALPHA, const void * A, const int LDA, const void * B, const int LDB, const void * BETA, void * C, const int LDC) -- Function: void cblas_zsymm (const enum CBLAS_ORDER ORDER, const enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int M, const int N, const void * ALPHA, const void * A, const int LDA, const void * B, const int LDB, const void * BETA, void * C, const int LDC) -- Function: void cblas_zsyrk (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const int N, const int K, const void * ALPHA, const void * A, const int LDA, const void * BETA, void * C, const int LDC) -- Function: void cblas_zsyr2k (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const int N, const int K, const void * ALPHA, const void * A, const int LDA, const void * B, const int LDB, const void * BETA, void * C, const int LDC) -- Function: void cblas_ztrmm (const enum CBLAS_ORDER ORDER, const enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int M, const int N, const void * ALPHA, const void * A, const int LDA, void * B, const int LDB) -- Function: void cblas_ztrsm (const enum CBLAS_ORDER ORDER, const enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int M, const int N, const void * ALPHA, const void * A, const int LDA, void * B, const int LDB) -- Function: void cblas_chemm (const enum CBLAS_ORDER ORDER, const enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int M, const int N, const void * ALPHA, const void * A, const int LDA, const void * B, const int LDB, const void * BETA, void * C, const int LDC) -- Function: void cblas_cherk (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const int N, const int K, const float ALPHA, const void * A, const int LDA, const float BETA, void * C, const int LDC) -- Function: void cblas_cher2k (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const int N, const int K, const void * ALPHA, const void * A, const int LDA, const void * B, const int LDB, const float BETA, void * C, const int LDC) -- Function: void cblas_zhemm (const enum CBLAS_ORDER ORDER, const enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int M, const int N, const void * ALPHA, const void * A, const int LDA, const void * B, const int LDB, const void * BETA, void * C, const int LDC) -- Function: void cblas_zherk (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const int N, const int K, const double ALPHA, const void * A, const int LDA, const double BETA, void * C, const int LDC) -- Function: void cblas_zher2k (const enum CBLAS_ORDER ORDER, const enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const int N, const int K, const void * ALPHA, const void * A, const int LDA, const void * B, const int LDB, const double BETA, void * C, const int LDC) -- Function: void cblas_xerbla (int P, const char * ROUT, const char * FORM, ...)  File: gsl-ref.info, Node: GSL CBLAS Examples, Prev: Level 3 CBLAS Functions, Up: GSL CBLAS Library D.4 Examples ============ The following program computes the product of two matrices using the Level-3 BLAS function SGEMM, [ 0.11 0.12 0.13 ] [ 1011 1012 ] [ 367.76 368.12 ] [ 0.21 0.22 0.23 ] [ 1021 1022 ] = [ 674.06 674.72 ] [ 1031 1032 ] The matrices are stored in row major order but could be stored in column major order if the first argument of the call to 'cblas_sgemm' was changed to 'CblasColMajor'. #include #include int main (void) { int lda = 3; float A[] = { 0.11, 0.12, 0.13, 0.21, 0.22, 0.23 }; int ldb = 2; float B[] = { 1011, 1012, 1021, 1022, 1031, 1032 }; int ldc = 2; float C[] = { 0.00, 0.00, 0.00, 0.00 }; /* Compute C = A B */ cblas_sgemm (CblasRowMajor, CblasNoTrans, CblasNoTrans, 2, 2, 3, 1.0, A, lda, B, ldb, 0.0, C, ldc); printf ("[ %g, %g\n", C[0], C[1]); printf (" %g, %g ]\n", C[2], C[3]); return 0; } To compile the program use the following command line, $ gcc -Wall demo.c -lgslcblas There is no need to link with the main library '-lgsl' in this case as the CBLAS library is an independent unit. Here is the output from the program, $ ./a.out [ 367.76, 368.12 674.06, 674.72 ]  File: gsl-ref.info, Node: GNU General Public License, Next: GNU Free Documentation License, Prev: GSL CBLAS Library, Up: Top GNU General Public License ************************** Version 3, 29 June 2007 Copyright (C) 2007 Free Software Foundation, Inc. Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. Preamble ======== The GNU General Public License is a free, copyleft license for software and other kinds of works. The licenses for most software and other practical works are designed to take away your freedom to share and change the works. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change all versions of a program-to make sure it remains free software for all its users. 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The precise terms and conditions for copying, distribution and modification follow. TERMS AND CONDITIONS ==================== 0. Definitions. "This License" refers to version 3 of the GNU General Public License. "Copyright" also means copyright-like laws that apply to other kinds of works, such as semiconductor masks. "The Program" refers to any copyrightable work licensed under this License. Each licensee is addressed as "you". "Licensees" and "recipients" may be individuals or organizations. To "modify" a work means to copy from or adapt all or part of the work in a fashion requiring copyright permission, other than the making of an exact copy. The resulting work is called a "modified version" of the earlier work or a work "based on" the earlier work. A "covered work" means either the unmodified Program or a work based on the Program. 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If the interface presents a list of user commands or options, such as a menu, a prominent item in the list meets this criterion. 1. Source Code. The "source code" for a work means the preferred form of the work for making modifications to it. "Object code" means any non-source form of a work. A "Standard Interface" means an interface that either is an official standard defined by a recognized standards body, or, in the case of interfaces specified for a particular programming language, one that is widely used among developers working in that language. The "System Libraries" of an executable work include anything, other than the work as a whole, that (a) is included in the normal form of packaging a Major Component, but which is not part of that Major Component, and (b) serves only to enable use of the work with that Major Component, or to implement a Standard Interface for which an implementation is available to the public in source code form. A "Major Component", in this context, means a major essential component (kernel, window system, and so on) of the specific operating system (if any) on which the executable work runs, or a compiler used to produce the work, or an object code interpreter used to run it. The "Corresponding Source" for a work in object code form means all the source code needed to generate, install, and (for an executable work) run the object code and to modify the work, including scripts to control those activities. However, it does not include the work's System Libraries, or general-purpose tools or generally available free programs which are used unmodified in performing those activities but which are not part of the work. For example, Corresponding Source includes interface definition files associated with source files for the work, and the source code for shared libraries and dynamically linked subprograms that the work is specifically designed to require, such as by intimate data communication or control flow between those subprograms and other parts of the work. The Corresponding Source need not include anything that users can regenerate automatically from other parts of the Corresponding Source. The Corresponding Source for a work in source code form is that same work. 2. Basic Permissions. All rights granted under this License are granted for the term of copyright on the Program, and are irrevocable provided the stated conditions are met. This License explicitly affirms your unlimited permission to run the unmodified Program. The output from running a covered work is covered by this License only if the output, given its content, constitutes a covered work. This License acknowledges your rights of fair use or other equivalent, as provided by copyright law. You may make, run and propagate covered works that you do not convey, without conditions so long as your license otherwise remains in force. You may convey covered works to others for the sole purpose of having them make modifications exclusively for you, or provide you with facilities for running those works, provided that you comply with the terms of this License in conveying all material for which you do not control copyright. Those thus making or running the covered works for you must do so exclusively on your behalf, under your direction and control, on terms that prohibit them from making any copies of your copyrighted material outside their relationship with you. Conveying under any other circumstances is permitted solely under the conditions stated below. Sublicensing is not allowed; section 10 makes it unnecessary. 3. Protecting Users' Legal Rights From Anti-Circumvention Law. No covered work shall be deemed part of an effective technological measure under any applicable law fulfilling obligations under article 11 of the WIPO copyright treaty adopted on 20 December 1996, or similar laws prohibiting or restricting circumvention of such measures. When you convey a covered work, you waive any legal power to forbid circumvention of technological measures to the extent such circumvention is effected by exercising rights under this License with respect to the covered work, and you disclaim any intention to limit operation or modification of the work as a means of enforcing, against the work's users, your or third parties' legal rights to forbid circumvention of technological measures. 4. Conveying Verbatim Copies. You may convey verbatim copies of the Program's source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice; keep intact all notices stating that this License and any non-permissive terms added in accord with section 7 apply to the code; keep intact all notices of the absence of any warranty; and give all recipients a copy of this License along with the Program. You may charge any price or no price for each copy that you convey, and you may offer support or warranty protection for a fee. 5. Conveying Modified Source Versions. You may convey a work based on the Program, or the modifications to produce it from the Program, in the form of source code under the terms of section 4, provided that you also meet all of these conditions: a. The work must carry prominent notices stating that you modified it, and giving a relevant date. b. The work must carry prominent notices stating that it is released under this License and any conditions added under section 7. This requirement modifies the requirement in section 4 to "keep intact all notices". c. You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy. This License will therefore apply, along with any applicable section 7 additional terms, to the whole of the work, and all its parts, regardless of how they are packaged. This License gives no permission to license the work in any other way, but it does not invalidate such permission if you have separately received it. d. If the work has interactive user interfaces, each must display Appropriate Legal Notices; however, if the Program has interactive interfaces that do not display Appropriate Legal Notices, your work need not make them do so. A compilation of a covered work with other separate and independent works, which are not by their nature extensions of the covered work, and which are not combined with it such as to form a larger program, in or on a volume of a storage or distribution medium, is called an "aggregate" if the compilation and its resulting copyright are not used to limit the access or legal rights of the compilation's users beyond what the individual works permit. Inclusion of a covered work in an aggregate does not cause this License to apply to the other parts of the aggregate. 6. Conveying Non-Source Forms. You may convey a covered work in object code form under the terms of sections 4 and 5, provided that you also convey the machine-readable Corresponding Source under the terms of this License, in one of these ways: a. Convey the object code in, or embodied in, a physical product (including a physical distribution medium), accompanied by the Corresponding Source fixed on a durable physical medium customarily used for software interchange. b. Convey the object code in, or embodied in, a physical product (including a physical distribution medium), accompanied by a written offer, valid for at least three years and valid for as long as you offer spare parts or customer support for that product model, to give anyone who possesses the object code either (1) a copy of the Corresponding Source for all the software in the product that is covered by this License, on a durable physical medium customarily used for software interchange, for a price no more than your reasonable cost of physically performing this conveying of source, or (2) access to copy the Corresponding Source from a network server at no charge. c. Convey individual copies of the object code with a copy of the written offer to provide the Corresponding Source. This alternative is allowed only occasionally and noncommercially, and only if you received the object code with such an offer, in accord with subsection 6b. d. Convey the object code by offering access from a designated place (gratis or for a charge), and offer equivalent access to the Corresponding Source in the same way through the same place at no further charge. You need not require recipients to copy the Corresponding Source along with the object code. If the place to copy the object code is a network server, the Corresponding Source may be on a different server (operated by you or a third party) that supports equivalent copying facilities, provided you maintain clear directions next to the object code saying where to find the Corresponding Source. Regardless of what server hosts the Corresponding Source, you remain obligated to ensure that it is available for as long as needed to satisfy these requirements. e. Convey the object code using peer-to-peer transmission, provided you inform other peers where the object code and Corresponding Source of the work are being offered to the general public at no charge under subsection 6d. A separable portion of the object code, whose source code is excluded from the Corresponding Source as a System Library, need not be included in conveying the object code work. A "User Product" is either (1) a "consumer product", which means any tangible personal property which is normally used for personal, family, or household purposes, or (2) anything designed or sold for incorporation into a dwelling. In determining whether a product is a consumer product, doubtful cases shall be resolved in favor of coverage. For a particular product received by a particular user, "normally used" refers to a typical or common use of that class of product, regardless of the status of the particular user or of the way in which the particular user actually uses, or expects or is expected to use, the product. A product is a consumer product regardless of whether the product has substantial commercial, industrial or non-consumer uses, unless such uses represent the only significant mode of use of the product. "Installation Information" for a User Product means any methods, procedures, authorization keys, or other information required to install and execute modified versions of a covered work in that User Product from a modified version of its Corresponding Source. The information must suffice to ensure that the continued functioning of the modified object code is in no case prevented or interfered with solely because modification has been made. If you convey an object code work under this section in, or with, or specifically for use in, a User Product, and the conveying occurs as part of a transaction in which the right of possession and use of the User Product is transferred to the recipient in perpetuity or for a fixed term (regardless of how the transaction is characterized), the Corresponding Source conveyed under this section must be accompanied by the Installation Information. But this requirement does not apply if neither you nor any third party retains the ability to install modified object code on the User Product (for example, the work has been installed in ROM). The requirement to provide Installation Information does not include a requirement to continue to provide support service, warranty, or updates for a work that has been modified or installed by the recipient, or for the User Product in which it has been modified or installed. Access to a network may be denied when the modification itself materially and adversely affects the operation of the network or violates the rules and protocols for communication across the network. Corresponding Source conveyed, and Installation Information provided, in accord with this section must be in a format that is publicly documented (and with an implementation available to the public in source code form), and must require no special password or key for unpacking, reading or copying. 7. Additional Terms. "Additional permissions" are terms that supplement the terms of this License by making exceptions from one or more of its conditions. Additional permissions that are applicable to the entire Program shall be treated as though they were included in this License, to the extent that they are valid under applicable law. If additional permissions apply only to part of the Program, that part may be used separately under those permissions, but the entire Program remains governed by this License without regard to the additional permissions. When you convey a copy of a covered work, you may at your option remove any additional permissions from that copy, or from any part of it. (Additional permissions may be written to require their own removal in certain cases when you modify the work.) You may place additional permissions on material, added by you to a covered work, for which you have or can give appropriate copyright permission. Notwithstanding any other provision of this License, for material you add to a covered work, you may (if authorized by the copyright holders of that material) supplement the terms of this License with terms: a. Disclaiming warranty or limiting liability differently from the terms of sections 15 and 16 of this License; or b. Requiring preservation of specified reasonable legal notices or author attributions in that material or in the Appropriate Legal Notices displayed by works containing it; or c. Prohibiting misrepresentation of the origin of that material, or requiring that modified versions of such material be marked in reasonable ways as different from the original version; or d. Limiting the use for publicity purposes of names of licensors or authors of the material; or e. Declining to grant rights under trademark law for use of some trade names, trademarks, or service marks; or f. Requiring indemnification of licensors and authors of that material by anyone who conveys the material (or modified versions of it) with contractual assumptions of liability to the recipient, for any liability that these contractual assumptions directly impose on those licensors and authors. All other non-permissive additional terms are considered "further restrictions" within the meaning of section 10. If the Program as you received it, or any part of it, contains a notice stating that it is governed by this License along with a term that is a further restriction, you may remove that term. If a license document contains a further restriction but permits relicensing or conveying under this License, you may add to a covered work material governed by the terms of that license document, provided that the further restriction does not survive such relicensing or conveying. If you add terms to a covered work in accord with this section, you must place, in the relevant source files, a statement of the additional terms that apply to those files, or a notice indicating where to find the applicable terms. Additional terms, permissive or non-permissive, may be stated in the form of a separately written license, or stated as exceptions; the above requirements apply either way. 8. Termination. You may not propagate or modify a covered work except as expressly provided under this License. Any attempt otherwise to propagate or modify it is void, and will automatically terminate your rights under this License (including any patent licenses granted under the third paragraph of section 11). However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation. Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice. Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, you do not qualify to receive new licenses for the same material under section 10. 9. Acceptance Not Required for Having Copies. You are not required to accept this License in order to receive or run a copy of the Program. Ancillary propagation of a covered work occurring solely as a consequence of using peer-to-peer transmission to receive a copy likewise does not require acceptance. However, nothing other than this License grants you permission to propagate or modify any covered work. These actions infringe copyright if you do not accept this License. Therefore, by modifying or propagating a covered work, you indicate your acceptance of this License to do so. 10. Automatic Licensing of Downstream Recipients. Each time you convey a covered work, the recipient automatically receives a license from the original licensors, to run, modify and propagate that work, subject to this License. You are not responsible for enforcing compliance by third parties with this License. An "entity transaction" is a transaction transferring control of an organization, or substantially all assets of one, or subdividing an organization, or merging organizations. If propagation of a covered work results from an entity transaction, each party to that transaction who receives a copy of the work also receives whatever licenses to the work the party's predecessor in interest had or could give under the previous paragraph, plus a right to possession of the Corresponding Source of the work from the predecessor in interest, if the predecessor has it or can get it with reasonable efforts. You may not impose any further restrictions on the exercise of the rights granted or affirmed under this License. For example, you may not impose a license fee, royalty, or other charge for exercise of rights granted under this License, and you may not initiate litigation (including a cross-claim or counterclaim in a lawsuit) alleging that any patent claim is infringed by making, using, selling, offering for sale, or importing the Program or any portion of it. 11. Patents. A "contributor" is a copyright holder who authorizes use under this License of the Program or a work on which the Program is based. The work thus licensed is called the contributor's "contributor version". A contributor's "essential patent claims" are all patent claims owned or controlled by the contributor, whether already acquired or hereafter acquired, that would be infringed by some manner, permitted by this License, of making, using, or selling its contributor version, but do not include claims that would be infringed only as a consequence of further modification of the contributor version. For purposes of this definition, "control" includes the right to grant patent sublicenses in a manner consistent with the requirements of this License. Each contributor grants you a non-exclusive, worldwide, royalty-free patent license under the contributor's essential patent claims, to make, use, sell, offer for sale, import and otherwise run, modify and propagate the contents of its contributor version. In the following three paragraphs, a "patent license" is any express agreement or commitment, however denominated, not to enforce a patent (such as an express permission to practice a patent or covenant not to sue for patent infringement). To "grant" such a patent license to a party means to make such an agreement or commitment not to enforce a patent against the party. If you convey a covered work, knowingly relying on a patent license, and the Corresponding Source of the work is not available for anyone to copy, free of charge and under the terms of this License, through a publicly available network server or other readily accessible means, then you must either (1) cause the Corresponding Source to be so available, or (2) arrange to deprive yourself of the benefit of the patent license for this particular work, or (3) arrange, in a manner consistent with the requirements of this License, to extend the patent license to downstream recipients. "Knowingly relying" means you have actual knowledge that, but for the patent license, your conveying the covered work in a country, or your recipient's use of the covered work in a country, would infringe one or more identifiable patents in that country that you have reason to believe are valid. If, pursuant to or in connection with a single transaction or arrangement, you convey, or propagate by procuring conveyance of, a covered work, and grant a patent license to some of the parties receiving the covered work authorizing them to use, propagate, modify or convey a specific copy of the covered work, then the patent license you grant is automatically extended to all recipients of the covered work and works based on it. A patent license is "discriminatory" if it does not include within the scope of its coverage, prohibits the exercise of, or is conditioned on the non-exercise of one or more of the rights that are specifically granted under this License. You may not convey a covered work if you are a party to an arrangement with a third party that is in the business of distributing software, under which you make payment to the third party based on the extent of your activity of conveying the work, and under which the third party grants, to any of the parties who would receive the covered work from you, a discriminatory patent license (a) in connection with copies of the covered work conveyed by you (or copies made from those copies), or (b) primarily for and in connection with specific products or compilations that contain the covered work, unless you entered into that arrangement, or that patent license was granted, prior to 28 March 2007. Nothing in this License shall be construed as excluding or limiting any implied license or other defenses to infringement that may otherwise be available to you under applicable patent law. 12. No Surrender of Others' Freedom. If conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot convey a covered work so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not convey it at all. For example, if you agree to terms that obligate you to collect a royalty for further conveying from those to whom you convey the Program, the only way you could satisfy both those terms and this License would be to refrain entirely from conveying the Program. 13. Use with the GNU Affero General Public License. Notwithstanding any other provision of this License, you have permission to link or combine any covered work with a work licensed under version 3 of the GNU Affero General Public License into a single combined work, and to convey the resulting work. The terms of this License will continue to apply to the part which is the covered work, but the special requirements of the GNU Affero General Public License, section 13, concerning interaction through a network will apply to the combination as such. 14. Revised Versions of this License. The Free Software Foundation may publish revised and/or new versions of the GNU General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. Each version is given a distinguishing version number. If the Program specifies that a certain numbered version of the GNU General Public License "or any later version" applies to it, you have the option of following the terms and conditions either of that numbered version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of the GNU General Public License, you may choose any version ever published by the Free Software Foundation. If the Program specifies that a proxy can decide which future versions of the GNU General Public License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Program. Later license versions may give you additional or different permissions. However, no additional obligations are imposed on any author or copyright holder as a result of your choosing to follow a later version. 15. Disclaimer of Warranty. THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 16. Limitation of Liability. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES. 17. Interpretation of Sections 15 and 16. If the disclaimer of warranty and limitation of liability provided above cannot be given local legal effect according to their terms, reviewing courts shall apply local law that most closely approximates an absolute waiver of all civil liability in connection with the Program, unless a warranty or assumption of liability accompanies a copy of the Program in return for a fee. END OF TERMS AND CONDITIONS =========================== How to Apply These Terms to Your New Programs ============================================= If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms. To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively state the exclusion of warranty; and each file should have at least the "copyright" line and a pointer to where the full notice is found. ONE LINE TO GIVE THE PROGRAM'S NAME AND A BRIEF IDEA OF WHAT IT DOES. Copyright (C) YEAR NAME OF AUTHOR This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . Also add information on how to contact you by electronic and paper mail. If the program does terminal interaction, make it output a short notice like this when it starts in an interactive mode: PROGRAM Copyright (C) YEAR NAME OF AUTHOR This program comes with ABSOLUTELY NO WARRANTY; for details type 'show w'. This is free software, and you are welcome to redistribute it under certain conditions; type 'show c' for details. The hypothetical commands 'show w' and 'show c' should show the appropriate parts of the General Public License. Of course, your program's commands might be different; for a GUI interface, you would use an "about box". You should also get your employer (if you work as a programmer) or school, if any, to sign a "copyright disclaimer" for the program, if necessary. For more information on this, and how to apply and follow the GNU GPL, see . The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. But first, please read .  File: gsl-ref.info, Node: GNU Free Documentation License, Next: Function Index, Prev: GNU General Public License, Up: Top GNU Free Documentation License ****************************** Version 1.3, 3 November 2008 Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc. Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. 0. PREAMBLE The purpose of this License is to make a manual, textbook, or other functional and useful document "free" in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others. This License is a kind of "copyleft", which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software. We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference. 1. APPLICABILITY AND DEFINITIONS This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions stated herein. The "Document", below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as "you". You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law. A "Modified Version" of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language. A "Secondary Section" is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document's overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (Thus, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them. The "Invariant Sections" are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License. If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant. The Document may contain zero Invariant Sections. If the Document does not identify any Invariant Sections then there are none. The "Cover Texts" are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License. A Front-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words. A "Transparent" copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, that is suitable for revising the document straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup, or absence of markup, has been arranged to thwart or discourage subsequent modification by readers is not Transparent. An image format is not Transparent if used for any substantial amount of text. A copy that is not "Transparent" is called "Opaque". Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo input format, LaTeX input format, SGML or XML using a publicly available DTD, and standard-conforming simple HTML, PostScript or PDF designed for human modification. Examples of transparent image formats include PNG, XCF and JPG. Opaque formats include proprietary formats that can be read and edited only by proprietary word processors, SGML or XML for which the DTD and/or processing tools are not generally available, and the machine-generated HTML, PostScript or PDF produced by some word processors for output purposes only. The "Title Page" means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page. For works in formats which do not have any title page as such, "Title Page" means the text near the most prominent appearance of the work's title, preceding the beginning of the body of the text. The "publisher" means any person or entity that distributes copies of the Document to the public. A section "Entitled XYZ" means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language. (Here XYZ stands for a specific section name mentioned below, such as "Acknowledgements", "Dedications", "Endorsements", or "History".) To "Preserve the Title" of such a section when you modify the Document means that it remains a section "Entitled XYZ" according to this definition. The Document may include Warranty Disclaimers next to the notice which states that this License applies to the Document. These Warranty Disclaimers are considered to be included by reference in this License, but only as regards disclaiming warranties: any other implication that these Warranty Disclaimers may have is void and has no effect on the meaning of this License. 2. VERBATIM COPYING You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section 3. 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ADDENDUM: How to use this License for your documents ==================================================== To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page: Copyright (C) YEAR YOUR NAME. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled ``GNU Free Documentation License''. If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the "with...Texts." line with this: with the Invariant Sections being LIST THEIR TITLES, with the Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST. If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation. If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.  File: gsl-ref.info, Node: Function Index, Next: Variable Index, Prev: GNU Free Documentation License, Up: Top Function Index ************** [index] * Menu: * cblas_caxpy: Level 1 CBLAS Functions. (line 90) * cblas_ccopy: Level 1 CBLAS Functions. (line 87) * cblas_cdotc_sub: Level 1 CBLAS Functions. (line 21) * cblas_cdotu_sub: Level 1 CBLAS Functions. (line 18) * cblas_cgbmv: Level 2 CBLAS Functions. (line 93) * cblas_cgemm: Level 3 CBLAS Functions. (line 76) * cblas_cgemv: Level 2 CBLAS Functions. (line 88) * cblas_cgerc: Level 2 CBLAS Functions. (line 264) * cblas_cgeru: Level 2 CBLAS Functions. (line 260) * cblas_chbmv: Level 2 CBLAS Functions. (line 250) * cblas_chemm: Level 3 CBLAS Functions. (line 146) * cblas_chemv: Level 2 CBLAS Functions. (line 245) * cblas_cher: Level 2 CBLAS Functions. (line 268) * cblas_cher2: Level 2 CBLAS Functions. (line 276) * cblas_cher2k: Level 3 CBLAS Functions. (line 157) * cblas_cherk: Level 3 CBLAS Functions. (line 152) * cblas_chpmv: Level 2 CBLAS Functions. (line 255) * cblas_chpr: Level 2 CBLAS Functions. (line 272) * cblas_chpr2: Level 2 CBLAS Functions. (line 281) * cblas_cscal: Level 1 CBLAS Functions. (line 132) * cblas_csscal: Level 1 CBLAS Functions. (line 138) * cblas_cswap: Level 1 CBLAS Functions. (line 84) * cblas_csymm: Level 3 CBLAS Functions. (line 82) * cblas_csyr2k: Level 3 CBLAS Functions. (line 93) * cblas_csyrk: Level 3 CBLAS Functions. (line 88) * cblas_ctbmv: Level 2 CBLAS Functions. (line 104) * cblas_ctbsv: Level 2 CBLAS Functions. (line 119) * cblas_ctpmv: Level 2 CBLAS Functions. (line 109) * cblas_ctpsv: Level 2 CBLAS Functions. (line 124) * cblas_ctrmm: Level 3 CBLAS Functions. (line 99) * cblas_ctrmv: Level 2 CBLAS Functions. (line 99) * cblas_ctrsm: Level 3 CBLAS Functions. (line 105) * cblas_ctrsv: Level 2 CBLAS Functions. (line 114) * cblas_dasum: Level 1 CBLAS Functions. (line 39) * cblas_daxpy: Level 1 CBLAS Functions. (line 81) * cblas_dcopy: Level 1 CBLAS Functions. (line 78) * cblas_ddot: Level 1 CBLAS Functions. (line 15) * cblas_dgbmv: Level 2 CBLAS Functions. (line 52) * cblas_dgemm: Level 3 CBLAS Functions. (line 41) * cblas_dgemv: Level 2 CBLAS Functions. (line 47) * cblas_dger: Level 2 CBLAS Functions. (line 222) * cblas_dnrm2: Level 1 CBLAS Functions. (line 36) * cblas_drot: Level 1 CBLAS Functions. (line 120) * cblas_drotg: Level 1 CBLAS Functions. (line 114) * cblas_drotm: Level 1 CBLAS Functions. (line 123) * cblas_drotmg: Level 1 CBLAS Functions. (line 117) * cblas_dsbmv: Level 2 CBLAS Functions. (line 212) * cblas_dscal: Level 1 CBLAS Functions. (line 129) * cblas_dsdot: Level 1 CBLAS Functions. (line 9) * cblas_dspmv: Level 2 CBLAS Functions. (line 217) * cblas_dspr: Level 2 CBLAS Functions. (line 231) * cblas_dspr2: Level 2 CBLAS Functions. (line 240) * cblas_dswap: Level 1 CBLAS Functions. (line 75) * cblas_dsymm: Level 3 CBLAS Functions. (line 47) * cblas_dsymv: Level 2 CBLAS Functions. (line 207) * cblas_dsyr: Level 2 CBLAS Functions. (line 227) * cblas_dsyr2: Level 2 CBLAS Functions. (line 235) * cblas_dsyr2k: Level 3 CBLAS Functions. (line 58) * cblas_dsyrk: Level 3 CBLAS Functions. (line 53) * cblas_dtbmv: Level 2 CBLAS Functions. (line 63) * cblas_dtbsv: Level 2 CBLAS Functions. (line 78) * cblas_dtpmv: Level 2 CBLAS Functions. (line 68) * cblas_dtpsv: Level 2 CBLAS Functions. (line 83) * cblas_dtrmm: Level 3 CBLAS Functions. (line 64) * cblas_dtrmv: Level 2 CBLAS Functions. (line 58) * cblas_dtrsm: Level 3 CBLAS Functions. (line 70) * cblas_dtrsv: Level 2 CBLAS Functions. (line 73) * cblas_dzasum: Level 1 CBLAS Functions. (line 51) * cblas_dznrm2: Level 1 CBLAS Functions. (line 48) * cblas_icamax: Level 1 CBLAS Functions. (line 60) * cblas_idamax: Level 1 CBLAS Functions. (line 57) * cblas_isamax: Level 1 CBLAS Functions. (line 54) * cblas_izamax: Level 1 CBLAS Functions. (line 63) * cblas_sasum: Level 1 CBLAS Functions. (line 33) * cblas_saxpy: Level 1 CBLAS Functions. (line 72) * cblas_scasum: Level 1 CBLAS Functions. (line 45) * cblas_scnrm2: Level 1 CBLAS Functions. (line 42) * cblas_scopy: Level 1 CBLAS Functions. (line 69) * cblas_sdot: Level 1 CBLAS Functions. (line 12) * cblas_sdsdot: Level 1 CBLAS Functions. (line 6) * cblas_sgbmv: Level 2 CBLAS Functions. (line 11) * cblas_sgemm: Level 3 CBLAS Functions. (line 6) * cblas_sgemv: Level 2 CBLAS Functions. (line 6) * cblas_sger: Level 2 CBLAS Functions. (line 185) * cblas_snrm2: Level 1 CBLAS Functions. (line 30) * cblas_srot: Level 1 CBLAS Functions. (line 108) * cblas_srotg: Level 1 CBLAS Functions. (line 102) * cblas_srotm: Level 1 CBLAS Functions. (line 111) * cblas_srotmg: Level 1 CBLAS Functions. (line 105) * cblas_ssbmv: Level 2 CBLAS Functions. (line 175) * cblas_sscal: Level 1 CBLAS Functions. (line 126) * cblas_sspmv: Level 2 CBLAS Functions. (line 180) * cblas_sspr: Level 2 CBLAS Functions. (line 194) * cblas_sspr2: Level 2 CBLAS Functions. (line 203) * cblas_sswap: Level 1 CBLAS Functions. (line 66) * cblas_ssymm: Level 3 CBLAS Functions. (line 12) * cblas_ssymv: Level 2 CBLAS Functions. (line 170) * cblas_ssyr: Level 2 CBLAS Functions. (line 190) * cblas_ssyr2: Level 2 CBLAS Functions. (line 198) * cblas_ssyr2k: Level 3 CBLAS Functions. (line 23) * cblas_ssyrk: Level 3 CBLAS Functions. (line 18) * cblas_stbmv: Level 2 CBLAS Functions. (line 22) * cblas_stbsv: Level 2 CBLAS Functions. (line 37) * cblas_stpmv: Level 2 CBLAS Functions. (line 27) * cblas_stpsv: Level 2 CBLAS Functions. (line 42) * cblas_strmm: Level 3 CBLAS Functions. (line 29) * cblas_strmv: Level 2 CBLAS Functions. (line 17) * cblas_strsm: Level 3 CBLAS Functions. (line 35) * cblas_strsv: Level 2 CBLAS Functions. (line 32) * cblas_xerbla: Level 3 CBLAS Functions. (line 180) * cblas_zaxpy: Level 1 CBLAS Functions. (line 99) * cblas_zcopy: Level 1 CBLAS Functions. (line 96) * cblas_zdotc_sub: Level 1 CBLAS Functions. (line 27) * cblas_zdotu_sub: Level 1 CBLAS Functions. (line 24) * cblas_zdscal: Level 1 CBLAS Functions. (line 141) * cblas_zgbmv: Level 2 CBLAS Functions. (line 134) * cblas_zgemm: Level 3 CBLAS Functions. (line 111) * cblas_zgemv: Level 2 CBLAS Functions. (line 129) * cblas_zgerc: Level 2 CBLAS Functions. (line 304) * cblas_zgeru: Level 2 CBLAS Functions. (line 300) * cblas_zhbmv: Level 2 CBLAS Functions. (line 290) * cblas_zhemm: Level 3 CBLAS Functions. (line 163) * cblas_zhemv: Level 2 CBLAS Functions. (line 285) * cblas_zher: Level 2 CBLAS Functions. (line 308) * cblas_zher2: Level 2 CBLAS Functions. (line 316) * cblas_zher2k: Level 3 CBLAS Functions. (line 174) * cblas_zherk: Level 3 CBLAS Functions. (line 169) * cblas_zhpmv: Level 2 CBLAS Functions. (line 295) * cblas_zhpr: Level 2 CBLAS Functions. (line 312) * cblas_zhpr2: Level 2 CBLAS Functions. (line 321) * cblas_zscal: Level 1 CBLAS Functions. (line 135) * cblas_zswap: Level 1 CBLAS Functions. (line 93) * cblas_zsymm: Level 3 CBLAS Functions. (line 117) * cblas_zsyr2k: Level 3 CBLAS Functions. (line 128) * cblas_zsyrk: Level 3 CBLAS Functions. (line 123) * cblas_ztbmv: Level 2 CBLAS Functions. (line 145) * cblas_ztbsv: Level 2 CBLAS Functions. (line 160) * cblas_ztpmv: Level 2 CBLAS Functions. (line 150) * cblas_ztpsv: Level 2 CBLAS Functions. (line 165) * cblas_ztrmm: Level 3 CBLAS Functions. (line 134) * cblas_ztrmv: Level 2 CBLAS Functions. (line 140) * cblas_ztrsm: Level 3 CBLAS Functions. (line 140) * cblas_ztrsv: Level 2 CBLAS Functions. (line 155) * gsl_acosh: Elementary Functions. (line 32) * gsl_asinh: Elementary Functions. (line 36) * gsl_atanh: Elementary Functions. (line 40) * gsl_blas_caxpy: Level 1 GSL BLAS Interface. (line 92) * gsl_blas_ccopy: Level 1 GSL BLAS Interface. (line 81) * gsl_blas_cdotc: Level 1 GSL BLAS Interface. (line 27) * gsl_blas_cdotu: Level 1 GSL BLAS Interface. (line 20) * gsl_blas_cgemm: Level 3 GSL BLAS Interface. (line 13) * gsl_blas_cgemv: Level 2 GSL BLAS Interface. (line 12) * gsl_blas_cgerc: Level 2 GSL BLAS Interface. (line 106) * gsl_blas_cgeru: Level 2 GSL BLAS Interface. (line 97) * gsl_blas_chemm: Level 3 GSL BLAS Interface. (line 47) * gsl_blas_chemv: Level 2 GSL BLAS Interface. (line 78) * gsl_blas_cher: Level 2 GSL BLAS Interface. (line 126) * gsl_blas_cher2: Level 2 GSL BLAS Interface. (line 150) * gsl_blas_cher2k: Level 3 GSL BLAS Interface. (line 168) * gsl_blas_cherk: Level 3 GSL BLAS Interface. (line 130) * gsl_blas_cscal: Level 1 GSL BLAS Interface. (line 101) * gsl_blas_csscal: Level 1 GSL BLAS Interface. (line 105) * gsl_blas_cswap: Level 1 GSL BLAS Interface. (line 72) * gsl_blas_csymm: Level 3 GSL BLAS Interface. (line 32) * gsl_blas_csyr2k: Level 3 GSL BLAS Interface. (line 151) * gsl_blas_csyrk: Level 3 GSL BLAS Interface. (line 115) * gsl_blas_ctrmm: Level 3 GSL BLAS Interface. (line 69) * gsl_blas_ctrmv: Level 2 GSL BLAS Interface. (line 30) * gsl_blas_ctrsm: Level 3 GSL BLAS Interface. (line 92) * gsl_blas_ctrsv: Level 2 GSL BLAS Interface. (line 51) * gsl_blas_dasum: Level 1 GSL BLAS Interface. (line 46) * gsl_blas_daxpy: Level 1 GSL BLAS Interface. (line 90) * gsl_blas_dcopy: Level 1 GSL BLAS Interface. (line 80) * gsl_blas_ddot: Level 1 GSL BLAS Interface. (line 15) * gsl_blas_dgemm: Level 3 GSL BLAS Interface. (line 10) * gsl_blas_dgemv: Level 2 GSL BLAS Interface. (line 9) * gsl_blas_dger: Level 2 GSL BLAS Interface. (line 95) * gsl_blas_dnrm2: Level 1 GSL BLAS Interface. (line 35) * gsl_blas_drot: Level 1 GSL BLAS Interface. (line 126) * gsl_blas_drotg: Level 1 GSL BLAS Interface. (line 114) * gsl_blas_drotm: Level 1 GSL BLAS Interface. (line 141) * gsl_blas_drotmg: Level 1 GSL BLAS Interface. (line 133) * gsl_blas_dscal: Level 1 GSL BLAS Interface. (line 100) * gsl_blas_dsdot: Level 1 GSL BLAS Interface. (line 13) * gsl_blas_dswap: Level 1 GSL BLAS Interface. (line 71) * gsl_blas_dsymm: Level 3 GSL BLAS Interface. (line 29) * gsl_blas_dsymv: Level 2 GSL BLAS Interface. (line 68) * gsl_blas_dsyr: Level 2 GSL BLAS Interface. (line 117) * gsl_blas_dsyr2: Level 2 GSL BLAS Interface. (line 141) * gsl_blas_dsyr2k: Level 3 GSL BLAS Interface. (line 148) * gsl_blas_dsyrk: Level 3 GSL BLAS Interface. (line 112) * gsl_blas_dtrmm: Level 3 GSL BLAS Interface. (line 66) * gsl_blas_dtrmv: Level 2 GSL BLAS Interface. (line 27) * gsl_blas_dtrsm: Level 3 GSL BLAS Interface. (line 89) * gsl_blas_dtrsv: Level 2 GSL BLAS Interface. (line 48) * gsl_blas_dzasum: Level 1 GSL BLAS Interface. (line 51) * gsl_blas_dznrm2: Level 1 GSL BLAS Interface. (line 40) * gsl_blas_icamax: Level 1 GSL BLAS Interface. (line 58) * gsl_blas_idamax: Level 1 GSL BLAS Interface. (line 57) * gsl_blas_isamax: Level 1 GSL BLAS Interface. (line 56) * gsl_blas_izamax: Level 1 GSL BLAS Interface. (line 60) * gsl_blas_sasum: Level 1 GSL BLAS Interface. (line 45) * gsl_blas_saxpy: Level 1 GSL BLAS Interface. (line 88) * gsl_blas_scasum: Level 1 GSL BLAS Interface. (line 50) * gsl_blas_scnrm2: Level 1 GSL BLAS Interface. (line 39) * gsl_blas_scopy: Level 1 GSL BLAS Interface. (line 78) * gsl_blas_sdot: Level 1 GSL BLAS Interface. (line 11) * gsl_blas_sdsdot: Level 1 GSL BLAS Interface. (line 6) * gsl_blas_sgemm: Level 3 GSL BLAS Interface. (line 6) * gsl_blas_sgemv: Level 2 GSL BLAS Interface. (line 6) * gsl_blas_sger: Level 2 GSL BLAS Interface. (line 93) * gsl_blas_snrm2: Level 1 GSL BLAS Interface. (line 34) * gsl_blas_srot: Level 1 GSL BLAS Interface. (line 124) * gsl_blas_srotg: Level 1 GSL BLAS Interface. (line 112) * gsl_blas_srotm: Level 1 GSL BLAS Interface. (line 139) * gsl_blas_srotmg: Level 1 GSL BLAS Interface. (line 131) * gsl_blas_sscal: Level 1 GSL BLAS Interface. (line 99) * gsl_blas_sswap: Level 1 GSL BLAS Interface. (line 69) * gsl_blas_ssymm: Level 3 GSL BLAS Interface. (line 26) * gsl_blas_ssymv: Level 2 GSL BLAS Interface. (line 65) * gsl_blas_ssyr: Level 2 GSL BLAS Interface. (line 115) * gsl_blas_ssyr2: Level 2 GSL BLAS Interface. (line 138) * gsl_blas_ssyr2k: Level 3 GSL BLAS Interface. (line 145) * gsl_blas_ssyrk: Level 3 GSL BLAS Interface. (line 109) * gsl_blas_strmm: Level 3 GSL BLAS Interface. (line 63) * gsl_blas_strmv: Level 2 GSL BLAS Interface. (line 24) * gsl_blas_strsm: Level 3 GSL BLAS Interface. (line 86) * gsl_blas_strsv: Level 2 GSL BLAS Interface. (line 45) * gsl_blas_zaxpy: Level 1 GSL BLAS Interface. (line 94) * gsl_blas_zcopy: Level 1 GSL BLAS Interface. (line 83) * gsl_blas_zdotc: Level 1 GSL BLAS Interface. (line 29) * gsl_blas_zdotu: Level 1 GSL BLAS Interface. (line 22) * gsl_blas_zdscal: Level 1 GSL BLAS Interface. (line 107) * gsl_blas_zgemm: Level 3 GSL BLAS Interface. (line 17) * gsl_blas_zgemv: Level 2 GSL BLAS Interface. (line 16) * gsl_blas_zgerc: Level 2 GSL BLAS Interface. (line 109) * gsl_blas_zgeru: Level 2 GSL BLAS Interface. (line 100) * gsl_blas_zhemm: Level 3 GSL BLAS Interface. (line 51) * gsl_blas_zhemv: Level 2 GSL BLAS Interface. (line 82) * gsl_blas_zher: Level 2 GSL BLAS Interface. (line 128) * gsl_blas_zher2: Level 2 GSL BLAS Interface. (line 154) * gsl_blas_zher2k: Level 3 GSL BLAS Interface. (line 172) * gsl_blas_zherk: Level 3 GSL BLAS Interface. (line 133) * gsl_blas_zscal: Level 1 GSL BLAS Interface. (line 103) * gsl_blas_zswap: Level 1 GSL BLAS Interface. (line 74) * gsl_blas_zsymm: Level 3 GSL BLAS Interface. (line 36) * gsl_blas_zsyr2k: Level 3 GSL BLAS Interface. (line 155) * gsl_blas_zsyrk: Level 3 GSL BLAS Interface. (line 119) * gsl_blas_ztrmm: Level 3 GSL BLAS Interface. (line 73) * gsl_blas_ztrmv: Level 2 GSL BLAS Interface. (line 33) * gsl_blas_ztrsm: Level 3 GSL BLAS Interface. (line 96) * gsl_blas_ztrsv: Level 2 GSL BLAS Interface. (line 54) * gsl_block_alloc: Block allocation. (line 14) * gsl_block_calloc: Block allocation. (line 24) * gsl_block_fprintf: Reading and writing blocks. (line 25) * gsl_block_fread: Reading and writing blocks. (line 16) * gsl_block_free: Block allocation. (line 28) * gsl_block_fscanf: Reading and writing blocks. (line 33) * gsl_block_fwrite: Reading and writing blocks. (line 9) * gsl_bspline_alloc: Initializing the B-splines solver. (line 10) * gsl_bspline_deriv_alloc: Initializing the B-splines solver. (line 20) * gsl_bspline_deriv_eval: Evaluation of B-spline basis function derivatives. (line 6) * gsl_bspline_deriv_eval_nonzero: Evaluation of B-spline basis function derivatives. (line 20) * gsl_bspline_deriv_free: Initializing the B-splines solver. (line 26) * gsl_bspline_eval: Evaluation of B-spline basis functions. (line 6) * gsl_bspline_eval_nonzero: Evaluation of B-spline basis functions. (line 16) * gsl_bspline_free: Initializing the B-splines solver. (line 17) * gsl_bspline_greville_abscissa: Working with the Greville abscissae. (line 14) * gsl_bspline_knots: Constructing the knots vector. (line 6) * gsl_bspline_knots_uniform: Constructing the knots vector. (line 11) * gsl_bspline_ncoeffs: Evaluation of B-spline basis functions. (line 27) * gsl_cdf_beta_P: The Beta Distribution. (line 21) * gsl_cdf_beta_Pinv: The Beta Distribution. (line 23) * gsl_cdf_beta_Q: The Beta Distribution. (line 22) * gsl_cdf_beta_Qinv: The Beta Distribution. (line 24) * gsl_cdf_binomial_P: The Binomial Distribution. (line 24) * gsl_cdf_binomial_Q: The Binomial Distribution. (line 26) * gsl_cdf_cauchy_P: The Cauchy Distribution. (line 22) * gsl_cdf_cauchy_Pinv: The Cauchy Distribution. (line 24) * gsl_cdf_cauchy_Q: The Cauchy Distribution. (line 23) * gsl_cdf_cauchy_Qinv: The Cauchy Distribution. (line 25) * gsl_cdf_chisq_P: The Chi-squared Distribution. (line 29) * gsl_cdf_chisq_Pinv: The Chi-squared Distribution. (line 31) * gsl_cdf_chisq_Q: The Chi-squared Distribution. (line 30) * gsl_cdf_chisq_Qinv: The Chi-squared Distribution. (line 32) * gsl_cdf_exponential_P: The Exponential Distribution. (line 20) * gsl_cdf_exponential_Pinv: The Exponential Distribution. (line 22) * gsl_cdf_exponential_Q: The Exponential Distribution. (line 21) * gsl_cdf_exponential_Qinv: The Exponential Distribution. (line 23) * gsl_cdf_exppow_P: The Exponential Power Distribution. (line 24) * gsl_cdf_exppow_Q: The Exponential Power Distribution. (line 25) * gsl_cdf_fdist_P: The F-distribution. (line 33) * gsl_cdf_fdist_Pinv: The F-distribution. (line 35) * gsl_cdf_fdist_Q: The F-distribution. (line 34) * gsl_cdf_fdist_Qinv: The F-distribution. (line 37) * gsl_cdf_flat_P: The Flat (Uniform) Distribution. (line 20) * gsl_cdf_flat_Pinv: The Flat (Uniform) Distribution. (line 22) * gsl_cdf_flat_Q: The Flat (Uniform) Distribution. (line 21) * gsl_cdf_flat_Qinv: The Flat (Uniform) Distribution. (line 23) * gsl_cdf_gamma_P: The Gamma Distribution. (line 33) * gsl_cdf_gamma_Pinv: The Gamma Distribution. (line 35) * gsl_cdf_gamma_Q: The Gamma Distribution. (line 34) * gsl_cdf_gamma_Qinv: The Gamma Distribution. (line 36) * gsl_cdf_gaussian_P: The Gaussian Distribution. (line 41) * gsl_cdf_gaussian_Pinv: The Gaussian Distribution. (line 43) * gsl_cdf_gaussian_Q: The Gaussian Distribution. (line 42) * gsl_cdf_gaussian_Qinv: The Gaussian Distribution. (line 44) * gsl_cdf_geometric_P: The Geometric Distribution. (line 25) * gsl_cdf_geometric_Q: The Geometric Distribution. (line 26) * gsl_cdf_gumbel1_P: The Type-1 Gumbel Distribution. (line 21) * gsl_cdf_gumbel1_Pinv: The Type-1 Gumbel Distribution. (line 23) * gsl_cdf_gumbel1_Q: The Type-1 Gumbel Distribution. (line 22) * gsl_cdf_gumbel1_Qinv: The Type-1 Gumbel Distribution. (line 24) * gsl_cdf_gumbel2_P: The Type-2 Gumbel Distribution. (line 21) * gsl_cdf_gumbel2_Pinv: The Type-2 Gumbel Distribution. (line 23) * gsl_cdf_gumbel2_Q: The Type-2 Gumbel Distribution. (line 22) * gsl_cdf_gumbel2_Qinv: The Type-2 Gumbel Distribution. (line 24) * gsl_cdf_hypergeometric_P: The Hypergeometric Distribution. (line 29) * gsl_cdf_hypergeometric_Q: The Hypergeometric Distribution. (line 31) * gsl_cdf_laplace_P: The Laplace Distribution. (line 19) * gsl_cdf_laplace_Pinv: The Laplace Distribution. (line 21) * gsl_cdf_laplace_Q: The Laplace Distribution. (line 20) * gsl_cdf_laplace_Qinv: The Laplace Distribution. (line 22) * gsl_cdf_logistic_P: The Logistic Distribution. (line 20) * gsl_cdf_logistic_Pinv: The Logistic Distribution. (line 22) * gsl_cdf_logistic_Q: The Logistic Distribution. (line 21) * gsl_cdf_logistic_Qinv: The Logistic Distribution. (line 23) * gsl_cdf_lognormal_P: The Lognormal Distribution. (line 22) * gsl_cdf_lognormal_Pinv: The Lognormal Distribution. (line 26) * gsl_cdf_lognormal_Q: The Lognormal Distribution. (line 24) * gsl_cdf_lognormal_Qinv: The Lognormal Distribution. (line 28) * gsl_cdf_negative_binomial_P: The Negative Binomial Distribution. (line 24) * gsl_cdf_negative_binomial_Q: The Negative Binomial Distribution. (line 26) * gsl_cdf_pareto_P: The Pareto Distribution. (line 21) * gsl_cdf_pareto_Pinv: The Pareto Distribution. (line 23) * gsl_cdf_pareto_Q: The Pareto Distribution. (line 22) * gsl_cdf_pareto_Qinv: The Pareto Distribution. (line 24) * gsl_cdf_pascal_P: The Pascal Distribution. (line 23) * gsl_cdf_pascal_Q: The Pascal Distribution. (line 25) * gsl_cdf_poisson_P: The Poisson Distribution. (line 21) * gsl_cdf_poisson_Q: The Poisson Distribution. (line 22) * gsl_cdf_rayleigh_P: The Rayleigh Distribution. (line 20) * gsl_cdf_rayleigh_Pinv: The Rayleigh Distribution. (line 22) * gsl_cdf_rayleigh_Q: The Rayleigh Distribution. (line 21) * gsl_cdf_rayleigh_Qinv: The Rayleigh Distribution. (line 23) * gsl_cdf_tdist_P: The t-distribution. (line 29) * gsl_cdf_tdist_Pinv: The t-distribution. (line 31) * gsl_cdf_tdist_Q: The t-distribution. (line 30) * gsl_cdf_tdist_Qinv: The t-distribution. (line 32) * gsl_cdf_ugaussian_P: The Gaussian Distribution. (line 49) * gsl_cdf_ugaussian_Pinv: The Gaussian Distribution. (line 51) * gsl_cdf_ugaussian_Q: The Gaussian Distribution. (line 50) * gsl_cdf_ugaussian_Qinv: The Gaussian Distribution. (line 52) * gsl_cdf_weibull_P: The Weibull Distribution. (line 21) * gsl_cdf_weibull_Pinv: The Weibull Distribution. (line 23) * gsl_cdf_weibull_Q: The Weibull Distribution. (line 22) * gsl_cdf_weibull_Qinv: The Weibull Distribution. (line 24) * gsl_cheb_alloc: Creation and Calculation of Chebyshev Series. (line 6) * gsl_cheb_calc_deriv: Derivatives and Integrals. (line 12) * gsl_cheb_calc_integ: Derivatives and Integrals. (line 18) * gsl_cheb_coeffs: Auxiliary Functions for Chebyshev Series. (line 13) * gsl_cheb_eval: Chebyshev Series Evaluation. (line 6) * gsl_cheb_eval_err: Chebyshev Series Evaluation. (line 10) * gsl_cheb_eval_n: Chebyshev Series Evaluation. (line 17) * gsl_cheb_eval_n_err: Chebyshev Series Evaluation. (line 22) * gsl_cheb_free: Creation and Calculation of Chebyshev Series. (line 10) * gsl_cheb_init: Creation and Calculation of Chebyshev Series. (line 13) * gsl_cheb_order: Auxiliary Functions for Chebyshev Series. (line 9) * gsl_cheb_size: Auxiliary Functions for Chebyshev Series. (line 12) * gsl_combination_alloc: Combination allocation. (line 6) * gsl_combination_calloc: Combination allocation. (line 15) * gsl_combination_data: Combination properties. (line 13) * gsl_combination_fprintf: Reading and writing combinations. (line 27) * gsl_combination_fread: Reading and writing combinations. (line 17) * gsl_combination_free: Combination allocation. (line 30) * gsl_combination_fscanf: Reading and writing combinations. (line 36) * gsl_combination_fwrite: Reading and writing combinations. (line 9) * gsl_combination_get: Accessing combination elements. (line 9) * gsl_combination_init_first: Combination allocation. (line 22) * gsl_combination_init_last: Combination allocation. (line 26) * gsl_combination_k: Combination properties. (line 9) * gsl_combination_memcpy: Combination allocation. (line 33) * gsl_combination_n: Combination properties. (line 6) * gsl_combination_next: Combination functions. (line 6) * gsl_combination_prev: Combination functions. (line 14) * gsl_combination_valid: Combination properties. (line 17) * gsl_complex_abs: Properties of complex numbers. (line 10) * gsl_complex_abs2: Properties of complex numbers. (line 13) * gsl_complex_add: Complex arithmetic operators. (line 6) * gsl_complex_add_imag: Complex arithmetic operators. (line 38) * gsl_complex_add_real: Complex arithmetic operators. (line 22) * gsl_complex_arccos: Inverse Complex Trigonometric Functions. (line 19) * gsl_complex_arccosh: Inverse Complex Hyperbolic Functions. (line 11) * gsl_complex_arccosh_real: Inverse Complex Hyperbolic Functions. (line 18) * gsl_complex_arccos_real: Inverse Complex Trigonometric Functions. (line 24) * gsl_complex_arccot: Inverse Complex Trigonometric Functions. (line 52) * gsl_complex_arccoth: Inverse Complex Hyperbolic Functions. (line 39) * gsl_complex_arccsc: Inverse Complex Trigonometric Functions. (line 44) * gsl_complex_arccsch: Inverse Complex Hyperbolic Functions. (line 35) * gsl_complex_arccsc_real: Inverse Complex Trigonometric Functions. (line 48) * gsl_complex_arcsec: Inverse Complex Trigonometric Functions. (line 36) * gsl_complex_arcsech: Inverse Complex Hyperbolic Functions. (line 31) * gsl_complex_arcsec_real: Inverse Complex Trigonometric Functions. (line 40) * gsl_complex_arcsin: Inverse Complex Trigonometric Functions. (line 6) * gsl_complex_arcsinh: Inverse Complex Hyperbolic Functions. (line 6) * gsl_complex_arcsin_real: Inverse Complex Trigonometric Functions. (line 11) * gsl_complex_arctan: Inverse Complex Trigonometric Functions. (line 31) * gsl_complex_arctanh: Inverse Complex Hyperbolic Functions. (line 22) * gsl_complex_arctanh_real: Inverse Complex Hyperbolic Functions. (line 27) * gsl_complex_arg: Properties of complex numbers. (line 6) * gsl_complex_conjugate: Complex arithmetic operators. (line 54) * gsl_complex_cos: Complex Trigonometric Functions. (line 10) * gsl_complex_cosh: Complex Hyperbolic Functions. (line 10) * gsl_complex_cot: Complex Trigonometric Functions. (line 26) * gsl_complex_coth: Complex Hyperbolic Functions. (line 26) * gsl_complex_csc: Complex Trigonometric Functions. (line 22) * gsl_complex_csch: Complex Hyperbolic Functions. (line 22) * gsl_complex_div: Complex arithmetic operators. (line 18) * gsl_complex_div_imag: Complex arithmetic operators. (line 50) * gsl_complex_div_real: Complex arithmetic operators. (line 34) * gsl_complex_exp: Elementary Complex Functions. (line 24) * gsl_complex_inverse: Complex arithmetic operators. (line 58) * gsl_complex_log: Elementary Complex Functions. (line 28) * gsl_complex_log10: Elementary Complex Functions. (line 33) * gsl_complex_logabs: Properties of complex numbers. (line 17) * gsl_complex_log_b: Elementary Complex Functions. (line 37) * gsl_complex_mul: Complex arithmetic operators. (line 14) * gsl_complex_mul_imag: Complex arithmetic operators. (line 46) * gsl_complex_mul_real: Complex arithmetic operators. (line 30) * gsl_complex_negative: Complex arithmetic operators. (line 62) * gsl_complex_polar: Representation of complex numbers. (line 29) * gsl_complex_poly_complex_eval: Polynomial Evaluation. (line 21) * gsl_complex_pow: Elementary Complex Functions. (line 15) * gsl_complex_pow_real: Elementary Complex Functions. (line 20) * gsl_complex_rect: Representation of complex numbers. (line 24) * gsl_complex_sec: Complex Trigonometric Functions. (line 18) * gsl_complex_sech: Complex Hyperbolic Functions. (line 18) * gsl_complex_sin: Complex Trigonometric Functions. (line 6) * gsl_complex_sinh: Complex Hyperbolic Functions. (line 6) * gsl_complex_sqrt: Elementary Complex Functions. (line 6) * gsl_complex_sqrt_real: Elementary Complex Functions. (line 11) * gsl_complex_sub: Complex arithmetic operators. (line 10) * gsl_complex_sub_imag: Complex arithmetic operators. (line 42) * gsl_complex_sub_real: Complex arithmetic operators. (line 26) * gsl_complex_tan: Complex Trigonometric Functions. (line 14) * gsl_complex_tanh: Complex Hyperbolic Functions. (line 14) * gsl_deriv_backward: Numerical Differentiation functions. (line 40) * gsl_deriv_central: Numerical Differentiation functions. (line 6) * gsl_deriv_forward: Numerical Differentiation functions. (line 22) * gsl_dht_alloc: Discrete Hankel Transform Functions. (line 6) * gsl_dht_apply: Discrete Hankel Transform Functions. (line 22) * gsl_dht_free: Discrete Hankel Transform Functions. (line 19) * gsl_dht_init: Discrete Hankel Transform Functions. (line 10) * gsl_dht_k_sample: Discrete Hankel Transform Functions. (line 36) * gsl_dht_new: Discrete Hankel Transform Functions. (line 14) * gsl_dht_x_sample: Discrete Hankel Transform Functions. (line 31) * gsl_eigen_gen: Real Generalized Nonsymmetric Eigensystems. (line 72) * gsl_eigen_genherm: Complex Generalized Hermitian-Definite Eigensystems. (line 27) * gsl_eigen_genhermv: Complex Generalized Hermitian-Definite Eigensystems. (line 45) * gsl_eigen_genhermv_alloc: Complex Generalized Hermitian-Definite Eigensystems. (line 35) * gsl_eigen_genhermv_free: Complex Generalized Hermitian-Definite Eigensystems. (line 41) * gsl_eigen_genhermv_sort: Sorting Eigenvalues and Eigenvectors. (line 45) * gsl_eigen_genherm_alloc: Complex Generalized Hermitian-Definite Eigensystems. (line 17) * gsl_eigen_genherm_free: Complex Generalized Hermitian-Definite Eigensystems. (line 23) * gsl_eigen_gensymm: Real Generalized Symmetric-Definite Eigensystems. (line 32) * gsl_eigen_gensymmv: Real Generalized Symmetric-Definite Eigensystems. (line 49) * gsl_eigen_gensymmv_alloc: Real Generalized Symmetric-Definite Eigensystems. (line 39) * gsl_eigen_gensymmv_free: Real Generalized Symmetric-Definite Eigensystems. (line 45) * gsl_eigen_gensymmv_sort: Sorting Eigenvalues and Eigenvectors. (line 38) * gsl_eigen_gensymm_alloc: Real Generalized Symmetric-Definite Eigensystems. (line 22) * gsl_eigen_gensymm_free: Real Generalized Symmetric-Definite Eigensystems. (line 28) * gsl_eigen_genv: Real Generalized Nonsymmetric Eigensystems. (line 105) * gsl_eigen_genv_alloc: Real Generalized Nonsymmetric Eigensystems. (line 96) * gsl_eigen_genv_free: Real Generalized Nonsymmetric Eigensystems. (line 102) * gsl_eigen_genv_QZ: Real Generalized Nonsymmetric Eigensystems. (line 120) * gsl_eigen_genv_sort: Sorting Eigenvalues and Eigenvectors. (line 52) * gsl_eigen_gen_alloc: Real Generalized Nonsymmetric Eigensystems. (line 41) * gsl_eigen_gen_free: Real Generalized Nonsymmetric Eigensystems. (line 47) * gsl_eigen_gen_params: Real Generalized Nonsymmetric Eigensystems. (line 50) * gsl_eigen_gen_QZ: Real Generalized Nonsymmetric Eigensystems. (line 89) * gsl_eigen_herm: Complex Hermitian Matrices. (line 18) * gsl_eigen_hermv: Complex Hermitian Matrices. (line 37) * gsl_eigen_hermv_alloc: Complex Hermitian Matrices. (line 28) * gsl_eigen_hermv_free: Complex Hermitian Matrices. (line 34) * gsl_eigen_hermv_sort: Sorting Eigenvalues and Eigenvectors. (line 22) * gsl_eigen_herm_alloc: Complex Hermitian Matrices. (line 9) * gsl_eigen_herm_free: Complex Hermitian Matrices. (line 15) * gsl_eigen_nonsymm: Real Nonsymmetric Matrices. (line 60) * gsl_eigen_nonsymmv: Real Nonsymmetric Matrices. (line 97) * gsl_eigen_nonsymmv_alloc: Real Nonsymmetric Matrices. (line 78) * gsl_eigen_nonsymmv_free: Real Nonsymmetric Matrices. (line 84) * gsl_eigen_nonsymmv_params: Real Nonsymmetric Matrices. (line 88) * gsl_eigen_nonsymmv_sort: Sorting Eigenvalues and Eigenvectors. (line 29) * gsl_eigen_nonsymmv_Z: Real Nonsymmetric Matrices. (line 111) * gsl_eigen_nonsymm_alloc: Real Nonsymmetric Matrices. (line 17) * gsl_eigen_nonsymm_free: Real Nonsymmetric Matrices. (line 23) * gsl_eigen_nonsymm_params: Real Nonsymmetric Matrices. (line 27) * gsl_eigen_nonsymm_Z: Real Nonsymmetric Matrices. (line 72) * gsl_eigen_symm: Real Symmetric Matrices. (line 21) * gsl_eigen_symmv: Real Symmetric Matrices. (line 39) * gsl_eigen_symmv_alloc: Real Symmetric Matrices. (line 30) * gsl_eigen_symmv_free: Real Symmetric Matrices. (line 36) * gsl_eigen_symmv_sort: Sorting Eigenvalues and Eigenvectors. (line 6) * gsl_eigen_symm_alloc: Real Symmetric Matrices. (line 12) * gsl_eigen_symm_free: Real Symmetric Matrices. (line 18) * GSL_ERROR: Using GSL error reporting in your own functions. (line 15) * GSL_ERROR_VAL: Using GSL error reporting in your own functions. (line 36) * gsl_expm1: Elementary Functions. (line 17) * gsl_fcmp: Approximate Comparison of Floating Point Numbers. (line 12) * gsl_fft_complex_backward: Mixed-radix FFT routines for complex data. (line 124) * gsl_fft_complex_forward: Mixed-radix FFT routines for complex data. (line 117) * gsl_fft_complex_inverse: Mixed-radix FFT routines for complex data. (line 127) * gsl_fft_complex_radix2_backward: Radix-2 FFT routines for complex data. (line 23) * gsl_fft_complex_radix2_dif_backward: Radix-2 FFT routines for complex data. (line 47) * gsl_fft_complex_radix2_dif_forward: Radix-2 FFT routines for complex data. (line 40) * gsl_fft_complex_radix2_dif_inverse: Radix-2 FFT routines for complex data. (line 50) * gsl_fft_complex_radix2_dif_transform: Radix-2 FFT routines for complex data. (line 43) * gsl_fft_complex_radix2_forward: Radix-2 FFT routines for complex data. (line 16) * gsl_fft_complex_radix2_inverse: Radix-2 FFT routines for complex data. (line 26) * gsl_fft_complex_radix2_transform: Radix-2 FFT routines for complex data. (line 19) * gsl_fft_complex_transform: Mixed-radix FFT routines for complex data. (line 120) * gsl_fft_complex_wavetable_alloc: Mixed-radix FFT routines for complex data. (line 44) * gsl_fft_complex_wavetable_free: Mixed-radix FFT routines for complex data. (line 63) * gsl_fft_complex_workspace_alloc: Mixed-radix FFT routines for complex data. (line 104) * gsl_fft_complex_workspace_free: Mixed-radix FFT routines for complex data. (line 109) * gsl_fft_halfcomplex_radix2_backward: Radix-2 FFT routines for real data. (line 58) * gsl_fft_halfcomplex_radix2_inverse: Radix-2 FFT routines for real data. (line 56) * gsl_fft_halfcomplex_radix2_unpack: Radix-2 FFT routines for real data. (line 66) * gsl_fft_halfcomplex_transform: Mixed-radix FFT routines for real data. (line 122) * gsl_fft_halfcomplex_unpack: Mixed-radix FFT routines for real data. (line 153) * gsl_fft_halfcomplex_wavetable_alloc: Mixed-radix FFT routines for real data. (line 74) * gsl_fft_halfcomplex_wavetable_free: Mixed-radix FFT routines for real data. (line 95) * gsl_fft_real_radix2_transform: Radix-2 FFT routines for real data. (line 13) * gsl_fft_real_transform: Mixed-radix FFT routines for real data. (line 119) * gsl_fft_real_unpack: Mixed-radix FFT routines for real data. (line 136) * gsl_fft_real_wavetable_alloc: Mixed-radix FFT routines for real data. (line 72) * gsl_fft_real_wavetable_free: Mixed-radix FFT routines for real data. (line 93) * gsl_fft_real_workspace_alloc: Mixed-radix FFT routines for real data. (line 104) * gsl_fft_real_workspace_free: Mixed-radix FFT routines for real data. (line 110) * gsl_finite: Infinities and Not-a-number. (line 25) * gsl_fit_linear: Linear regression. (line 9) * gsl_fit_linear_est: Linear regression. (line 42) * gsl_fit_mul: Linear fitting without a constant term. (line 10) * gsl_fit_mul_est: Linear fitting without a constant term. (line 36) * gsl_fit_wlinear: Linear regression. (line 25) * gsl_fit_wmul: Linear fitting without a constant term. (line 21) * gsl_frexp: Elementary Functions. (line 48) * gsl_heapsort: Sorting objects. (line 16) * gsl_heapsort_index: Sorting objects. (line 58) * gsl_histogram2d_accumulate: Updating and accessing 2D histogram elements. (line 25) * gsl_histogram2d_add: 2D Histogram Operations. (line 11) * gsl_histogram2d_alloc: 2D Histogram allocation. (line 14) * gsl_histogram2d_clone: Copying 2D Histograms. (line 12) * gsl_histogram2d_cov: 2D Histogram Statistics. (line 52) * gsl_histogram2d_div: 2D Histogram Operations. (line 31) * gsl_histogram2d_equal_bins_p: 2D Histogram Operations. (line 6) * gsl_histogram2d_find: Searching 2D histogram ranges. (line 9) * gsl_histogram2d_fprintf: Reading and writing 2D histograms. (line 27) * gsl_histogram2d_fread: Reading and writing 2D histograms. (line 17) * gsl_histogram2d_free: 2D Histogram allocation. (line 37) * gsl_histogram2d_fscanf: Reading and writing 2D histograms. (line 64) * gsl_histogram2d_fwrite: Reading and writing 2D histograms. (line 9) * gsl_histogram2d_get: Updating and accessing 2D histogram elements. (line 31) * gsl_histogram2d_get_xrange: Updating and accessing 2D histogram elements. (line 38) * gsl_histogram2d_get_yrange: Updating and accessing 2D histogram elements. (line 40) * gsl_histogram2d_increment: Updating and accessing 2D histogram elements. (line 12) * gsl_histogram2d_max_bin: 2D Histogram Statistics. (line 10) * gsl_histogram2d_max_val: 2D Histogram Statistics. (line 6) * gsl_histogram2d_memcpy: Copying 2D Histograms. (line 6) * gsl_histogram2d_min_bin: 2D Histogram Statistics. (line 21) * gsl_histogram2d_min_val: 2D Histogram Statistics. (line 17) * gsl_histogram2d_mul: 2D Histogram Operations. (line 24) * gsl_histogram2d_nx: Updating and accessing 2D histogram elements. (line 55) * gsl_histogram2d_ny: Updating and accessing 2D histogram elements. (line 58) * gsl_histogram2d_pdf_alloc: Resampling from 2D histograms. (line 38) * gsl_histogram2d_pdf_free: Resampling from 2D histograms. (line 54) * gsl_histogram2d_pdf_init: Resampling from 2D histograms. (line 46) * gsl_histogram2d_pdf_sample: Resampling from 2D histograms. (line 58) * gsl_histogram2d_reset: Updating and accessing 2D histogram elements. (line 64) * gsl_histogram2d_scale: 2D Histogram Operations. (line 38) * gsl_histogram2d_set_ranges: 2D Histogram allocation. (line 24) * gsl_histogram2d_set_ranges_uniform: 2D Histogram allocation. (line 31) * gsl_histogram2d_shift: 2D Histogram Operations. (line 43) * gsl_histogram2d_sub: 2D Histogram Operations. (line 17) * gsl_histogram2d_sum: 2D Histogram Statistics. (line 58) * gsl_histogram2d_xmax: Updating and accessing 2D histogram elements. (line 53) * gsl_histogram2d_xmean: 2D Histogram Statistics. (line 28) * gsl_histogram2d_xmin: Updating and accessing 2D histogram elements. (line 54) * gsl_histogram2d_xsigma: 2D Histogram Statistics. (line 40) * gsl_histogram2d_ymax: Updating and accessing 2D histogram elements. (line 56) * gsl_histogram2d_ymean: 2D Histogram Statistics. (line 34) * gsl_histogram2d_ymin: Updating and accessing 2D histogram elements. (line 57) * gsl_histogram2d_ysigma: 2D Histogram Statistics. (line 46) * gsl_histogram_accumulate: Updating and accessing histogram elements. (line 25) * gsl_histogram_add: Histogram Operations. (line 11) * gsl_histogram_alloc: Histogram allocation. (line 14) * gsl_histogram_bins: Updating and accessing histogram elements. (line 53) * gsl_histogram_clone: Copying Histograms. (line 12) * gsl_histogram_div: Histogram Operations. (line 30) * gsl_histogram_equal_bins_p: Histogram Operations. (line 6) * gsl_histogram_find: Searching histogram ranges. (line 9) * gsl_histogram_fprintf: Reading and writing histograms. (line 26) * gsl_histogram_fread: Reading and writing histograms. (line 17) * gsl_histogram_free: Histogram allocation. (line 60) * gsl_histogram_fscanf: Reading and writing histograms. (line 51) * gsl_histogram_fwrite: Reading and writing histograms. (line 9) * gsl_histogram_get: Updating and accessing histogram elements. (line 31) * gsl_histogram_get_range: Updating and accessing histogram elements. (line 38) * gsl_histogram_increment: Updating and accessing histogram elements. (line 11) * gsl_histogram_max: Updating and accessing histogram elements. (line 51) * gsl_histogram_max_bin: Histogram Statistics. (line 10) * gsl_histogram_max_val: Histogram Statistics. (line 6) * gsl_histogram_mean: Histogram Statistics. (line 24) * gsl_histogram_memcpy: Copying Histograms. (line 6) * gsl_histogram_min: Updating and accessing histogram elements. (line 52) * gsl_histogram_min_bin: Histogram Statistics. (line 19) * gsl_histogram_min_val: Histogram Statistics. (line 15) * gsl_histogram_mul: Histogram Operations. (line 23) * gsl_histogram_pdf_alloc: The histogram probability distribution struct. (line 33) * gsl_histogram_pdf_free: The histogram probability distribution struct. (line 47) * gsl_histogram_pdf_init: The histogram probability distribution struct. (line 40) * gsl_histogram_pdf_sample: The histogram probability distribution struct. (line 51) * gsl_histogram_reset: Updating and accessing histogram elements. (line 59) * gsl_histogram_scale: Histogram Operations. (line 37) * gsl_histogram_set_ranges: Histogram allocation. (line 23) * gsl_histogram_set_ranges_uniform: Histogram allocation. (line 48) * gsl_histogram_shift: Histogram Operations. (line 41) * gsl_histogram_sigma: Histogram Statistics. (line 30) * gsl_histogram_sub: Histogram Operations. (line 17) * gsl_histogram_sum: Histogram Statistics. (line 37) * gsl_hypot: Elementary Functions. (line 22) * gsl_hypot3: Elementary Functions. (line 27) * gsl_ieee_env_setup: Setting up your IEEE environment. (line 23) * gsl_ieee_fprintf_double: Representation of floating point numbers. (line 55) * gsl_ieee_fprintf_float: Representation of floating point numbers. (line 53) * gsl_ieee_printf_double: Representation of floating point numbers. (line 82) * gsl_ieee_printf_float: Representation of floating point numbers. (line 81) * GSL_IMAG: Representation of complex numbers. (line 35) * gsl_integration_cquad: CQUAD doubly-adaptive integration. (line 33) * gsl_integration_cquad_workspace_alloc: CQUAD doubly-adaptive integration. (line 20) * gsl_integration_cquad_workspace_free: CQUAD doubly-adaptive integration. (line 29) * gsl_integration_glfixed: Fixed order Gauss-Legendre integration. (line 22) * gsl_integration_glfixed_point: Fixed order Gauss-Legendre integration. (line 27) * gsl_integration_glfixed_table_alloc: Fixed order Gauss-Legendre integration. (line 14) * gsl_integration_glfixed_table_free: Fixed order Gauss-Legendre integration. (line 35) * gsl_integration_qag: QAG adaptive integration. (line 27) * gsl_integration_qagi: QAGI adaptive integration on infinite intervals. (line 6) * gsl_integration_qagil: QAGI adaptive integration on infinite intervals. (line 37) * gsl_integration_qagiu: QAGI adaptive integration on infinite intervals. (line 22) * gsl_integration_qagp: QAGP adaptive integration with known singular points. (line 6) * gsl_integration_qags: QAGS adaptive integration with singularities. (line 15) * gsl_integration_qawc: QAWC adaptive integration for Cauchy principal values. (line 6) * gsl_integration_qawf: QAWF adaptive integration for Fourier integrals. (line 6) * gsl_integration_qawo: QAWO adaptive integration for oscillatory functions. (line 53) * gsl_integration_qawo_table_alloc: QAWO adaptive integration for oscillatory functions. (line 11) * gsl_integration_qawo_table_free: QAWO adaptive integration for oscillatory functions. (line 49) * gsl_integration_qawo_table_set: QAWO adaptive integration for oscillatory functions. (line 38) * gsl_integration_qawo_table_set_length: QAWO adaptive integration for oscillatory functions. (line 44) * gsl_integration_qaws: QAWS adaptive integration for singular functions. (line 48) * gsl_integration_qaws_table_alloc: QAWS adaptive integration for singular functions. (line 11) * gsl_integration_qaws_table_free: QAWS adaptive integration for singular functions. (line 43) * gsl_integration_qaws_table_set: QAWS adaptive integration for singular functions. (line 37) * gsl_integration_qng: QNG non-adaptive Gauss-Kronrod integration. (line 10) * gsl_integration_workspace_alloc: QAG adaptive integration. (line 15) * gsl_integration_workspace_free: QAG adaptive integration. (line 23) * gsl_interp_accel_alloc: Index Look-up and Acceleration. (line 19) * gsl_interp_accel_find: Index Look-up and Acceleration. (line 25) * gsl_interp_accel_free: Index Look-up and Acceleration. (line 39) * gsl_interp_accel_reset: Index Look-up and Acceleration. (line 34) * gsl_interp_akima: Interpolation Types. (line 34) * gsl_interp_akima_periodic: Interpolation Types. (line 38) * gsl_interp_alloc: Interpolation Functions. (line 9) * gsl_interp_bsearch: Index Look-up and Acceleration. (line 12) * gsl_interp_cspline: Interpolation Types. (line 19) * gsl_interp_cspline_periodic: Interpolation Types. (line 25) * gsl_interp_eval: Evaluation of Interpolating Functions. (line 6) * gsl_interp_eval_deriv: Evaluation of Interpolating Functions. (line 17) * gsl_interp_eval_deriv2: Evaluation of Interpolating Functions. (line 27) * gsl_interp_eval_deriv2_e: Evaluation of Interpolating Functions. (line 30) * gsl_interp_eval_deriv_e: Evaluation of Interpolating Functions. (line 20) * gsl_interp_eval_e: Evaluation of Interpolating Functions. (line 9) * gsl_interp_eval_integ: Evaluation of Interpolating Functions. (line 37) * gsl_interp_eval_integ_e: Evaluation of Interpolating Functions. (line 40) * gsl_interp_free: Interpolation Functions. (line 24) * gsl_interp_init: Interpolation Functions. (line 14) * gsl_interp_linear: Interpolation Types. (line 8) * gsl_interp_min_size: Interpolation Types. (line 55) * gsl_interp_name: Interpolation Types. (line 44) * gsl_interp_polynomial: Interpolation Types. (line 12) * gsl_interp_type_min_size: Interpolation Types. (line 57) * gsl_isinf: Infinities and Not-a-number. (line 21) * gsl_isnan: Infinities and Not-a-number. (line 18) * GSL_IS_EVEN: Testing for Odd and Even Numbers. (line 10) * GSL_IS_ODD: Testing for Odd and Even Numbers. (line 6) * gsl_ldexp: Elementary Functions. (line 44) * gsl_linalg_balance_matrix: Balancing. (line 17) * gsl_linalg_bidiag_decomp: Bidiagonalization. (line 15) * gsl_linalg_bidiag_unpack: Bidiagonalization. (line 26) * gsl_linalg_bidiag_unpack2: Bidiagonalization. (line 35) * gsl_linalg_bidiag_unpack_B: Bidiagonalization. (line 42) * gsl_linalg_cholesky_decomp: Cholesky Decomposition. (line 19) * gsl_linalg_cholesky_invert: Cholesky Decomposition. (line 56) * gsl_linalg_cholesky_solve: Cholesky Decomposition. (line 36) * gsl_linalg_cholesky_svx: Cholesky Decomposition. (line 46) * gsl_linalg_complex_cholesky_decomp: Cholesky Decomposition. (line 20) * gsl_linalg_complex_cholesky_invert: Cholesky Decomposition. (line 57) * gsl_linalg_complex_cholesky_solve: Cholesky Decomposition. (line 38) * gsl_linalg_complex_cholesky_svx: Cholesky Decomposition. (line 48) * gsl_linalg_complex_householder_hm: Householder Transformations. (line 27) * gsl_linalg_complex_householder_hv: Householder Transformations. (line 43) * gsl_linalg_complex_householder_mh: Householder Transformations. (line 35) * gsl_linalg_complex_householder_transform: Householder Transformations. (line 18) * gsl_linalg_complex_LU_decomp: LU Decomposition. (line 20) * gsl_linalg_complex_LU_det: LU Decomposition. (line 83) * gsl_linalg_complex_LU_invert: LU Decomposition. (line 71) * gsl_linalg_complex_LU_lndet: LU Decomposition. (line 91) * gsl_linalg_complex_LU_refine: LU Decomposition. (line 60) * gsl_linalg_complex_LU_sgndet: LU Decomposition. (line 99) * gsl_linalg_complex_LU_solve: LU Decomposition. (line 41) * gsl_linalg_complex_LU_svx: LU Decomposition. (line 50) * gsl_linalg_hermtd_decomp: Tridiagonal Decomposition of Hermitian Matrices. (line 14) * gsl_linalg_hermtd_unpack: Tridiagonal Decomposition of Hermitian Matrices. (line 26) * gsl_linalg_hermtd_unpack_T: Tridiagonal Decomposition of Hermitian Matrices. (line 34) * gsl_linalg_hessenberg_decomp: Hessenberg Decomposition of Real Matrices. (line 16) * gsl_linalg_hessenberg_set_zero: Hessenberg Decomposition of Real Matrices. (line 42) * gsl_linalg_hessenberg_unpack: Hessenberg Decomposition of Real Matrices. (line 27) * gsl_linalg_hessenberg_unpack_accum: Hessenberg Decomposition of Real Matrices. (line 33) * gsl_linalg_hesstri_decomp: Hessenberg-Triangular Decomposition of Real Matrices. (line 17) * gsl_linalg_HH_solve: Householder solver for linear systems. (line 6) * gsl_linalg_HH_svx: Householder solver for linear systems. (line 13) * gsl_linalg_householder_hm: Householder Transformations. (line 25) * gsl_linalg_householder_hv: Householder Transformations. (line 41) * gsl_linalg_householder_mh: Householder Transformations. (line 33) * gsl_linalg_householder_transform: Householder Transformations. (line 17) * gsl_linalg_LU_decomp: LU Decomposition. (line 18) * gsl_linalg_LU_det: LU Decomposition. (line 82) * gsl_linalg_LU_invert: LU Decomposition. (line 69) * gsl_linalg_LU_lndet: LU Decomposition. (line 90) * gsl_linalg_LU_refine: LU Decomposition. (line 57) * gsl_linalg_LU_sgndet: LU Decomposition. (line 98) * gsl_linalg_LU_solve: LU Decomposition. (line 39) * gsl_linalg_LU_svx: LU Decomposition. (line 48) * gsl_linalg_QRPT_decomp: QR Decomposition with Column Pivoting. (line 18) * gsl_linalg_QRPT_decomp2: QR Decomposition with Column Pivoting. (line 39) * gsl_linalg_QRPT_QRsolve: QR Decomposition with Column Pivoting. (line 60) * gsl_linalg_QRPT_Rsolve: QR Decomposition with Column Pivoting. (line 76) * gsl_linalg_QRPT_Rsvx: QR Decomposition with Column Pivoting. (line 81) * gsl_linalg_QRPT_solve: QR Decomposition with Column Pivoting. (line 46) * gsl_linalg_QRPT_svx: QR Decomposition with Column Pivoting. (line 53) * gsl_linalg_QRPT_update: QR Decomposition with Column Pivoting. (line 67) * gsl_linalg_QR_decomp: QR Decomposition. (line 19) * gsl_linalg_QR_lssolve: QR Decomposition. (line 50) * gsl_linalg_QR_QRsolve: QR Decomposition. (line 102) * gsl_linalg_QR_QTmat: QR Decomposition. (line 76) * gsl_linalg_QR_QTvec: QR Decomposition. (line 61) * gsl_linalg_QR_Qvec: QR Decomposition. (line 69) * gsl_linalg_QR_Rsolve: QR Decomposition. (line 84) * gsl_linalg_QR_Rsvx: QR Decomposition. (line 90) * gsl_linalg_QR_solve: QR Decomposition. (line 35) * gsl_linalg_QR_svx: QR Decomposition. (line 42) * gsl_linalg_QR_unpack: QR Decomposition. (line 97) * gsl_linalg_QR_update: QR Decomposition. (line 108) * gsl_linalg_R_solve: QR Decomposition. (line 115) * gsl_linalg_R_svx: QR Decomposition. (line 120) * gsl_linalg_solve_cyc_tridiag: Tridiagonal Systems. (line 39) * gsl_linalg_solve_symm_cyc_tridiag: Tridiagonal Systems. (line 53) * gsl_linalg_solve_symm_tridiag: Tridiagonal Systems. (line 26) * gsl_linalg_solve_tridiag: Tridiagonal Systems. (line 13) * gsl_linalg_SV_decomp: Singular Value Decomposition. (line 37) * gsl_linalg_SV_decomp_jacobi: Singular Value Decomposition. (line 56) * gsl_linalg_SV_decomp_mod: Singular Value Decomposition. (line 50) * gsl_linalg_SV_leverage: Singular Value Decomposition. (line 79) * gsl_linalg_SV_solve: Singular Value Decomposition. (line 63) * gsl_linalg_symmtd_decomp: Tridiagonal Decomposition of Real Symmetric Matrices. (line 13) * gsl_linalg_symmtd_unpack: Tridiagonal Decomposition of Real Symmetric Matrices. (line 24) * gsl_linalg_symmtd_unpack_T: Tridiagonal Decomposition of Real Symmetric Matrices. (line 32) * gsl_log1p: Elementary Functions. (line 12) * gsl_matrix_add: Matrix operations. (line 8) * gsl_matrix_add_constant: Matrix operations. (line 38) * gsl_matrix_alloc: Matrix allocation. (line 14) * gsl_matrix_calloc: Matrix allocation. (line 21) * gsl_matrix_column: Creating row and column views. (line 24) * gsl_matrix_const_column: Creating row and column views. (line 26) * gsl_matrix_const_diagonal: Creating row and column views. (line 63) * gsl_matrix_const_ptr: Accessing matrix elements. (line 36) * gsl_matrix_const_row: Creating row and column views. (line 14) * gsl_matrix_const_subcolumn: Creating row and column views. (line 51) * gsl_matrix_const_subdiagonal: Creating row and column views. (line 76) * gsl_matrix_const_submatrix: Matrix views. (line 20) * gsl_matrix_const_subrow: Creating row and column views. (line 38) * gsl_matrix_const_superdiagonal: Creating row and column views. (line 88) * gsl_matrix_const_view_array: Matrix views. (line 51) * gsl_matrix_const_view_array_with_tda: Matrix views. (line 75) * gsl_matrix_const_view_vector: Matrix views. (line 101) * gsl_matrix_const_view_vector_with_tda: Matrix views. (line 125) * gsl_matrix_diagonal: Creating row and column views. (line 62) * gsl_matrix_div_elements: Matrix operations. (line 27) * gsl_matrix_equal: Matrix properties. (line 20) * gsl_matrix_fprintf: Reading and writing matrices. (line 26) * gsl_matrix_fread: Reading and writing matrices. (line 17) * gsl_matrix_free: Matrix allocation. (line 25) * gsl_matrix_fscanf: Reading and writing matrices. (line 34) * gsl_matrix_fwrite: Reading and writing matrices. (line 9) * gsl_matrix_get: Accessing matrix elements. (line 20) * gsl_matrix_get_col: Copying rows and columns. (line 19) * gsl_matrix_get_row: Copying rows and columns. (line 13) * gsl_matrix_isneg: Matrix properties. (line 12) * gsl_matrix_isnonneg: Matrix properties. (line 13) * gsl_matrix_isnull: Matrix properties. (line 10) * gsl_matrix_ispos: Matrix properties. (line 11) * gsl_matrix_max: Finding maximum and minimum elements of matrices. (line 8) * gsl_matrix_max_index: Finding maximum and minimum elements of matrices. (line 19) * gsl_matrix_memcpy: Copying matrices. (line 6) * gsl_matrix_min: Finding maximum and minimum elements of matrices. (line 11) * gsl_matrix_minmax: Finding maximum and minimum elements of matrices. (line 14) * gsl_matrix_minmax_index: Finding maximum and minimum elements of matrices. (line 33) * gsl_matrix_min_index: Finding maximum and minimum elements of matrices. (line 26) * gsl_matrix_mul_elements: Matrix operations. (line 20) * gsl_matrix_ptr: Accessing matrix elements. (line 34) * gsl_matrix_row: Creating row and column views. (line 13) * gsl_matrix_scale: Matrix operations. (line 34) * gsl_matrix_set: Accessing matrix elements. (line 27) * gsl_matrix_set_all: Initializing matrix elements. (line 6) * gsl_matrix_set_col: Copying rows and columns. (line 31) * gsl_matrix_set_identity: Initializing matrix elements. (line 12) * gsl_matrix_set_row: Copying rows and columns. (line 25) * gsl_matrix_set_zero: Initializing matrix elements. (line 9) * gsl_matrix_sub: Matrix operations. (line 14) * gsl_matrix_subcolumn: Creating row and column views. (line 49) * gsl_matrix_subdiagonal: Creating row and column views. (line 74) * gsl_matrix_submatrix: Matrix views. (line 18) * gsl_matrix_subrow: Creating row and column views. (line 36) * gsl_matrix_superdiagonal: Creating row and column views. (line 86) * gsl_matrix_swap: Copying matrices. (line 11) * gsl_matrix_swap_columns: Exchanging rows and columns. (line 14) * gsl_matrix_swap_rowcol: Exchanging rows and columns. (line 19) * gsl_matrix_swap_rows: Exchanging rows and columns. (line 9) * gsl_matrix_transpose: Exchanging rows and columns. (line 32) * gsl_matrix_transpose_memcpy: Exchanging rows and columns. (line 25) * gsl_matrix_view_array: Matrix views. (line 49) * gsl_matrix_view_array_with_tda: Matrix views. (line 73) * gsl_matrix_view_vector: Matrix views. (line 99) * gsl_matrix_view_vector_with_tda: Matrix views. (line 123) * GSL_MAX: Maximum and Minimum functions. (line 10) * GSL_MAX_DBL: Maximum and Minimum functions. (line 18) * GSL_MAX_INT: Maximum and Minimum functions. (line 32) * GSL_MAX_LDBL: Maximum and Minimum functions. (line 39) * GSL_MIN: Maximum and Minimum functions. (line 14) * GSL_MIN_DBL: Maximum and Minimum functions. (line 25) * gsl_min_fminimizer_alloc: Initializing the Minimizer. (line 6) * gsl_min_fminimizer_brent: Minimization Algorithms. (line 30) * gsl_min_fminimizer_free: Initializing the Minimizer. (line 39) * gsl_min_fminimizer_f_lower: Minimization Iteration. (line 45) * gsl_min_fminimizer_f_minimum: Minimization Iteration. (line 41) * gsl_min_fminimizer_f_upper: Minimization Iteration. (line 43) * gsl_min_fminimizer_goldensection: Minimization Algorithms. (line 13) * gsl_min_fminimizer_iterate: Minimization Iteration. (line 12) * gsl_min_fminimizer_name: Initializing the Minimizer. (line 42) * gsl_min_fminimizer_quad_golden: Minimization Algorithms. (line 46) * gsl_min_fminimizer_set: Initializing the Minimizer. (line 21) * gsl_min_fminimizer_set_with_values: Initializing the Minimizer. (line 31) * gsl_min_fminimizer_x_lower: Minimization Iteration. (line 36) * gsl_min_fminimizer_x_minimum: Minimization Iteration. (line 29) * gsl_min_fminimizer_x_upper: Minimization Iteration. (line 34) * GSL_MIN_INT: Maximum and Minimum functions. (line 33) * GSL_MIN_LDBL: Maximum and Minimum functions. (line 41) * gsl_min_test_interval: Minimization Stopping Parameters. (line 18) * gsl_monte_miser_alloc: MISER. (line 45) * gsl_monte_miser_free: MISER. (line 68) * gsl_monte_miser_init: MISER. (line 50) * gsl_monte_miser_integrate: MISER. (line 55) * gsl_monte_miser_params_get: MISER. (line 75) * gsl_monte_miser_params_set: MISER. (line 80) * gsl_monte_plain_alloc: PLAIN Monte Carlo. (line 29) * gsl_monte_plain_free: PLAIN Monte Carlo. (line 51) * gsl_monte_plain_init: PLAIN Monte Carlo. (line 33) * gsl_monte_plain_integrate: PLAIN Monte Carlo. (line 38) * gsl_monte_vegas_alloc: VEGAS. (line 50) * gsl_monte_vegas_chisq: VEGAS. (line 113) * gsl_monte_vegas_free: VEGAS. (line 77) * gsl_monte_vegas_init: VEGAS. (line 55) * gsl_monte_vegas_integrate: VEGAS. (line 60) * gsl_monte_vegas_params_get: VEGAS. (line 131) * gsl_monte_vegas_params_set: VEGAS. (line 136) * gsl_monte_vegas_runval: VEGAS. (line 122) * gsl_multifit_covar: Computing the covariance matrix of best fit parameters. (line 6) * gsl_multifit_fdfsolver_alloc: Initializing the Nonlinear Least-Squares Solver. (line 16) * gsl_multifit_fdfsolver_dif_df: Finite Difference Jacobian. (line 15) * gsl_multifit_fdfsolver_dif_fdf: Finite Difference Jacobian. (line 27) * gsl_multifit_fdfsolver_driver: High Level Driver. (line 12) * gsl_multifit_fdfsolver_free: Initializing the Nonlinear Least-Squares Solver. (line 46) * gsl_multifit_fdfsolver_iterate: Iteration of the Minimization Algorithm. (line 14) * gsl_multifit_fdfsolver_lmder: Minimization Algorithms using Derivatives. (line 64) * gsl_multifit_fdfsolver_lmsder: Minimization Algorithms using Derivatives. (line 12) * gsl_multifit_fdfsolver_name: Initializing the Nonlinear Least-Squares Solver. (line 52) * gsl_multifit_fdfsolver_position: Iteration of the Minimization Algorithm. (line 43) * gsl_multifit_fdfsolver_set: Initializing the Nonlinear Least-Squares Solver. (line 40) * gsl_multifit_fsolver_alloc: Initializing the Nonlinear Least-Squares Solver. (line 6) * gsl_multifit_fsolver_driver: High Level Driver. (line 9) * gsl_multifit_fsolver_free: Initializing the Nonlinear Least-Squares Solver. (line 45) * gsl_multifit_fsolver_iterate: Iteration of the Minimization Algorithm. (line 12) * gsl_multifit_fsolver_name: Initializing the Nonlinear Least-Squares Solver. (line 50) * gsl_multifit_fsolver_position: Iteration of the Minimization Algorithm. (line 41) * gsl_multifit_fsolver_set: Initializing the Nonlinear Least-Squares Solver. (line 35) * gsl_multifit_gradient: Search Stopping Parameters for Minimization Algorithms. (line 45) * gsl_multifit_linear: Multi-parameter fitting. (line 50) * gsl_multifit_linear_alloc: Multi-parameter fitting. (line 41) * gsl_multifit_linear_est: Multi-parameter fitting. (line 108) * gsl_multifit_linear_free: Multi-parameter fitting. (line 46) * gsl_multifit_linear_residuals: Multi-parameter fitting. (line 116) * gsl_multifit_linear_svd: Multi-parameter fitting. (line 85) * gsl_multifit_linear_usvd: Multi-parameter fitting. (line 97) * gsl_multifit_robust: Robust linear regression. (line 159) * gsl_multifit_robust_alloc: Robust linear regression. (line 70) * gsl_multifit_robust_bisquare: Robust linear regression. (line 81) * gsl_multifit_robust_cauchy: Robust linear regression. (line 90) * gsl_multifit_robust_default: Robust linear regression. (line 76) * gsl_multifit_robust_est: Robust linear regression. (line 173) * gsl_multifit_robust_fair: Robust linear regression. (line 101) * gsl_multifit_robust_free: Robust linear regression. (line 142) * gsl_multifit_robust_huber: Robust linear regression. (line 110) * gsl_multifit_robust_name: Robust linear regression. (line 146) * gsl_multifit_robust_ols: Robust linear regression. (line 123) * gsl_multifit_robust_statistics: Robust linear regression. (line 181) * gsl_multifit_robust_tune: Robust linear regression. (line 151) * gsl_multifit_robust_welsch: Robust linear regression. (line 133) * gsl_multifit_test_delta: Search Stopping Parameters for Minimization Algorithms. (line 19) * gsl_multifit_test_gradient: Search Stopping Parameters for Minimization Algorithms. (line 31) * gsl_multifit_wlinear: Multi-parameter fitting. (line 70) * gsl_multifit_wlinear_svd: Multi-parameter fitting. (line 89) * gsl_multifit_wlinear_usvd: Multi-parameter fitting. (line 101) * gsl_multimin_fdfminimizer_alloc: Initializing the Multidimensional Minimizer. (line 10) * gsl_multimin_fdfminimizer_conjugate_fr: Multimin Algorithms with Derivatives. (line 11) * gsl_multimin_fdfminimizer_conjugate_pr: Multimin Algorithms with Derivatives. (line 28) * gsl_multimin_fdfminimizer_free: Initializing the Multidimensional Minimizer. (line 46) * gsl_multimin_fdfminimizer_gradient: Multimin Iteration. (line 34) * gsl_multimin_fdfminimizer_iterate: Multimin Iteration. (line 11) * gsl_multimin_fdfminimizer_minimum: Multimin Iteration. (line 30) * gsl_multimin_fdfminimizer_name: Initializing the Multidimensional Minimizer. (line 52) * gsl_multimin_fdfminimizer_restart: Multimin Iteration. (line 42) * gsl_multimin_fdfminimizer_set: Initializing the Multidimensional Minimizer. (line 21) * gsl_multimin_fdfminimizer_steepest_descent: Multimin Algorithms with Derivatives. (line 55) * gsl_multimin_fdfminimizer_vector_bfgs: Multimin Algorithms with Derivatives. (line 36) * gsl_multimin_fdfminimizer_vector_bfgs2: Multimin Algorithms with Derivatives. (line 35) * gsl_multimin_fdfminimizer_x: Multimin Iteration. (line 26) * gsl_multimin_fminimizer_alloc: Initializing the Multidimensional Minimizer. (line 13) * gsl_multimin_fminimizer_free: Initializing the Multidimensional Minimizer. (line 48) * gsl_multimin_fminimizer_iterate: Multimin Iteration. (line 13) * gsl_multimin_fminimizer_minimum: Multimin Iteration. (line 32) * gsl_multimin_fminimizer_name: Initializing the Multidimensional Minimizer. (line 54) * gsl_multimin_fminimizer_nmsimplex: Multimin Algorithms without Derivatives. (line 10) * gsl_multimin_fminimizer_nmsimplex2: Multimin Algorithms without Derivatives. (line 9) * gsl_multimin_fminimizer_nmsimplex2rand: Multimin Algorithms without Derivatives. (line 49) * gsl_multimin_fminimizer_set: Initializing the Multidimensional Minimizer. (line 24) * gsl_multimin_fminimizer_size: Multimin Iteration. (line 36) * gsl_multimin_fminimizer_x: Multimin Iteration. (line 28) * gsl_multimin_test_gradient: Multimin Stopping Criteria. (line 18) * gsl_multimin_test_size: Multimin Stopping Criteria. (line 32) * gsl_multiroot_fdfsolver_alloc: Initializing the Multidimensional Solver. (line 27) * gsl_multiroot_fdfsolver_dx: Iteration of the multidimensional solver. (line 48) * gsl_multiroot_fdfsolver_f: Iteration of the multidimensional solver. (line 41) * gsl_multiroot_fdfsolver_free: Initializing the Multidimensional Solver. (line 54) * gsl_multiroot_fdfsolver_gnewton: Algorithms using Derivatives. (line 90) * gsl_multiroot_fdfsolver_hybridj: Algorithms using Derivatives. (line 64) * gsl_multiroot_fdfsolver_hybridsj: Algorithms using Derivatives. (line 13) * gsl_multiroot_fdfsolver_iterate: Iteration of the multidimensional solver. (line 14) * gsl_multiroot_fdfsolver_name: Initializing the Multidimensional Solver. (line 60) * gsl_multiroot_fdfsolver_newton: Algorithms using Derivatives. (line 70) * gsl_multiroot_fdfsolver_root: Iteration of the multidimensional solver. (line 34) * gsl_multiroot_fdfsolver_set: Initializing the Multidimensional Solver. (line 45) * gsl_multiroot_fsolver_alloc: Initializing the Multidimensional Solver. (line 11) * gsl_multiroot_fsolver_broyden: Algorithms without Derivatives. (line 45) * gsl_multiroot_fsolver_dnewton: Algorithms without Derivatives. (line 25) * gsl_multiroot_fsolver_dx: Iteration of the multidimensional solver. (line 46) * gsl_multiroot_fsolver_f: Iteration of the multidimensional solver. (line 39) * gsl_multiroot_fsolver_free: Initializing the Multidimensional Solver. (line 52) * gsl_multiroot_fsolver_hybrid: Algorithms without Derivatives. (line 21) * gsl_multiroot_fsolver_hybrids: Algorithms without Derivatives. (line 13) * gsl_multiroot_fsolver_iterate: Iteration of the multidimensional solver. (line 12) * gsl_multiroot_fsolver_name: Initializing the Multidimensional Solver. (line 58) * gsl_multiroot_fsolver_root: Iteration of the multidimensional solver. (line 32) * gsl_multiroot_fsolver_set: Initializing the Multidimensional Solver. (line 43) * gsl_multiroot_test_delta: Search Stopping Parameters for the multidimensional solver. (line 20) * gsl_multiroot_test_residual: Search Stopping Parameters for the multidimensional solver. (line 32) * gsl_multiset_alloc: Multiset allocation. (line 6) * gsl_multiset_calloc: Multiset allocation. (line 14) * gsl_multiset_data: Multiset properties. (line 12) * gsl_multiset_fprintf: Reading and writing multisets. (line 26) * gsl_multiset_fread: Reading and writing multisets. (line 17) * gsl_multiset_free: Multiset allocation. (line 28) * gsl_multiset_fscanf: Reading and writing multisets. (line 35) * gsl_multiset_fwrite: Reading and writing multisets. (line 9) * gsl_multiset_get: Accessing multiset elements. (line 8) * gsl_multiset_init_first: Multiset allocation. (line 20) * gsl_multiset_init_last: Multiset allocation. (line 24) * gsl_multiset_k: Multiset properties. (line 9) * gsl_multiset_memcpy: Multiset allocation. (line 31) * gsl_multiset_n: Multiset properties. (line 6) * gsl_multiset_next: Multiset functions. (line 6) * gsl_multiset_prev: Multiset functions. (line 14) * gsl_multiset_valid: Multiset properties. (line 16) * gsl_ntuple_bookdata: Writing ntuples. (line 10) * gsl_ntuple_close: Closing an ntuple file. (line 6) * gsl_ntuple_create: Creating ntuples. (line 6) * gsl_ntuple_open: Opening an existing ntuple file. (line 6) * gsl_ntuple_project: Histogramming ntuple values. (line 35) * gsl_ntuple_read: Reading ntuples. (line 6) * gsl_ntuple_write: Writing ntuples. (line 6) * gsl_odeiv2_control_alloc: Adaptive Step-size Control. (line 73) * gsl_odeiv2_control_errlevel: Adaptive Step-size Control. (line 115) * gsl_odeiv2_control_free: Adaptive Step-size Control. (line 86) * gsl_odeiv2_control_hadjust: Adaptive Step-size Control. (line 90) * gsl_odeiv2_control_init: Adaptive Step-size Control. (line 80) * gsl_odeiv2_control_name: Adaptive Step-size Control. (line 105) * gsl_odeiv2_control_scaled_new: Adaptive Step-size Control. (line 60) * gsl_odeiv2_control_set_driver: Adaptive Step-size Control. (line 122) * gsl_odeiv2_control_standard_new: Adaptive Step-size Control. (line 10) * gsl_odeiv2_control_yp_new: Adaptive Step-size Control. (line 52) * gsl_odeiv2_control_y_new: Adaptive Step-size Control. (line 44) * gsl_odeiv2_driver_alloc_scaled_new: Driver. (line 19) * gsl_odeiv2_driver_alloc_standard_new: Driver. (line 15) * gsl_odeiv2_driver_alloc_yp_new: Driver. (line 12) * gsl_odeiv2_driver_alloc_y_new: Driver. (line 9) * gsl_odeiv2_driver_apply: Driver. (line 46) * gsl_odeiv2_driver_apply_fixed_step: Driver. (line 61) * gsl_odeiv2_driver_free: Driver. (line 78) * gsl_odeiv2_driver_reset: Driver. (line 69) * gsl_odeiv2_driver_reset_hstart: Driver. (line 72) * gsl_odeiv2_driver_set_hmax: Driver. (line 36) * gsl_odeiv2_driver_set_hmin: Driver. (line 31) * gsl_odeiv2_driver_set_nmax: Driver. (line 41) * gsl_odeiv2_evolve_alloc: Evolution. (line 10) * gsl_odeiv2_evolve_apply: Evolution. (line 14) * gsl_odeiv2_evolve_apply_fixed_step: Evolution. (line 52) * gsl_odeiv2_evolve_free: Evolution. (line 68) * gsl_odeiv2_evolve_reset: Evolution. (line 63) * gsl_odeiv2_evolve_set_driver: Evolution. (line 72) * gsl_odeiv2_step_alloc: Stepping Functions. (line 10) * gsl_odeiv2_step_apply: Stepping Functions. (line 51) * gsl_odeiv2_step_bsimp: Stepping Functions. (line 122) * gsl_odeiv2_step_free: Stepping Functions. (line 23) * gsl_odeiv2_step_msadams: Stepping Functions. (line 127) * gsl_odeiv2_step_msbdf: Stepping Functions. (line 135) * gsl_odeiv2_step_name: Stepping Functions. (line 27) * gsl_odeiv2_step_order: Stepping Functions. (line 37) * gsl_odeiv2_step_reset: Stepping Functions. (line 18) * gsl_odeiv2_step_rk1imp: Stepping Functions. (line 104) * gsl_odeiv2_step_rk2: Stepping Functions. (line 86) * gsl_odeiv2_step_rk2imp: Stepping Functions. (line 110) * gsl_odeiv2_step_rk4: Stepping Functions. (line 89) * gsl_odeiv2_step_rk4imp: Stepping Functions. (line 116) * gsl_odeiv2_step_rk8pd: Stepping Functions. (line 101) * gsl_odeiv2_step_rkck: Stepping Functions. (line 98) * gsl_odeiv2_step_rkf45: Stepping Functions. (line 94) * gsl_odeiv2_step_set_driver: Stepping Functions. (line 43) * gsl_permutation_alloc: Permutation allocation. (line 6) * gsl_permutation_calloc: Permutation allocation. (line 14) * gsl_permutation_canonical_cycles: Permutations in cyclic form. (line 67) * gsl_permutation_canonical_to_linear: Permutations in cyclic form. (line 49) * gsl_permutation_data: Permutation properties. (line 9) * gsl_permutation_fprintf: Reading and writing permutations. (line 27) * gsl_permutation_fread: Reading and writing permutations. (line 17) * gsl_permutation_free: Permutation allocation. (line 23) * gsl_permutation_fscanf: Reading and writing permutations. (line 36) * gsl_permutation_fwrite: Reading and writing permutations. (line 9) * gsl_permutation_get: Accessing permutation elements. (line 9) * gsl_permutation_init: Permutation allocation. (line 19) * gsl_permutation_inverse: Permutation functions. (line 9) * gsl_permutation_inversions: Permutations in cyclic form. (line 54) * gsl_permutation_linear_cycles: Permutations in cyclic form. (line 62) * gsl_permutation_linear_to_canonical: Permutations in cyclic form. (line 44) * gsl_permutation_memcpy: Permutation allocation. (line 26) * gsl_permutation_mul: Applying Permutations. (line 35) * gsl_permutation_next: Permutation functions. (line 14) * gsl_permutation_prev: Permutation functions. (line 22) * gsl_permutation_reverse: Permutation functions. (line 6) * gsl_permutation_size: Permutation properties. (line 6) * gsl_permutation_swap: Accessing permutation elements. (line 16) * gsl_permutation_valid: Permutation properties. (line 13) * gsl_permute: Applying Permutations. (line 6) * gsl_permute_inverse: Applying Permutations. (line 11) * gsl_permute_vector: Applying Permutations. (line 16) * gsl_permute_vector_inverse: Applying Permutations. (line 25) * gsl_poly_complex_eval: Polynomial Evaluation. (line 16) * gsl_poly_complex_solve: General Polynomial Equations. (line 25) * gsl_poly_complex_solve_cubic: Cubic Equations. (line 24) * gsl_poly_complex_solve_quadratic: Quadratic Equations. (line 28) * gsl_poly_complex_workspace_alloc: General Polynomial Equations. (line 11) * gsl_poly_complex_workspace_free: General Polynomial Equations. (line 21) * gsl_poly_dd_eval: Divided Difference Representation of Polynomials. (line 38) * gsl_poly_dd_hermite_init: Divided Difference Representation of Polynomials. (line 54) * gsl_poly_dd_init: Divided Difference Representation of Polynomials. (line 30) * gsl_poly_dd_taylor: Divided Difference Representation of Polynomials. (line 45) * gsl_poly_eval: Polynomial Evaluation. (line 11) * gsl_poly_eval_derivs: Polynomial Evaluation. (line 26) * gsl_poly_solve_cubic: Cubic Equations. (line 6) * gsl_poly_solve_quadratic: Quadratic Equations. (line 6) * gsl_pow_2: Small integer powers. (line 19) * gsl_pow_3: Small integer powers. (line 20) * gsl_pow_4: Small integer powers. (line 21) * gsl_pow_5: Small integer powers. (line 22) * gsl_pow_6: Small integer powers. (line 23) * gsl_pow_7: Small integer powers. (line 24) * gsl_pow_8: Small integer powers. (line 25) * gsl_pow_9: Small integer powers. (line 26) * gsl_pow_int: Small integer powers. (line 11) * gsl_pow_uint: Small integer powers. (line 12) * gsl_qrng_alloc: Quasi-random number generator initialization. (line 6) * gsl_qrng_clone: Saving and resorting quasi-random number generator state. (line 12) * gsl_qrng_free: Quasi-random number generator initialization. (line 14) * gsl_qrng_get: Sampling from a quasi-random number generator. (line 6) * gsl_qrng_halton: Quasi-random number generator algorithms. (line 18) * gsl_qrng_init: Quasi-random number generator initialization. (line 17) * gsl_qrng_memcpy: Saving and resorting quasi-random number generator state. (line 6) * gsl_qrng_name: Auxiliary quasi-random number generator functions. (line 6) * gsl_qrng_niederreiter_2: Quasi-random number generator algorithms. (line 8) * gsl_qrng_reversehalton: Quasi-random number generator algorithms. (line 19) * gsl_qrng_size: Auxiliary quasi-random number generator functions. (line 9) * gsl_qrng_sobol: Quasi-random number generator algorithms. (line 13) * gsl_qrng_state: Auxiliary quasi-random number generator functions. (line 10) * gsl_ran_bernoulli: The Bernoulli Distribution. (line 6) * gsl_ran_bernoulli_pdf: The Bernoulli Distribution. (line 15) * gsl_ran_beta: The Beta Distribution. (line 6) * gsl_ran_beta_pdf: The Beta Distribution. (line 15) * gsl_ran_binomial: The Binomial Distribution. (line 6) * gsl_ran_binomial_pdf: The Binomial Distribution. (line 17) * gsl_ran_bivariate_gaussian: The Bivariate Gaussian Distribution. (line 6) * gsl_ran_bivariate_gaussian_pdf: The Bivariate Gaussian Distribution. (line 18) * gsl_ran_cauchy: The Cauchy Distribution. (line 6) * gsl_ran_cauchy_pdf: The Cauchy Distribution. (line 16) * gsl_ran_chisq: The Chi-squared Distribution. (line 14) * gsl_ran_chisq_pdf: The Chi-squared Distribution. (line 23) * gsl_ran_choose: Shuffling and Sampling. (line 33) * gsl_ran_dirichlet: The Dirichlet Distribution. (line 6) * gsl_ran_dirichlet_lnpdf: The Dirichlet Distribution. (line 30) * gsl_ran_dirichlet_pdf: The Dirichlet Distribution. (line 24) * gsl_ran_dir_2d: Spherical Vector Distributions. (line 10) * gsl_ran_dir_2d_trig_method: Spherical Vector Distributions. (line 12) * gsl_ran_dir_3d: Spherical Vector Distributions. (line 31) * gsl_ran_dir_nd: Spherical Vector Distributions. (line 40) * gsl_ran_discrete: General Discrete Distributions. (line 59) * gsl_ran_discrete_free: General Discrete Distributions. (line 72) * gsl_ran_discrete_pdf: General Discrete Distributions. (line 64) * gsl_ran_discrete_preproc: General Discrete Distributions. (line 49) * gsl_ran_exponential: The Exponential Distribution. (line 6) * gsl_ran_exponential_pdf: The Exponential Distribution. (line 14) * gsl_ran_exppow: The Exponential Power Distribution. (line 6) * gsl_ran_exppow_pdf: The Exponential Power Distribution. (line 18) * gsl_ran_fdist: The F-distribution. (line 13) * gsl_ran_fdist_pdf: The F-distribution. (line 26) * gsl_ran_flat: The Flat (Uniform) Distribution. (line 6) * gsl_ran_flat_pdf: The Flat (Uniform) Distribution. (line 15) * gsl_ran_gamma: The Gamma Distribution. (line 6) * gsl_ran_gamma_knuth: The Gamma Distribution. (line 22) * gsl_ran_gamma_pdf: The Gamma Distribution. (line 27) * gsl_ran_gaussian: The Gaussian Distribution. (line 6) * gsl_ran_gaussian_pdf: The Gaussian Distribution. (line 19) * gsl_ran_gaussian_ratio_method: The Gaussian Distribution. (line 27) * gsl_ran_gaussian_tail: The Gaussian Tail Distribution. (line 6) * gsl_ran_gaussian_tail_pdf: The Gaussian Tail Distribution. (line 23) * gsl_ran_gaussian_ziggurat: The Gaussian Distribution. (line 25) * gsl_ran_geometric: The Geometric Distribution. (line 6) * gsl_ran_geometric_pdf: The Geometric Distribution. (line 19) * gsl_ran_gumbel1: The Type-1 Gumbel Distribution. (line 6) * gsl_ran_gumbel1_pdf: The Type-1 Gumbel Distribution. (line 15) * gsl_ran_gumbel2: The Type-2 Gumbel Distribution. (line 6) * gsl_ran_gumbel2_pdf: The Type-2 Gumbel Distribution. (line 15) * gsl_ran_hypergeometric: The Hypergeometric Distribution. (line 6) * gsl_ran_hypergeometric_pdf: The Hypergeometric Distribution. (line 22) * gsl_ran_landau: The Landau Distribution. (line 6) * gsl_ran_landau_pdf: The Landau Distribution. (line 17) * gsl_ran_laplace: The Laplace Distribution. (line 6) * gsl_ran_laplace_pdf: The Laplace Distribution. (line 14) * gsl_ran_levy: The Levy alpha-Stable Distributions. (line 6) * gsl_ran_levy_skew: The Levy skew alpha-Stable Distribution. (line 6) * gsl_ran_logarithmic: The Logarithmic Distribution. (line 6) * gsl_ran_logarithmic_pdf: The Logarithmic Distribution. (line 16) * gsl_ran_logistic: The Logistic Distribution. (line 6) * gsl_ran_logistic_pdf: The Logistic Distribution. (line 14) * gsl_ran_lognormal: The Lognormal Distribution. (line 6) * gsl_ran_lognormal_pdf: The Lognormal Distribution. (line 15) * gsl_ran_multinomial: The Multinomial Distribution. (line 6) * gsl_ran_multinomial_lnpdf: The Multinomial Distribution. (line 34) * gsl_ran_multinomial_pdf: The Multinomial Distribution. (line 28) * gsl_ran_negative_binomial: The Negative Binomial Distribution. (line 6) * gsl_ran_negative_binomial_pdf: The Negative Binomial Distribution. (line 17) * gsl_ran_pareto: The Pareto Distribution. (line 6) * gsl_ran_pareto_pdf: The Pareto Distribution. (line 15) * gsl_ran_pascal: The Pascal Distribution. (line 6) * gsl_ran_pascal_pdf: The Pascal Distribution. (line 16) * gsl_ran_poisson: The Poisson Distribution. (line 6) * gsl_ran_poisson_pdf: The Poisson Distribution. (line 16) * gsl_ran_rayleigh: The Rayleigh Distribution. (line 6) * gsl_ran_rayleigh_pdf: The Rayleigh Distribution. (line 14) * gsl_ran_rayleigh_tail: The Rayleigh Tail Distribution. (line 6) * gsl_ran_rayleigh_tail_pdf: The Rayleigh Tail Distribution. (line 16) * gsl_ran_sample: Shuffling and Sampling. (line 59) * gsl_ran_shuffle: Shuffling and Sampling. (line 13) * gsl_ran_tdist: The t-distribution. (line 14) * gsl_ran_tdist_pdf: The t-distribution. (line 23) * gsl_ran_ugaussian: The Gaussian Distribution. (line 34) * gsl_ran_ugaussian_pdf: The Gaussian Distribution. (line 35) * gsl_ran_ugaussian_ratio_method: The Gaussian Distribution. (line 36) * gsl_ran_ugaussian_tail: The Gaussian Tail Distribution. (line 30) * gsl_ran_ugaussian_tail_pdf: The Gaussian Tail Distribution. (line 32) * gsl_ran_weibull: The Weibull Distribution. (line 6) * gsl_ran_weibull_pdf: The Weibull Distribution. (line 15) * GSL_REAL: Representation of complex numbers. (line 34) * gsl_rng_alloc: Random number generator initialization. (line 6) * gsl_rng_borosh13: Other random number generators. (line 174) * gsl_rng_clone: Copying random number generator state. (line 16) * gsl_rng_cmrg: Random number generator algorithms. (line 98) * gsl_rng_coveyou: Other random number generators. (line 201) * gsl_rng_env_setup: Random number environment variables. (line 11) * gsl_rng_fishman18: Other random number generators. (line 175) * gsl_rng_fishman20: Other random number generators. (line 176) * gsl_rng_fishman2x: Other random number generators. (line 191) * gsl_rng_fread: Reading and writing random number generator state. (line 17) * gsl_rng_free: Random number generator initialization. (line 45) * gsl_rng_fwrite: Reading and writing random number generator state. (line 9) * gsl_rng_get: Sampling from a random number generator. (line 11) * gsl_rng_gfsr4: Random number generator algorithms. (line 173) * gsl_rng_knuthran: Other random number generators. (line 167) * gsl_rng_knuthran2: Other random number generators. (line 157) * gsl_rng_knuthran2002: Other random number generators. (line 166) * gsl_rng_lecuyer21: Other random number generators. (line 177) * gsl_rng_max: Auxiliary random number generator functions. (line 19) * gsl_rng_memcpy: Copying random number generator state. (line 11) * gsl_rng_min: Auxiliary random number generator functions. (line 23) * gsl_rng_minstd: Other random number generators. (line 113) * gsl_rng_mrg: Random number generator algorithms. (line 120) * gsl_rng_mt19937: Random number generator algorithms. (line 19) * gsl_rng_name: Auxiliary random number generator functions. (line 10) * gsl_rng_r250: Other random number generators. (line 59) * gsl_rng_rand: Unix random number generators. (line 17) * gsl_rng_rand48: Unix random number generators. (line 57) * gsl_rng_random_bsd: Unix random number generators. (line 26) * gsl_rng_random_glibc2: Unix random number generators. (line 28) * gsl_rng_random_libc5: Unix random number generators. (line 27) * gsl_rng_randu: Other random number generators. (line 104) * gsl_rng_ranf: Other random number generators. (line 22) * gsl_rng_ranlux: Random number generator algorithms. (line 71) * gsl_rng_ranlux389: Random number generator algorithms. (line 72) * gsl_rng_ranlxd1: Random number generator algorithms. (line 64) * gsl_rng_ranlxd2: Random number generator algorithms. (line 65) * gsl_rng_ranlxs0: Random number generator algorithms. (line 44) * gsl_rng_ranlxs1: Random number generator algorithms. (line 45) * gsl_rng_ranlxs2: Random number generator algorithms. (line 46) * gsl_rng_ranmar: Other random number generators. (line 53) * gsl_rng_set: Random number generator initialization. (line 25) * gsl_rng_size: Auxiliary random number generator functions. (line 30) * gsl_rng_slatec: Other random number generators. (line 140) * gsl_rng_state: Auxiliary random number generator functions. (line 29) * gsl_rng_taus: Random number generator algorithms. (line 137) * gsl_rng_taus2: Random number generator algorithms. (line 138) * gsl_rng_transputer: Other random number generators. (line 95) * gsl_rng_tt800: Other random number generators. (line 74) * gsl_rng_types_setup: Auxiliary random number generator functions. (line 40) * gsl_rng_uni: Other random number generators. (line 133) * gsl_rng_uni32: Other random number generators. (line 134) * gsl_rng_uniform: Sampling from a random number generator. (line 18) * gsl_rng_uniform_int: Sampling from a random number generator. (line 36) * gsl_rng_uniform_pos: Sampling from a random number generator. (line 28) * gsl_rng_vax: Other random number generators. (line 86) * gsl_rng_waterman14: Other random number generators. (line 178) * gsl_rng_zuf: Other random number generators. (line 144) * gsl_root_fdfsolver_alloc: Initializing the Solver. (line 21) * gsl_root_fdfsolver_free: Initializing the Solver. (line 48) * gsl_root_fdfsolver_iterate: Root Finding Iteration. (line 13) * gsl_root_fdfsolver_name: Initializing the Solver. (line 53) * gsl_root_fdfsolver_newton: Root Finding Algorithms using Derivatives. (line 15) * gsl_root_fdfsolver_root: Root Finding Iteration. (line 33) * gsl_root_fdfsolver_secant: Root Finding Algorithms using Derivatives. (line 29) * gsl_root_fdfsolver_set: Initializing the Solver. (line 42) * gsl_root_fdfsolver_steffenson: Root Finding Algorithms using Derivatives. (line 60) * gsl_root_fsolver_alloc: Initializing the Solver. (line 6) * gsl_root_fsolver_bisection: Root Bracketing Algorithms. (line 16) * gsl_root_fsolver_brent: Root Bracketing Algorithms. (line 49) * gsl_root_fsolver_falsepos: Root Bracketing Algorithms. (line 32) * gsl_root_fsolver_free: Initializing the Solver. (line 47) * gsl_root_fsolver_iterate: Root Finding Iteration. (line 12) * gsl_root_fsolver_name: Initializing the Solver. (line 51) * gsl_root_fsolver_root: Root Finding Iteration. (line 32) * gsl_root_fsolver_set: Initializing the Solver. (line 36) * gsl_root_fsolver_x_lower: Root Finding Iteration. (line 38) * gsl_root_fsolver_x_upper: Root Finding Iteration. (line 40) * gsl_root_test_delta: Search Stopping Parameters. (line 41) * gsl_root_test_interval: Search Stopping Parameters. (line 19) * gsl_root_test_residual: Search Stopping Parameters. (line 52) * GSL_SET_COMPLEX: Representation of complex numbers. (line 39) * gsl_set_error_handler: Error Handlers. (line 43) * gsl_set_error_handler_off: Error Handlers. (line 69) * GSL_SET_IMAG: Representation of complex numbers. (line 49) * GSL_SET_REAL: Representation of complex numbers. (line 48) * gsl_sf_airy_Ai: Airy Functions. (line 6) * gsl_sf_airy_Ai_deriv: Derivatives of Airy Functions. (line 6) * gsl_sf_airy_Ai_deriv_e: Derivatives of Airy Functions. (line 7) * gsl_sf_airy_Ai_deriv_scaled: Derivatives of Airy Functions. (line 18) * gsl_sf_airy_Ai_deriv_scaled_e: Derivatives of Airy Functions. (line 20) * gsl_sf_airy_Ai_e: Airy Functions. (line 7) * gsl_sf_airy_Ai_scaled: Airy Functions. (line 18) * gsl_sf_airy_Ai_scaled_e: Airy Functions. (line 19) * gsl_sf_airy_Bi: Airy Functions. (line 12) * gsl_sf_airy_Bi_deriv: Derivatives of Airy Functions. (line 12) * gsl_sf_airy_Bi_deriv_e: Derivatives of Airy Functions. (line 13) * gsl_sf_airy_Bi_deriv_scaled: Derivatives of Airy Functions. (line 26) * gsl_sf_airy_Bi_deriv_scaled_e: Derivatives of Airy Functions. (line 28) * gsl_sf_airy_Bi_e: Airy Functions. (line 13) * gsl_sf_airy_Bi_scaled: Airy Functions. (line 25) * gsl_sf_airy_Bi_scaled_e: Airy Functions. (line 26) * gsl_sf_airy_zero_Ai: Zeros of Airy Functions. (line 6) * gsl_sf_airy_zero_Ai_deriv: Zeros of Derivatives of Airy Functions. (line 6) * gsl_sf_airy_zero_Ai_deriv_e: Zeros of Derivatives of Airy Functions. (line 7) * gsl_sf_airy_zero_Ai_e: Zeros of Airy Functions. (line 7) * gsl_sf_airy_zero_Bi: Zeros of Airy Functions. (line 12) * gsl_sf_airy_zero_Bi_deriv: Zeros of Derivatives of Airy Functions. (line 12) * gsl_sf_airy_zero_Bi_deriv_e: Zeros of Derivatives of Airy Functions. (line 13) * gsl_sf_airy_zero_Bi_e: Zeros of Airy Functions. (line 13) * gsl_sf_angle_restrict_pos: Restriction Functions. (line 15) * gsl_sf_angle_restrict_pos_e: Restriction Functions. (line 16) * gsl_sf_angle_restrict_symm: Restriction Functions. (line 6) * gsl_sf_angle_restrict_symm_e: Restriction Functions. (line 7) * gsl_sf_atanint: Arctangent Integral. (line 6) * gsl_sf_atanint_e: Arctangent Integral. (line 7) * gsl_sf_bessel_I0: Regular Modified Cylindrical Bessel Functions. (line 6) * gsl_sf_bessel_I0_e: Regular Modified Cylindrical Bessel Functions. (line 7) * gsl_sf_bessel_I0_scaled: Regular Modified Cylindrical Bessel Functions. (line 31) * gsl_sf_bessel_i0_scaled: Regular Modified Spherical Bessel Functions. (line 10) * gsl_sf_bessel_I0_scaled_e: Regular Modified Cylindrical Bessel Functions. (line 32) * gsl_sf_bessel_i0_scaled_e: Regular Modified Spherical Bessel Functions. (line 11) * gsl_sf_bessel_I1: Regular Modified Cylindrical Bessel Functions. (line 11) * gsl_sf_bessel_I1_e: Regular Modified Cylindrical Bessel Functions. (line 12) * gsl_sf_bessel_I1_scaled: Regular Modified Cylindrical Bessel Functions. (line 37) * gsl_sf_bessel_i1_scaled: Regular Modified Spherical Bessel Functions. (line 16) * gsl_sf_bessel_I1_scaled_e: Regular Modified Cylindrical Bessel Functions. (line 38) * gsl_sf_bessel_i1_scaled_e: Regular Modified Spherical Bessel Functions. (line 17) * gsl_sf_bessel_i2_scaled: Regular Modified Spherical Bessel Functions. (line 22) * gsl_sf_bessel_i2_scaled_e: Regular Modified Spherical Bessel Functions. (line 23) * gsl_sf_bessel_il_scaled: Regular Modified Spherical Bessel Functions. (line 28) * gsl_sf_bessel_il_scaled_array: Regular Modified Spherical Bessel Functions. (line 34) * gsl_sf_bessel_il_scaled_e: Regular Modified Spherical Bessel Functions. (line 29) * gsl_sf_bessel_In: Regular Modified Cylindrical Bessel Functions. (line 16) * gsl_sf_bessel_Inu: Regular Modified Bessel Functions - Fractional Order. (line 6) * gsl_sf_bessel_Inu_e: Regular Modified Bessel Functions - Fractional Order. (line 7) * gsl_sf_bessel_Inu_scaled: Regular Modified Bessel Functions - Fractional Order. (line 12) * gsl_sf_bessel_Inu_scaled_e: Regular Modified Bessel Functions - Fractional Order. (line 13) * gsl_sf_bessel_In_array: Regular Modified Cylindrical Bessel Functions. (line 22) * gsl_sf_bessel_In_e: Regular Modified Cylindrical Bessel Functions. (line 17) * gsl_sf_bessel_In_scaled: Regular Modified Cylindrical Bessel Functions. (line 43) * gsl_sf_bessel_In_scaled_array: Regular Modified Cylindrical Bessel Functions. (line 49) * gsl_sf_bessel_In_scaled_e: Regular Modified Cylindrical Bessel Functions. (line 44) * gsl_sf_bessel_J0: Regular Cylindrical Bessel Functions. (line 6) * gsl_sf_bessel_j0: Regular Spherical Bessel Functions. (line 6) * gsl_sf_bessel_J0_e: Regular Cylindrical Bessel Functions. (line 7) * gsl_sf_bessel_j0_e: Regular Spherical Bessel Functions. (line 7) * gsl_sf_bessel_J1: Regular Cylindrical Bessel Functions. (line 11) * gsl_sf_bessel_j1: Regular Spherical Bessel Functions. (line 11) * gsl_sf_bessel_J1_e: Regular Cylindrical Bessel Functions. (line 12) * gsl_sf_bessel_j1_e: Regular Spherical Bessel Functions. (line 12) * gsl_sf_bessel_j2: Regular Spherical Bessel Functions. (line 16) * gsl_sf_bessel_j2_e: Regular Spherical Bessel Functions. (line 17) * gsl_sf_bessel_jl: Regular Spherical Bessel Functions. (line 21) * gsl_sf_bessel_jl_array: Regular Spherical Bessel Functions. (line 27) * gsl_sf_bessel_jl_e: Regular Spherical Bessel Functions. (line 22) * gsl_sf_bessel_jl_steed_array: Regular Spherical Bessel Functions. (line 35) * gsl_sf_bessel_Jn: Regular Cylindrical Bessel Functions. (line 16) * gsl_sf_bessel_Jnu: Regular Bessel Function - Fractional Order. (line 6) * gsl_sf_bessel_Jnu_e: Regular Bessel Function - Fractional Order. (line 7) * gsl_sf_bessel_Jn_array: Regular Cylindrical Bessel Functions. (line 22) * gsl_sf_bessel_Jn_e: Regular Cylindrical Bessel Functions. (line 17) * gsl_sf_bessel_K0: Irregular Modified Cylindrical Bessel Functions. (line 6) * gsl_sf_bessel_K0_e: Irregular Modified Cylindrical Bessel Functions. (line 7) * gsl_sf_bessel_K0_scaled: Irregular Modified Cylindrical Bessel Functions. (line 32) * gsl_sf_bessel_k0_scaled: Irregular Modified Spherical Bessel Functions. (line 10) * gsl_sf_bessel_K0_scaled_e: Irregular Modified Cylindrical Bessel Functions. (line 33) * gsl_sf_bessel_k0_scaled_e: Irregular Modified Spherical Bessel Functions. (line 11) * gsl_sf_bessel_K1: Irregular Modified Cylindrical Bessel Functions. (line 11) * gsl_sf_bessel_K1_e: Irregular Modified Cylindrical Bessel Functions. (line 12) * gsl_sf_bessel_K1_scaled: Irregular Modified Cylindrical Bessel Functions. (line 38) * gsl_sf_bessel_k1_scaled: Irregular Modified Spherical Bessel Functions. (line 16) * gsl_sf_bessel_K1_scaled_e: Irregular Modified Cylindrical Bessel Functions. (line 39) * gsl_sf_bessel_k1_scaled_e: Irregular Modified Spherical Bessel Functions. (line 17) * gsl_sf_bessel_k2_scaled: Irregular Modified Spherical Bessel Functions. (line 22) * gsl_sf_bessel_k2_scaled_e: Irregular Modified Spherical Bessel Functions. (line 23) * gsl_sf_bessel_kl_scaled: Irregular Modified Spherical Bessel Functions. (line 28) * gsl_sf_bessel_kl_scaled_array: Irregular Modified Spherical Bessel Functions. (line 34) * gsl_sf_bessel_kl_scaled_e: Irregular Modified Spherical Bessel Functions. (line 29) * gsl_sf_bessel_Kn: Irregular Modified Cylindrical Bessel Functions. (line 16) * gsl_sf_bessel_Knu: Irregular Modified Bessel Functions - Fractional Order. (line 6) * gsl_sf_bessel_Knu_e: Irregular Modified Bessel Functions - Fractional Order. (line 7) * gsl_sf_bessel_Knu_scaled: Irregular Modified Bessel Functions - Fractional Order. (line 19) * gsl_sf_bessel_Knu_scaled_e: Irregular Modified Bessel Functions - Fractional Order. (line 20) * gsl_sf_bessel_Kn_array: Irregular Modified Cylindrical Bessel Functions. (line 22) * gsl_sf_bessel_Kn_e: Irregular Modified Cylindrical Bessel Functions. (line 17) * gsl_sf_bessel_Kn_scaled: Irregular Modified Cylindrical Bessel Functions. (line 44) * gsl_sf_bessel_Kn_scaled_array: Irregular Modified Cylindrical Bessel Functions. (line 50) * gsl_sf_bessel_Kn_scaled_e: Irregular Modified Cylindrical Bessel Functions. (line 45) * gsl_sf_bessel_lnKnu: Irregular Modified Bessel Functions - Fractional Order. (line 12) * gsl_sf_bessel_lnKnu_e: Irregular Modified Bessel Functions - Fractional Order. (line 13) * gsl_sf_bessel_sequence_Jnu_e: Regular Bessel Function - Fractional Order. (line 12) * gsl_sf_bessel_Y0: Irregular Cylindrical Bessel Functions. (line 6) * gsl_sf_bessel_y0: Irregular Spherical Bessel Functions. (line 6) * gsl_sf_bessel_Y0_e: Irregular Cylindrical Bessel Functions. (line 7) * gsl_sf_bessel_y0_e: Irregular Spherical Bessel Functions. (line 7) * gsl_sf_bessel_Y1: Irregular Cylindrical Bessel Functions. (line 11) * gsl_sf_bessel_y1: Irregular Spherical Bessel Functions. (line 11) * gsl_sf_bessel_Y1_e: Irregular Cylindrical Bessel Functions. (line 12) * gsl_sf_bessel_y1_e: Irregular Spherical Bessel Functions. (line 12) * gsl_sf_bessel_y2: Irregular Spherical Bessel Functions. (line 16) * gsl_sf_bessel_y2_e: Irregular Spherical Bessel Functions. (line 17) * gsl_sf_bessel_yl: Irregular Spherical Bessel Functions. (line 21) * gsl_sf_bessel_yl_array: Irregular Spherical Bessel Functions. (line 27) * gsl_sf_bessel_yl_e: Irregular Spherical Bessel Functions. (line 22) * gsl_sf_bessel_Yn: Irregular Cylindrical Bessel Functions. (line 16) * gsl_sf_bessel_Ynu: Irregular Bessel Functions - Fractional Order. (line 6) * gsl_sf_bessel_Ynu_e: Irregular Bessel Functions - Fractional Order. (line 7) * gsl_sf_bessel_Yn_array: Irregular Cylindrical Bessel Functions. (line 22) * gsl_sf_bessel_Yn_e: Irregular Cylindrical Bessel Functions. (line 17) * gsl_sf_bessel_zero_J0: Zeros of Regular Bessel Functions. (line 6) * gsl_sf_bessel_zero_J0_e: Zeros of Regular Bessel Functions. (line 7) * gsl_sf_bessel_zero_J1: Zeros of Regular Bessel Functions. (line 12) * gsl_sf_bessel_zero_J1_e: Zeros of Regular Bessel Functions. (line 13) * gsl_sf_bessel_zero_Jnu: Zeros of Regular Bessel Functions. (line 18) * gsl_sf_bessel_zero_Jnu_e: Zeros of Regular Bessel Functions. (line 19) * gsl_sf_beta: Beta Functions. (line 6) * gsl_sf_beta_e: Beta Functions. (line 7) * gsl_sf_beta_inc: Incomplete Beta Function. (line 6) * gsl_sf_beta_inc_e: Incomplete Beta Function. (line 7) * gsl_sf_Chi: Hyperbolic Integrals. (line 11) * gsl_sf_Chi_e: Hyperbolic Integrals. (line 12) * gsl_sf_choose: Factorials. (line 38) * gsl_sf_choose_e: Factorials. (line 39) * gsl_sf_Ci: Trigonometric Integrals. (line 11) * gsl_sf_Ci_e: Trigonometric Integrals. (line 12) * gsl_sf_clausen: Clausen Functions. (line 14) * gsl_sf_clausen_e: Clausen Functions. (line 15) * gsl_sf_complex_cos_e: Trigonometric Functions for Complex Arguments. (line 11) * gsl_sf_complex_dilog_e: Complex Argument. (line 6) * gsl_sf_complex_logsin_e: Trigonometric Functions for Complex Arguments. (line 16) * gsl_sf_complex_log_e: Logarithm and Related Functions. (line 19) * gsl_sf_complex_sin_e: Trigonometric Functions for Complex Arguments. (line 6) * gsl_sf_conicalP_0: Conical Functions. (line 22) * gsl_sf_conicalP_0_e: Conical Functions. (line 23) * gsl_sf_conicalP_1: Conical Functions. (line 28) * gsl_sf_conicalP_1_e: Conical Functions. (line 29) * gsl_sf_conicalP_cyl_reg: Conical Functions. (line 41) * gsl_sf_conicalP_cyl_reg_e: Conical Functions. (line 43) * gsl_sf_conicalP_half: Conical Functions. (line 10) * gsl_sf_conicalP_half_e: Conical Functions. (line 11) * gsl_sf_conicalP_mhalf: Conical Functions. (line 16) * gsl_sf_conicalP_mhalf_e: Conical Functions. (line 17) * gsl_sf_conicalP_sph_reg: Conical Functions. (line 34) * gsl_sf_conicalP_sph_reg_e: Conical Functions. (line 36) * gsl_sf_cos: Circular Trigonometric Functions. (line 10) * gsl_sf_cos_e: Circular Trigonometric Functions. (line 11) * gsl_sf_cos_err_e: Trigonometric Functions With Error Estimates. (line 13) * gsl_sf_coulomb_CL_array: Coulomb Wave Function Normalization Constant. (line 14) * gsl_sf_coulomb_CL_e: Coulomb Wave Function Normalization Constant. (line 9) * gsl_sf_coulomb_wave_FGp_array: Coulomb Wave Functions. (line 47) * gsl_sf_coulomb_wave_FG_array: Coulomb Wave Functions. (line 39) * gsl_sf_coulomb_wave_FG_e: Coulomb Wave Functions. (line 19) * gsl_sf_coulomb_wave_F_array: Coulomb Wave Functions. (line 33) * gsl_sf_coulomb_wave_sphF_array: Coulomb Wave Functions. (line 57) * gsl_sf_coupling_3j: 3-j Symbols. (line 6) * gsl_sf_coupling_3j_e: 3-j Symbols. (line 8) * gsl_sf_coupling_6j: 6-j Symbols. (line 6) * gsl_sf_coupling_6j_e: 6-j Symbols. (line 8) * gsl_sf_coupling_9j: 9-j Symbols. (line 6) * gsl_sf_coupling_9j_e: 9-j Symbols. (line 9) * gsl_sf_dawson: Dawson Function. (line 11) * gsl_sf_dawson_e: Dawson Function. (line 12) * gsl_sf_debye_1: Debye Functions. (line 13) * gsl_sf_debye_1_e: Debye Functions. (line 14) * gsl_sf_debye_2: Debye Functions. (line 18) * gsl_sf_debye_2_e: Debye Functions. (line 19) * gsl_sf_debye_3: Debye Functions. (line 23) * gsl_sf_debye_3_e: Debye Functions. (line 24) * gsl_sf_debye_4: Debye Functions. (line 28) * gsl_sf_debye_4_e: Debye Functions. (line 29) * gsl_sf_debye_5: Debye Functions. (line 33) * gsl_sf_debye_5_e: Debye Functions. (line 34) * gsl_sf_debye_6: Debye Functions. (line 38) * gsl_sf_debye_6_e: Debye Functions. (line 39) * gsl_sf_dilog: Real Argument. (line 6) * gsl_sf_dilog_e: Real Argument. (line 7) * gsl_sf_doublefact: Factorials. (line 18) * gsl_sf_doublefact_e: Factorials. (line 19) * gsl_sf_ellint_D: Legendre Form of Incomplete Elliptic Integrals. (line 34) * gsl_sf_ellint_D_e: Legendre Form of Incomplete Elliptic Integrals. (line 36) * gsl_sf_ellint_E: Legendre Form of Incomplete Elliptic Integrals. (line 15) * gsl_sf_ellint_Ecomp: Legendre Form of Complete Elliptic Integrals. (line 13) * gsl_sf_ellint_Ecomp_e: Legendre Form of Complete Elliptic Integrals. (line 14) * gsl_sf_ellint_E_e: Legendre Form of Incomplete Elliptic Integrals. (line 17) * gsl_sf_ellint_F: Legendre Form of Incomplete Elliptic Integrals. (line 6) * gsl_sf_ellint_F_e: Legendre Form of Incomplete Elliptic Integrals. (line 8) * gsl_sf_ellint_Kcomp: Legendre Form of Complete Elliptic Integrals. (line 6) * gsl_sf_ellint_Kcomp_e: Legendre Form of Complete Elliptic Integrals. (line 7) * gsl_sf_ellint_P: Legendre Form of Incomplete Elliptic Integrals. (line 24) * gsl_sf_ellint_Pcomp: Legendre Form of Complete Elliptic Integrals. (line 20) * gsl_sf_ellint_Pcomp_e: Legendre Form of Complete Elliptic Integrals. (line 22) * gsl_sf_ellint_P_e: Legendre Form of Incomplete Elliptic Integrals. (line 26) * gsl_sf_ellint_RC: Carlson Forms. (line 6) * gsl_sf_ellint_RC_e: Carlson Forms. (line 8) * gsl_sf_ellint_RD: Carlson Forms. (line 13) * gsl_sf_ellint_RD_e: Carlson Forms. (line 15) * gsl_sf_ellint_RF: Carlson Forms. (line 20) * gsl_sf_ellint_RF_e: Carlson Forms. (line 22) * gsl_sf_ellint_RJ: Carlson Forms. (line 27) * gsl_sf_ellint_RJ_e: Carlson Forms. (line 29) * gsl_sf_elljac_e: Elliptic Functions (Jacobi). (line 10) * gsl_sf_erf: Error Function. (line 6) * gsl_sf_erfc: Complementary Error Function. (line 6) * gsl_sf_erfc_e: Complementary Error Function. (line 7) * gsl_sf_erf_e: Error Function. (line 7) * gsl_sf_erf_Q: Probability functions. (line 14) * gsl_sf_erf_Q_e: Probability functions. (line 15) * gsl_sf_erf_Z: Probability functions. (line 9) * gsl_sf_erf_Z_e: Probability functions. (line 10) * gsl_sf_eta: Eta Function. (line 12) * gsl_sf_eta_e: Eta Function. (line 13) * gsl_sf_eta_int: Eta Function. (line 8) * gsl_sf_eta_int_e: Eta Function. (line 9) * gsl_sf_exp: Exponential Function. (line 6) * gsl_sf_expint_3: Ei_3(x). (line 6) * gsl_sf_expint_3_e: Ei_3(x). (line 7) * gsl_sf_expint_E1: Exponential Integral. (line 6) * gsl_sf_expint_E1_e: Exponential Integral. (line 7) * gsl_sf_expint_E2: Exponential Integral. (line 12) * gsl_sf_expint_E2_e: Exponential Integral. (line 13) * gsl_sf_expint_Ei: Ei(x). (line 6) * gsl_sf_expint_Ei_e: Ei(x). (line 7) * gsl_sf_expint_En: Exponential Integral. (line 19) * gsl_sf_expint_En_e: Exponential Integral. (line 20) * gsl_sf_expm1: Relative Exponential Functions. (line 6) * gsl_sf_expm1_e: Relative Exponential Functions. (line 7) * gsl_sf_exprel: Relative Exponential Functions. (line 11) * gsl_sf_exprel_2: Relative Exponential Functions. (line 18) * gsl_sf_exprel_2_e: Relative Exponential Functions. (line 19) * gsl_sf_exprel_e: Relative Exponential Functions. (line 12) * gsl_sf_exprel_n: Relative Exponential Functions. (line 25) * gsl_sf_exprel_n_e: Relative Exponential Functions. (line 26) * gsl_sf_exp_e: Exponential Function. (line 7) * gsl_sf_exp_e10_e: Exponential Function. (line 11) * gsl_sf_exp_err_e: Exponentiation With Error Estimate. (line 6) * gsl_sf_exp_err_e10_e: Exponentiation With Error Estimate. (line 10) * gsl_sf_exp_mult: Exponential Function. (line 18) * gsl_sf_exp_mult_e: Exponential Function. (line 19) * gsl_sf_exp_mult_e10_e: Exponential Function. (line 24) * gsl_sf_exp_mult_err_e: Exponentiation With Error Estimate. (line 16) * gsl_sf_exp_mult_err_e10_e: Exponentiation With Error Estimate. (line 21) * gsl_sf_fact: Factorials. (line 11) * gsl_sf_fact_e: Factorials. (line 12) * gsl_sf_fermi_dirac_0: Complete Fermi-Dirac Integrals. (line 19) * gsl_sf_fermi_dirac_0_e: Complete Fermi-Dirac Integrals. (line 20) * gsl_sf_fermi_dirac_1: Complete Fermi-Dirac Integrals. (line 25) * gsl_sf_fermi_dirac_1_e: Complete Fermi-Dirac Integrals. (line 26) * gsl_sf_fermi_dirac_2: Complete Fermi-Dirac Integrals. (line 31) * gsl_sf_fermi_dirac_2_e: Complete Fermi-Dirac Integrals. (line 32) * gsl_sf_fermi_dirac_3half: Complete Fermi-Dirac Integrals. (line 56) * gsl_sf_fermi_dirac_3half_e: Complete Fermi-Dirac Integrals. (line 57) * gsl_sf_fermi_dirac_half: Complete Fermi-Dirac Integrals. (line 50) * gsl_sf_fermi_dirac_half_e: Complete Fermi-Dirac Integrals. (line 51) * gsl_sf_fermi_dirac_inc_0: Incomplete Fermi-Dirac Integrals. (line 10) * gsl_sf_fermi_dirac_inc_0_e: Incomplete Fermi-Dirac Integrals. (line 11) * gsl_sf_fermi_dirac_int: Complete Fermi-Dirac Integrals. (line 37) * gsl_sf_fermi_dirac_int_e: Complete Fermi-Dirac Integrals. (line 38) * gsl_sf_fermi_dirac_m1: Complete Fermi-Dirac Integrals. (line 12) * gsl_sf_fermi_dirac_m1_e: Complete Fermi-Dirac Integrals. (line 13) * gsl_sf_fermi_dirac_mhalf: Complete Fermi-Dirac Integrals. (line 44) * gsl_sf_fermi_dirac_mhalf_e: Complete Fermi-Dirac Integrals. (line 45) * gsl_sf_gamma: Gamma Functions. (line 14) * gsl_sf_gammainv: Gamma Functions. (line 49) * gsl_sf_gammainv_e: Gamma Functions. (line 50) * gsl_sf_gammastar: Gamma Functions. (line 40) * gsl_sf_gammastar_e: Gamma Functions. (line 41) * gsl_sf_gamma_e: Gamma Functions. (line 15) * gsl_sf_gamma_inc: Incomplete Gamma Functions. (line 6) * gsl_sf_gamma_inc_e: Incomplete Gamma Functions. (line 7) * gsl_sf_gamma_inc_P: Incomplete Gamma Functions. (line 20) * gsl_sf_gamma_inc_P_e: Incomplete Gamma Functions. (line 21) * gsl_sf_gamma_inc_Q: Incomplete Gamma Functions. (line 13) * gsl_sf_gamma_inc_Q_e: Incomplete Gamma Functions. (line 14) * gsl_sf_gegenpoly_1: Gegenbauer Functions. (line 11) * gsl_sf_gegenpoly_1_e: Gegenbauer Functions. (line 14) * gsl_sf_gegenpoly_2: Gegenbauer Functions. (line 12) * gsl_sf_gegenpoly_2_e: Gegenbauer Functions. (line 16) * gsl_sf_gegenpoly_3: Gegenbauer Functions. (line 13) * gsl_sf_gegenpoly_3_e: Gegenbauer Functions. (line 18) * gsl_sf_gegenpoly_array: Gegenbauer Functions. (line 30) * gsl_sf_gegenpoly_n: Gegenbauer Functions. (line 23) * gsl_sf_gegenpoly_n_e: Gegenbauer Functions. (line 24) * gsl_sf_hazard: Probability functions. (line 27) * gsl_sf_hazard_e: Probability functions. (line 28) * gsl_sf_hydrogenicR: Normalized Hydrogenic Bound States. (line 12) * gsl_sf_hydrogenicR_1: Normalized Hydrogenic Bound States. (line 6) * gsl_sf_hydrogenicR_1_e: Normalized Hydrogenic Bound States. (line 7) * gsl_sf_hydrogenicR_e: Normalized Hydrogenic Bound States. (line 14) * gsl_sf_hyperg_0F1: Hypergeometric Functions. (line 10) * gsl_sf_hyperg_0F1_e: Hypergeometric Functions. (line 11) * gsl_sf_hyperg_1F1: Hypergeometric Functions. (line 21) * gsl_sf_hyperg_1F1_e: Hypergeometric Functions. (line 22) * gsl_sf_hyperg_1F1_int: Hypergeometric Functions. (line 15) * gsl_sf_hyperg_1F1_int_e: Hypergeometric Functions. (line 16) * gsl_sf_hyperg_2F0: Hypergeometric Functions. (line 85) * gsl_sf_hyperg_2F0_e: Hypergeometric Functions. (line 86) * gsl_sf_hyperg_2F1: Hypergeometric Functions. (line 51) * gsl_sf_hyperg_2F1_conj: Hypergeometric Functions. (line 63) * gsl_sf_hyperg_2F1_conj_e: Hypergeometric Functions. (line 65) * gsl_sf_hyperg_2F1_conj_renorm: Hypergeometric Functions. (line 77) * gsl_sf_hyperg_2F1_conj_renorm_e: Hypergeometric Functions. (line 79) * gsl_sf_hyperg_2F1_e: Hypergeometric Functions. (line 53) * gsl_sf_hyperg_2F1_renorm: Hypergeometric Functions. (line 70) * gsl_sf_hyperg_2F1_renorm_e: Hypergeometric Functions. (line 72) * gsl_sf_hyperg_U: Hypergeometric Functions. (line 39) * gsl_sf_hyperg_U_e: Hypergeometric Functions. (line 40) * gsl_sf_hyperg_U_e10_e: Hypergeometric Functions. (line 45) * gsl_sf_hyperg_U_int: Hypergeometric Functions. (line 27) * gsl_sf_hyperg_U_int_e: Hypergeometric Functions. (line 28) * gsl_sf_hyperg_U_int_e10_e: Hypergeometric Functions. (line 33) * gsl_sf_hypot: Circular Trigonometric Functions. (line 14) * gsl_sf_hypot_e: Circular Trigonometric Functions. (line 15) * gsl_sf_hzeta: Hurwitz Zeta Function. (line 9) * gsl_sf_hzeta_e: Hurwitz Zeta Function. (line 10) * gsl_sf_laguerre_1: Laguerre Functions. (line 16) * gsl_sf_laguerre_1_e: Laguerre Functions. (line 19) * gsl_sf_laguerre_2: Laguerre Functions. (line 17) * gsl_sf_laguerre_2_e: Laguerre Functions. (line 21) * gsl_sf_laguerre_3: Laguerre Functions. (line 18) * gsl_sf_laguerre_3_e: Laguerre Functions. (line 23) * gsl_sf_laguerre_n: Laguerre Functions. (line 28) * gsl_sf_laguerre_n_e: Laguerre Functions. (line 30) * gsl_sf_lambert_W0: Lambert W Functions. (line 13) * gsl_sf_lambert_W0_e: Lambert W Functions. (line 14) * gsl_sf_lambert_Wm1: Lambert W Functions. (line 18) * gsl_sf_lambert_Wm1_e: Lambert W Functions. (line 19) * gsl_sf_legendre_array_size: Associated Legendre Polynomials and Spherical Harmonics. (line 47) * gsl_sf_legendre_H3d: Radial Functions for Hyperbolic Space. (line 30) * gsl_sf_legendre_H3d_0: Radial Functions for Hyperbolic Space. (line 11) * gsl_sf_legendre_H3d_0_e: Radial Functions for Hyperbolic Space. (line 12) * gsl_sf_legendre_H3d_1: Radial Functions for Hyperbolic Space. (line 20) * gsl_sf_legendre_H3d_1_e: Radial Functions for Hyperbolic Space. (line 21) * gsl_sf_legendre_H3d_array: Radial Functions for Hyperbolic Space. (line 39) * gsl_sf_legendre_H3d_e: Radial Functions for Hyperbolic Space. (line 32) * gsl_sf_legendre_P1: Legendre Polynomials. (line 6) * gsl_sf_legendre_P1_e: Legendre Polynomials. (line 9) * gsl_sf_legendre_P2: Legendre Polynomials. (line 7) * gsl_sf_legendre_P2_e: Legendre Polynomials. (line 11) * gsl_sf_legendre_P3: Legendre Polynomials. (line 8) * gsl_sf_legendre_P3_e: Legendre Polynomials. (line 13) * gsl_sf_legendre_Pl: Legendre Polynomials. (line 18) * gsl_sf_legendre_Plm: Associated Legendre Polynomials and Spherical Harmonics. (line 17) * gsl_sf_legendre_Plm_array: Associated Legendre Polynomials and Spherical Harmonics. (line 23) * gsl_sf_legendre_Plm_deriv_array: Associated Legendre Polynomials and Spherical Harmonics. (line 25) * gsl_sf_legendre_Plm_e: Associated Legendre Polynomials and Spherical Harmonics. (line 18) * gsl_sf_legendre_Pl_array: Legendre Polynomials. (line 24) * gsl_sf_legendre_Pl_deriv_array: Legendre Polynomials. (line 26) * gsl_sf_legendre_Pl_e: Legendre Polynomials. (line 19) * gsl_sf_legendre_Q0: Legendre Polynomials. (line 32) * gsl_sf_legendre_Q0_e: Legendre Polynomials. (line 33) * gsl_sf_legendre_Q1: Legendre Polynomials. (line 38) * gsl_sf_legendre_Q1_e: Legendre Polynomials. (line 39) * gsl_sf_legendre_Ql: Legendre Polynomials. (line 44) * gsl_sf_legendre_Ql_e: Legendre Polynomials. (line 45) * gsl_sf_legendre_sphPlm: Associated Legendre Polynomials and Spherical Harmonics. (line 30) * gsl_sf_legendre_sphPlm_array: Associated Legendre Polynomials and Spherical Harmonics. (line 39) * gsl_sf_legendre_sphPlm_deriv_array: Associated Legendre Polynomials and Spherical Harmonics. (line 41) * gsl_sf_legendre_sphPlm_e: Associated Legendre Polynomials and Spherical Harmonics. (line 31) * gsl_sf_lnbeta: Beta Functions. (line 13) * gsl_sf_lnbeta_e: Beta Functions. (line 14) * gsl_sf_lnchoose: Factorials. (line 44) * gsl_sf_lnchoose_e: Factorials. (line 45) * gsl_sf_lncosh: Hyperbolic Trigonometric Functions. (line 10) * gsl_sf_lncosh_e: Hyperbolic Trigonometric Functions. (line 11) * gsl_sf_lndoublefact: Factorials. (line 32) * gsl_sf_lndoublefact_e: Factorials. (line 33) * gsl_sf_lnfact: Factorials. (line 25) * gsl_sf_lnfact_e: Factorials. (line 26) * gsl_sf_lngamma: Gamma Functions. (line 22) * gsl_sf_lngamma_complex_e: Gamma Functions. (line 54) * gsl_sf_lngamma_e: Gamma Functions. (line 23) * gsl_sf_lngamma_sgn_e: Gamma Functions. (line 30) * gsl_sf_lnpoch: Pochhammer Symbol. (line 14) * gsl_sf_lnpoch_e: Pochhammer Symbol. (line 15) * gsl_sf_lnpoch_sgn_e: Pochhammer Symbol. (line 20) * gsl_sf_lnsinh: Hyperbolic Trigonometric Functions. (line 6) * gsl_sf_lnsinh_e: Hyperbolic Trigonometric Functions. (line 7) * gsl_sf_log: Logarithm and Related Functions. (line 10) * gsl_sf_log_1plusx: Logarithm and Related Functions. (line 25) * gsl_sf_log_1plusx_e: Logarithm and Related Functions. (line 26) * gsl_sf_log_1plusx_mx: Logarithm and Related Functions. (line 30) * gsl_sf_log_1plusx_mx_e: Logarithm and Related Functions. (line 31) * gsl_sf_log_abs: Logarithm and Related Functions. (line 14) * gsl_sf_log_abs_e: Logarithm and Related Functions. (line 15) * gsl_sf_log_e: Logarithm and Related Functions. (line 11) * gsl_sf_log_erfc: Log Complementary Error Function. (line 6) * gsl_sf_log_erfc_e: Log Complementary Error Function. (line 7) * gsl_sf_mathieu_a: Mathieu Function Characteristic Values. (line 6) * gsl_sf_mathieu_alloc: Mathieu Function Workspace. (line 10) * gsl_sf_mathieu_a_array: Mathieu Function Characteristic Values. (line 13) * gsl_sf_mathieu_b: Mathieu Function Characteristic Values. (line 8) * gsl_sf_mathieu_b_array: Mathieu Function Characteristic Values. (line 16) * gsl_sf_mathieu_ce: Angular Mathieu Functions. (line 6) * gsl_sf_mathieu_ce_array: Angular Mathieu Functions. (line 13) * gsl_sf_mathieu_free: Mathieu Function Workspace. (line 17) * gsl_sf_mathieu_Mc: Radial Mathieu Functions. (line 6) * gsl_sf_mathieu_Mc_array: Radial Mathieu Functions. (line 17) * gsl_sf_mathieu_Ms: Radial Mathieu Functions. (line 8) * gsl_sf_mathieu_Ms_array: Radial Mathieu Functions. (line 20) * gsl_sf_mathieu_se: Angular Mathieu Functions. (line 8) * gsl_sf_mathieu_se_array: Angular Mathieu Functions. (line 16) * gsl_sf_multiply_e: Elementary Operations. (line 10) * gsl_sf_multiply_err_e: Elementary Operations. (line 15) * gsl_sf_poch: Pochhammer Symbol. (line 6) * gsl_sf_pochrel: Pochhammer Symbol. (line 27) * gsl_sf_pochrel_e: Pochhammer Symbol. (line 28) * gsl_sf_poch_e: Pochhammer Symbol. (line 7) * gsl_sf_polar_to_rect: Conversion Functions. (line 6) * gsl_sf_pow_int: Power Function. (line 10) * gsl_sf_pow_int_e: Power Function. (line 11) * gsl_sf_psi: Digamma Function. (line 11) * gsl_sf_psi_1: Trigamma Function. (line 11) * gsl_sf_psi_1piy: Digamma Function. (line 16) * gsl_sf_psi_1piy_e: Digamma Function. (line 17) * gsl_sf_psi_1_e: Trigamma Function. (line 12) * gsl_sf_psi_1_int: Trigamma Function. (line 6) * gsl_sf_psi_1_int_e: Trigamma Function. (line 7) * gsl_sf_psi_e: Digamma Function. (line 12) * gsl_sf_psi_int: Digamma Function. (line 6) * gsl_sf_psi_int_e: Digamma Function. (line 7) * gsl_sf_psi_n: Polygamma Function. (line 6) * gsl_sf_psi_n_e: Polygamma Function. (line 7) * gsl_sf_rect_to_polar: Conversion Functions. (line 12) * gsl_sf_Shi: Hyperbolic Integrals. (line 6) * gsl_sf_Shi_e: Hyperbolic Integrals. (line 7) * gsl_sf_Si: Trigonometric Integrals. (line 6) * gsl_sf_sin: Circular Trigonometric Functions. (line 6) * gsl_sf_sinc: Circular Trigonometric Functions. (line 20) * gsl_sf_sinc_e: Circular Trigonometric Functions. (line 21) * gsl_sf_sin_e: Circular Trigonometric Functions. (line 7) * gsl_sf_sin_err_e: Trigonometric Functions With Error Estimates. (line 6) * gsl_sf_Si_e: Trigonometric Integrals. (line 7) * gsl_sf_synchrotron_1: Synchrotron Functions. (line 9) * gsl_sf_synchrotron_1_e: Synchrotron Functions. (line 10) * gsl_sf_synchrotron_2: Synchrotron Functions. (line 15) * gsl_sf_synchrotron_2_e: Synchrotron Functions. (line 16) * gsl_sf_taylorcoeff: Factorials. (line 50) * gsl_sf_taylorcoeff_e: Factorials. (line 51) * gsl_sf_transport_2: Transport Functions. (line 10) * gsl_sf_transport_2_e: Transport Functions. (line 11) * gsl_sf_transport_3: Transport Functions. (line 15) * gsl_sf_transport_3_e: Transport Functions. (line 16) * gsl_sf_transport_4: Transport Functions. (line 20) * gsl_sf_transport_4_e: Transport Functions. (line 21) * gsl_sf_transport_5: Transport Functions. (line 25) * gsl_sf_transport_5_e: Transport Functions. (line 26) * gsl_sf_zeta: Riemann Zeta Function. (line 14) * gsl_sf_zetam1: Riemann Zeta Function Minus One. (line 14) * gsl_sf_zetam1_e: Riemann Zeta Function Minus One. (line 15) * gsl_sf_zetam1_int: Riemann Zeta Function Minus One. (line 10) * gsl_sf_zetam1_int_e: Riemann Zeta Function Minus One. (line 11) * gsl_sf_zeta_e: Riemann Zeta Function. (line 15) * gsl_sf_zeta_int: Riemann Zeta Function. (line 9) * gsl_sf_zeta_int_e: Riemann Zeta Function. (line 10) * GSL_SIGN: Testing the Sign of Numbers. (line 6) * gsl_siman_solve: Simulated Annealing functions. (line 6) * gsl_sort: Sorting vectors. (line 23) * gsl_sort2: Sorting vectors. (line 28) * gsl_sort_index: Sorting vectors. (line 44) * gsl_sort_largest: Selecting the k smallest or largest elements. (line 23) * gsl_sort_largest_index: Selecting the k smallest or largest elements. (line 49) * gsl_sort_smallest: Selecting the k smallest or largest elements. (line 16) * gsl_sort_smallest_index: Selecting the k smallest or largest elements. (line 41) * gsl_sort_vector: Sorting vectors. (line 35) * gsl_sort_vector2: Sorting vectors. (line 39) * gsl_sort_vector_index: Sorting vectors. (line 54) * gsl_sort_vector_largest: Selecting the k smallest or largest elements. (line 32) * gsl_sort_vector_largest_index: Selecting the k smallest or largest elements. (line 59) * gsl_sort_vector_smallest: Selecting the k smallest or largest elements. (line 30) * gsl_sort_vector_smallest_index: Selecting the k smallest or largest elements. (line 57) * gsl_spline_alloc: Higher-level Interface. (line 13) * gsl_spline_eval: Higher-level Interface. (line 26) * gsl_spline_eval_deriv: Higher-level Interface. (line 31) * gsl_spline_eval_deriv2: Higher-level Interface. (line 36) * gsl_spline_eval_deriv2_e: Higher-level Interface. (line 38) * gsl_spline_eval_deriv_e: Higher-level Interface. (line 33) * gsl_spline_eval_e: Higher-level Interface. (line 28) * gsl_spline_eval_integ: Higher-level Interface. (line 41) * gsl_spline_eval_integ_e: Higher-level Interface. (line 43) * gsl_spline_free: Higher-level Interface. (line 19) * gsl_spline_init: Higher-level Interface. (line 16) * gsl_spline_min_size: Higher-level Interface. (line 23) * gsl_spline_name: Higher-level Interface. (line 21) * gsl_stats_absdev: Absolute deviation. (line 6) * gsl_stats_absdev_m: Absolute deviation. (line 19) * gsl_stats_correlation: Correlation. (line 6) * gsl_stats_covariance: Covariance. (line 6) * gsl_stats_covariance_m: Covariance. (line 14) * gsl_stats_kurtosis: Higher moments (skewness and kurtosis). (line 29) * gsl_stats_kurtosis_m_sd: Higher moments (skewness and kurtosis). (line 40) * gsl_stats_lag1_autocorrelation: Autocorrelation. (line 6) * gsl_stats_lag1_autocorrelation_m: Autocorrelation. (line 15) * gsl_stats_max: Maximum and Minimum values. (line 12) * gsl_stats_max_index: Maximum and Minimum values. (line 37) * gsl_stats_mean: Mean and standard deviation and variance. (line 6) * gsl_stats_median_from_sorted_data: Median and Percentiles. (line 10) * gsl_stats_min: Maximum and Minimum values. (line 22) * gsl_stats_minmax: Maximum and Minimum values. (line 32) * gsl_stats_minmax_index: Maximum and Minimum values. (line 53) * gsl_stats_min_index: Maximum and Minimum values. (line 45) * gsl_stats_quantile_from_sorted_data: Median and Percentiles. (line 25) * gsl_stats_sd: Mean and standard deviation and variance. (line 44) * gsl_stats_sd_m: Mean and standard deviation and variance. (line 46) * gsl_stats_sd_with_fixed_mean: Mean and standard deviation and variance. (line 73) * gsl_stats_skew: Higher moments (skewness and kurtosis). (line 6) * gsl_stats_skew_m_sd: Higher moments (skewness and kurtosis). (line 19) * gsl_stats_spearman: Correlation. (line 18) * gsl_stats_tss: Mean and standard deviation and variance. (line 52) * gsl_stats_tss_m: Mean and standard deviation and variance. (line 54) * gsl_stats_variance: Mean and standard deviation and variance. (line 18) * gsl_stats_variance_m: Mean and standard deviation and variance. (line 36) * gsl_stats_variance_with_fixed_mean: Mean and standard deviation and variance. (line 63) * gsl_stats_wabsdev: Weighted Samples. (line 83) * gsl_stats_wabsdev_m: Weighted Samples. (line 91) * gsl_stats_wkurtosis: Weighted Samples. (line 110) * gsl_stats_wkurtosis_m_sd: Weighted Samples. (line 116) * gsl_stats_wmean: Weighted Samples. (line 14) * gsl_stats_wsd: Weighted Samples. (line 42) * gsl_stats_wsd_m: Weighted Samples. (line 48) * gsl_stats_wsd_with_fixed_mean: Weighted Samples. (line 64) * gsl_stats_wskew: Weighted Samples. (line 97) * gsl_stats_wskew_m_sd: Weighted Samples. (line 103) * gsl_stats_wtss: Weighted Samples. (line 71) * gsl_stats_wtss_m: Weighted Samples. (line 73) * gsl_stats_wvariance: Weighted Samples. (line 22) * gsl_stats_wvariance_m: Weighted Samples. (line 36) * gsl_stats_wvariance_with_fixed_mean: Weighted Samples. (line 53) * gsl_strerror: Error Codes. (line 37) * gsl_sum_levin_utrunc_accel: Acceleration functions without error estimation. (line 32) * gsl_sum_levin_utrunc_alloc: Acceleration functions without error estimation. (line 22) * gsl_sum_levin_utrunc_free: Acceleration functions without error estimation. (line 28) * gsl_sum_levin_u_accel: Acceleration functions. (line 31) * gsl_sum_levin_u_alloc: Acceleration functions. (line 23) * gsl_sum_levin_u_free: Acceleration functions. (line 28) * gsl_vector_add: Vector operations. (line 6) * gsl_vector_add_constant: Vector operations. (line 32) * gsl_vector_alloc: Vector allocation. (line 14) * gsl_vector_calloc: Vector allocation. (line 21) * gsl_vector_complex_const_imag: Vector views. (line 108) * gsl_vector_complex_const_real: Vector views. (line 97) * gsl_vector_complex_imag: Vector views. (line 106) * gsl_vector_complex_real: Vector views. (line 95) * gsl_vector_const_ptr: Accessing vector elements. (line 52) * gsl_vector_const_subvector: Vector views. (line 29) * gsl_vector_const_subvector_with_stride: Vector views. (line 59) * gsl_vector_const_view_array: Vector views. (line 119) * gsl_vector_const_view_array_with_stride: Vector views. (line 142) * gsl_vector_div: Vector operations. (line 23) * gsl_vector_equal: Vector properties. (line 18) * gsl_vector_fprintf: Reading and writing vectors. (line 26) * gsl_vector_fread: Reading and writing vectors. (line 17) * gsl_vector_free: Vector allocation. (line 25) * gsl_vector_fscanf: Reading and writing vectors. (line 34) * gsl_vector_fwrite: Reading and writing vectors. (line 9) * gsl_vector_get: Accessing vector elements. (line 39) * gsl_vector_isneg: Vector properties. (line 12) * gsl_vector_isnonneg: Vector properties. (line 13) * gsl_vector_isnull: Vector properties. (line 10) * gsl_vector_ispos: Vector properties. (line 11) * gsl_vector_max: Finding maximum and minimum elements of vectors. (line 8) * gsl_vector_max_index: Finding maximum and minimum elements of vectors. (line 19) * gsl_vector_memcpy: Copying vectors. (line 12) * gsl_vector_min: Finding maximum and minimum elements of vectors. (line 11) * gsl_vector_minmax: Finding maximum and minimum elements of vectors. (line 14) * gsl_vector_minmax_index: Finding maximum and minimum elements of vectors. (line 29) * gsl_vector_min_index: Finding maximum and minimum elements of vectors. (line 24) * gsl_vector_mul: Vector operations. (line 17) * gsl_vector_ptr: Accessing vector elements. (line 51) * gsl_vector_reverse: Exchanging elements. (line 14) * gsl_vector_scale: Vector operations. (line 28) * gsl_vector_set: Accessing vector elements. (line 45) * gsl_vector_set_all: Initializing vector elements. (line 6) * gsl_vector_set_basis: Initializing vector elements. (line 12) * gsl_vector_set_zero: Initializing vector elements. (line 9) * gsl_vector_sub: Vector operations. (line 11) * gsl_vector_subvector: Vector views. (line 27) * gsl_vector_subvector_with_stride: Vector views. (line 57) * gsl_vector_swap: Copying vectors. (line 17) * gsl_vector_swap_elements: Exchanging elements. (line 9) * gsl_vector_view_array: Vector views. (line 117) * gsl_vector_view_array_with_stride: Vector views. (line 140) * gsl_wavelet2d_nstransform: DWT in two dimension. (line 64) * gsl_wavelet2d_nstransform_forward: DWT in two dimension. (line 67) * gsl_wavelet2d_nstransform_inverse: DWT in two dimension. (line 70) * gsl_wavelet2d_nstransform_matrix: DWT in two dimension. (line 76) * gsl_wavelet2d_nstransform_matrix_forward: DWT in two dimension. (line 79) * gsl_wavelet2d_nstransform_matrix_inverse: DWT in two dimension. (line 81) * gsl_wavelet2d_transform: DWT in two dimension. (line 29) * gsl_wavelet2d_transform_forward: DWT in two dimension. (line 32) * gsl_wavelet2d_transform_inverse: DWT in two dimension. (line 35) * gsl_wavelet2d_transform_matrix: DWT in two dimension. (line 54) * gsl_wavelet2d_transform_matrix_forward: DWT in two dimension. (line 57) * gsl_wavelet2d_transform_matrix_inverse: DWT in two dimension. (line 59) * gsl_wavelet_alloc: DWT Initialization. (line 9) * gsl_wavelet_bspline: DWT Initialization. (line 29) * gsl_wavelet_bspline_centered: DWT Initialization. (line 30) * gsl_wavelet_daubechies: DWT Initialization. (line 18) * gsl_wavelet_daubechies_centered: DWT Initialization. (line 19) * gsl_wavelet_free: DWT Initialization. (line 44) * gsl_wavelet_haar: DWT Initialization. (line 24) * gsl_wavelet_haar_centered: DWT Initialization. (line 25) * gsl_wavelet_name: DWT Initialization. (line 40) * gsl_wavelet_transform: DWT in one dimension. (line 6) * gsl_wavelet_transform_forward: DWT in one dimension. (line 9) * gsl_wavelet_transform_inverse: DWT in one dimension. (line 12) * gsl_wavelet_workspace_alloc: DWT Initialization. (line 51) * gsl_wavelet_workspace_free: DWT Initialization. (line 61)