Consider the problem of finding a solution to a system of nonlinear equations whose Jacobian is sparse. The dimension of the problem in this example is 1000. The goal is to find x such that F(x) = 0. Assuming n = 1000, the nonlinear equations are
To solve a large nonlinear system of equations, F(x)
= 0, you can use the trust-region reflective algorithm available in fsolve, a large-scale algorithm (Large-Scale vs. Medium-Scale Algorithms).
function [F,J] = nlsf1(x) % Evaluate the vector function n = length(x); F = zeros(n,1); i = 2:(n-1); F(i) = (3-2*x(i)).*x(i)-x(i-1)-2*x(i+1) + 1; F(n) = (3-2*x(n)).*x(n)-x(n-1) + 1; F(1) = (3-2*x(1)).*x(1)-2*x(2) + 1; % Evaluate the Jacobian if nargout > 1 if nargout > 1 d = -4*x + 3*ones(n,1); D = sparse(1:n,1:n,d,n,n); c = -2*ones(n-1,1); C = sparse(1:n-1,2:n,c,n,n); e = -ones(n-1,1); E = sparse(2:n,1:n-1,e,n,n); J = C + D + E; end
xstart = -ones(1000,1);
fun = @nlsf1;
options = optimoptions(@fsolve,'Display','iter',...
'Algorithm','trust-region-reflective',...
'Jacobian','on','PrecondBandWidth',0);
[x,fval,exitflag,output] = fsolve(fun,xstart,options);A starting point is given as well as the function name. The
default method for fsolve is trust-region-dogleg,
so it is necessary to specify 'Algorithm' as 'trust-region-reflective' in
the options argument in order to select the trust-region-reflective
algorithm. Setting the Display option to 'iter' causes fsolve to
display the output at each iteration. Setting Jacobian to 'on',
causes fsolve to use the Jacobian information available
in nlsf1.m.
The commands display this output:
Norm of First-order
Iteration Func-count f(x) step optimality
0 1 1011 19
1 2 16.1942 7.91898 2.35
2 3 0.0228027 1.33142 0.291
3 4 0.000103359 0.0433329 0.0201
4 5 7.3792e-07 0.0022606 0.000946
5 6 4.02299e-10 0.000268381 4.12e-05
Equation solved, inaccuracy possible.
The vector of function values is near zero, as measured by the default value
of the function tolerance. However, the last step was ineffective.A linear system is (approximately) solved in each major iteration
using the preconditioned conjugate gradient method. Setting PrecondBandWidth to
0 in options means a diagonal preconditioner is used. (PrecondBandWidth specifies
the bandwidth of the preconditioning matrix. A bandwidth of 0 means
there is only one diagonal in the matrix.)
From the first-order optimality values, fast linear convergence occurs. The number of conjugate gradient (CG) iterations required per major iteration is low, at most five for a problem of 1000 dimensions, implying that the linear systems are not very difficult to solve in this case (though more work is required as convergence progresses).
If you want to use a tridiagonal preconditioner, i.e., a preconditioning
matrix with three diagonals (or bandwidth of one), set PrecondBandWidth to
the value 1:
options = optimoptions(@fsolve,'Display','iter','Jacobian','on',... 'Algorithm','trust-region-reflective','PrecondBandWidth',1); [x,fval,exitflag,output] = fsolve(fun,xstart,options);
In this case the output is
Norm of First-order
Iteration Func-count f(x) step optimality
0 1 1011 19
1 2 16.0839 7.92496 1.92
2 3 0.0458181 1.3279 0.579
3 4 0.000101184 0.0631898 0.0203
4 5 3.16615e-07 0.00273698 0.00079
5 6 9.72481e-10 0.00018111 5.82e-05
Equation solved, inaccuracy possible.
The vector of function values is near zero, as measured by the default value
of the function tolerance. However, the last step was ineffective.Note that although the same number of iterations takes place, the number of PCG iterations has dropped, so less work is being done per iteration. See Preconditioned Conjugate Gradient Method.
Setting PrecondBandWidth to Inf (this
is the default) means that the solver uses Cholesky factorization
rather than PCG.