SUBROUTINE sla_SVDSOL (M, N, MP, NP, B, U, W, V, WORK, X) *+ * - - - - - - - * S V D S O L * - - - - - - - * * From a given vector and the SVD of a matrix (as obtained from * the SVD routine), obtain the solution vector (double precision) * * This routine solves the equation: * * A . x = b * * where: * * A is a given M (rows) x N (columns) matrix, where M.GE.N * x is the N-vector we wish to find * b is a given M-vector * * by means of the Singular Value Decomposition method (SVD). In * this method, the matrix A is first factorised (for example by * the routine sla_SVD) into the following components: * * A = U x W x VT * * where: * * A is the M (rows) x N (columns) matrix * U is an M x N column-orthogonal matrix * W is an N x N diagonal matrix with W(I,I).GE.0 * VT is the transpose of an NxN orthogonal matrix * * Note that M and N, above, are the LOGICAL dimensions of the * matrices and vectors concerned, which can be located in * arrays of larger PHYSICAL dimensions MP and NP. * * The solution is found from the expression: * * x = V . [diag(1/Wj)] . (transpose(U) . b) * * Notes: * * 1) If matrix A is square, and if the diagonal matrix W is not * adjusted, the method is equivalent to conventional solution * of simultaneous equations. * * 2) If M>N, the result is a least-squares fit. * * 3) If the solution is poorly determined, this shows up in the * SVD factorisation as very small or zero Wj values. Where * a Wj value is small but non-zero it can be set to zero to * avoid ill effects. The present routine detects such zero * Wj values and produces a sensible solution, with highly * correlated terms kept under control rather than being allowed * to elope to infinity, and with meaningful values for the * other terms. * * Given: * M,N i numbers of rows and columns in matrix A * MP,NP i physical dimensions of array containing matrix A * B d(M) known vector b * U d(MP,NP) array containing MxN matrix U * W d(N) NxN diagonal matrix W (diagonal elements only) * V d(NP,NP) array containing NxN orthogonal matrix V * * Returned: * WORK d(N) workspace * X d(N) unknown vector x * * Reference: * Numerical Recipes, section 2.9. * * P.T.Wallace Starlink 29 October 1993 * * Copyright (C) 1995 Rutherford Appleton Laboratory * * License: * 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 2 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 (see SLA_CONDITIONS); if not, write to the * Free Software Foundation, Inc., 59 Temple Place, Suite 330, * Boston, MA 02111-1307 USA * *- IMPLICIT NONE INTEGER M,N,MP,NP DOUBLE PRECISION B(M),U(MP,NP),W(N),V(NP,NP),WORK(N),X(N) INTEGER J,I,JJ DOUBLE PRECISION S * Calculate [diag(1/Wj)] . transpose(U) . b (or zero for zero Wj) DO J=1,N S=0D0 IF (W(J).NE.0D0) THEN DO I=1,M S=S+U(I,J)*B(I) END DO S=S/W(J) END IF WORK(J)=S END DO * Multiply by matrix V to get result DO J=1,N S=0D0 DO JJ=1,N S=S+V(J,JJ)*WORK(JJ) END DO X(J)=S END DO END