/*
* Copyright (c) 2002-2006 Samit Basu
*
* 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; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*
*/
#include "LinearEqSolver.hpp"
#include "LAPACK.hpp"
#include <stdlib.h>
#include <stdio.h>
#include <iostream>
#include "Malloc.hpp"
#define MSGBUFLEN 2048
/***************************************************************************
* Linear equation solver for real matrices
***************************************************************************/
// Solve A*C = B, where A is m x m, and B is m x n, all quantities are real.
void doubleSolveLinEq(Interpreter* eval, int m, int n, double *c, double* a, double *b) {
char msgBuffer[MSGBUFLEN];
if ((m == 0) || (n == 0)) return;
// Here are the comments from the LAPACK routine used:
//SUBROUTINE DGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
//$ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
//$ WORK, IWORK, INFO )
//*
//* -- LAPACK driver routine (version 3.0) --
//* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
//* Courant Institute, Argonne National Lab, and Rice University
//* June 30, 1999
//*
//* .. Scalar Arguments ..
// CHARACTER EQUED, FACT, TRANS
// INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
// DOUBLE PRECISION RCOND
//* ..
//* .. Array Arguments ..
// INTEGER IPIV( * ), IWORK( * )
// DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
// $ BERR( * ), C( * ), FERR( * ), R( * ),
// $ WORK( * ), X( LDX, * )
//* ..
//*
//* Purpose
//* =======
//*
//* DGESVX uses the LU factorization to compute the solution to a real
//* system of linear equations
//* A * X = B,
//* where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
//*
//* Error bounds on the solution and a condition estimate are also
//* provided.
//*
//* Description
//* ===========
//*
//* The following steps are performed:
//*
//* 1. If FACT = 'E', real scaling factors are computed to equilibrate
//* the system:
//* TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
//* TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
//* TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
//* Whether or not the system will be equilibrated depends on the
//* scaling of the matrix A, but if equilibration is used, A is
//* overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
//* or diag(C)*B (if TRANS = 'T' or 'C').
//*
//* 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
//* matrix A (after equilibration if FACT = 'E') as
//* A = P * L * U,
//* where P is a permutation matrix, L is a unit lower triangular
//* matrix, and U is upper triangular.
//*
//* 3. If some U(i,i)=0, so that U is exactly singular, then the routine
//* returns with INFO = i. Otherwise, the factored form of A is used
//* to estimate the condition number of the matrix A. If the
//* reciprocal of the condition number is less than machine precision,
//* INFO = N+1 is returned as a warning, but the routine still goes on
//* to solve for X and compute error bounds as described below.
//*
//* 4. The system of equations is solved for X using the factored form
//* of A.
//*
//* 5. Iterative refinement is applied to improve the computed solution
//* matrix and calculate error bounds and backward error estimates
//* for it.
//*
//* 6. If equilibration was used, the matrix X is premultiplied by
//* diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
//* that it solves the original system before equilibration.
//*
//* Arguments
//* =========
//*
//* FACT (input) CHARACTER*1
//* Specifies whether or not the factored form of the matrix A is
//* supplied on entry, and if not, whether the matrix A should be
//* equilibrated before it is factored.
//* = 'F': On entry, AF and IPIV contain the factored form of A.
//* If EQUED is not 'N', the matrix A has been
//* equilibrated with scaling factors given by R and C.
//* A, AF, and IPIV are not modified.
//* = 'N': The matrix A will be copied to AF and factored.
//* = 'E': The matrix A will be equilibrated if necessary, then
//* copied to AF and factored.
char FACT = 'E';
//*
//* TRANS (input) CHARACTER*1
//* Specifies the form of the system of equations:
//* = 'N': A * X = B (No transpose)
//* = 'T': A**T * X = B (Transpose)
//* = 'C': A**H * X = B (Transpose)
char TRANS = 'N';
//*
//* N (input) INTEGER
//* The number of linear equations, i.e., the order of the
//* matrix A. N >= 0.
//*
int N = m;
//* NRHS (input) INTEGER
//* The number of right hand sides, i.e., the number of columns
//* of the matrices B and X. NRHS >= 0.
int NRHS = n;
//*
//* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
//* On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
//* not 'N', then A must have been equilibrated by the scaling
//* factors in R and/or C. A is not modified if FACT = 'F' or
//* 'N', or if FACT = 'E' and EQUED = 'N' on exit.
//*
//* On exit, if EQUED .ne. 'N', A is scaled as follows:
//* EQUED = 'R': A := diag(R) * A
//* EQUED = 'C': A := A * diag(C)
//* EQUED = 'B': A := diag(R) * A * diag(C).
double* A = a;
//* LDA (input) INTEGER
//* The leading dimension of the array A. LDA >= max(1,N).
//*
int LDA = m;
//* AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N)
//* If FACT = 'F', then AF is an input argument and on entry
//* contains the factors L and U from the factorization
//* A = P*L*U as computed by DGETRF. If EQUED .ne. 'N', then
//* AF is the factored form of the equilibrated matrix A.
//*
//* If FACT = 'N', then AF is an output argument and on exit
//* returns the factors L and U from the factorization A = P*L*U
//* of the original matrix A.
//*
//* If FACT = 'E', then AF is an output argument and on exit
//* returns the factors L and U from the factorization A = P*L*U
//* of the equilibrated matrix A (see the description of A for
//* the form of the equilibrated matrix).
double *AF = (double*) Malloc(sizeof(double)*LDA*N);
//* LDAF (input) INTEGER
//* The leading dimension of the array AF. LDAF >= max(1,N).
//*
int LDAF = m;
//* IPIV (input or output) INTEGER array, dimension (N)
//* If FACT = 'F', then IPIV is an input argument and on entry
//* contains the pivot indices from the factorization A = P*L*U
//* as computed by DGETRF; row i of the matrix was interchanged
//* with row IPIV(i).
//*
//* If FACT = 'N', then IPIV is an output argument and on exit
//* contains the pivot indices from the factorization A = P*L*U
//* of the original matrix A.
//*
//* If FACT = 'E', then IPIV is an output argument and on exit
//* contains the pivot indices from the factorization A = P*L*U
//* of the equilibrated matrix A.
int *IPIV = (int*) Malloc(sizeof(int)*N);
//* EQUED (input or output) CHARACTER*1
//* Specifies the form of equilibration that was done.
//* = 'N': No equilibration (always true if FACT = 'N').
//* = 'R': Row equilibration, i.e., A has been premultiplied by
//* diag(R).
//* = 'C': Column equilibration, i.e., A has been postmultiplied
//* by diag(C).
//* = 'B': Both row and column equilibration, i.e., A has been
//* replaced by diag(R) * A * diag(C).
//* EQUED is an input argument if FACT = 'F'; otherwise, it is an
//* output argument.
char EQUED;
//* R (input or output) DOUBLE PRECISION array, dimension (N)
//* The row scale factors for A. If EQUED = 'R' or 'B', A is
//* multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
//* is not accessed. R is an input argument if FACT = 'F';
//* otherwise, R is an output argument. If FACT = 'F' and
//* EQUED = 'R' or 'B', each element of R must be positive.
double *R = (double*) Malloc(sizeof(double)*N);
//* C (input or output) DOUBLE PRECISION array, dimension (N)
//* The column scale factors for A. If EQUED = 'C' or 'B', A is
//* multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
//* is not accessed. C is an input argument if FACT = 'F';
//* otherwise, C is an output argument. If FACT = 'F' and
//* EQUED = 'C' or 'B', each element of C must be positive.
double *C = (double*) Malloc(sizeof(double)*N);
//* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
//* On entry, the N-by-NRHS right hand side matrix B.
//* On exit,
//* if EQUED = 'N', B is not modified;
//* if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
//* diag(R)*B;
//* if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
//* overwritten by diag(C)*B.
double *B = b;
//* LDB (input) INTEGER
//* The leading dimension of the array B. LDB >= max(1,N).
int LDB = m;
//* X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
//* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
//* to the original system of equations. Note that A and B are
//* modified on exit if EQUED .ne. 'N', and the solution to the
//* equilibrated system is inv(diag(C))*X if TRANS = 'N' and
//* EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
//* and EQUED = 'R' or 'B'.
double *X = c;
//* LDX (input) INTEGER
//* The leading dimension of the array X. LDX >= max(1,N).
int LDX = m;
//* RCOND (output) DOUBLE PRECISION
//* The estimate of the reciprocal condition number of the matrix
//* A after equilibration (if done). If RCOND is less than the
//* machine precision (in particular, if RCOND = 0), the matrix
//* is singular to working precision. This condition is
//* indicated by a return code of INFO > 0.
double RCOND;
//* FERR (output) DOUBLE PRECISION array, dimension (NRHS)
//* The estimated forward error bound for each solution vector
//* X(j) (the j-th column of the solution matrix X).
//* If XTRUE is the true solution corresponding to X(j), FERR(j)
//* is an estimated upper bound for the magnitude of the largest
//* element in (X(j) - XTRUE) divided by the magnitude of the
//* largest element in X(j). The estimate is as reliable as
//* the estimate for RCOND, and is almost always a slight
//* overestimate of the true error.
double *FERR = (double*) Malloc(n*sizeof(double));
//* BERR (output) DOUBLE PRECISION array, dimension (NRHS)
//* The componentwise relative backward error of each solution
//* vector X(j) (i.e., the smallest relative change in
//* any element of A or B that makes X(j) an exact solution).
double *BERR = (double*) Malloc(n*sizeof(double));
//* WORK (workspace/output) DOUBLE PRECISION array, dimension (4*N)
//* On exit, WORK(1) contains the reciprocal pivot growth
//* factor norm(A)/norm(U). The "max absolute element" norm is
//* used. If WORK(1) is much less than 1, then the stability
//* of the LU factorization of the (equilibrated) matrix A
//* could be poor. This also means that the solution X, condition
//* estimator RCOND, and forward error bound FERR could be
//* unreliable. If factorization fails with 0<INFO<=N, then
//* WORK(1) contains the reciprocal pivot growth factor for the
//* leading INFO columns of A.
double *WORK = (double*) Malloc(4*N*sizeof(double));
//* IWORK (workspace) INTEGER array, dimension (N)
int *IWORK = (int*) Malloc(4*N*sizeof(int));
//* INFO (output) INTEGER
//* = 0: successful exit
//* < 0: if INFO = -i, the i-th argument had an illegal value
//* > 0: if INFO = i, and i is
//* <= N: U(i,i) is exactly zero. The factorization has
//* been completed, but the factor U is exactly
//* singular, so the solution and error bounds
//* could not be computed. RCOND = 0 is returned.
//* = N+1: U is nonsingular, but RCOND is less than machine
//* precision, meaning that the matrix is singular
//* to working precision. Nevertheless, the
//* solution and error bounds are computed because
//* there are a number of situations where the
//* computed solution can be more accurate than the
//* value of RCOND would suggest.
//*
//* =====================================================================
int INFO;
dgesvx_(&FACT, &TRANS, &N, &NRHS, A, &LDA, AF, &LDAF, IPIV, &EQUED, R, C, B,
&LDB, X, &LDX, &RCOND, FERR, BERR, WORK, IWORK, &INFO);
if ((INFO == N) || (INFO == N+1) || (RCOND < getEPS())) {
snprintf(msgBuffer,MSGBUFLEN,"Matrix is singular to working precision. RCOND = %e\n",RCOND);
eval->warningMessage(msgBuffer);
}
// Free the allocated arrays...
Free(AF);
Free(IPIV);
Free(R);
Free(C);
Free(WORK);
Free(IWORK);
Free(FERR);
Free(BERR);
}
// Solve A*C = B, where A is m x m, and B is m x n, all quantities are real.
void dcomplexSolveLinEq(Interpreter* eval, int m, int n, double *c, double* a, double *b) {
if ((m == 0) || (n == 0)) return;
char msgBuffer[MSGBUFLEN];
// Here are the comments from the LAPACK routine used:
//SUBROUTINE ZGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
//$ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
//$ WORK, IWORK, INFO )
//*
//* -- LAPACK driver routine (version 3.0) --
//* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
//* Courant Institute, Argonne National Lab, and Rice University
//* June 30, 1999
//*
//* .. Scalar Arguments ..
// CHARACTER EQUED, FACT, TRANS
// INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
// DOUBLE PRECISION RCOND
//* ..
//* .. Array Arguments ..
// INTEGER IPIV( * )
// DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
// $ RWORK( * )
// COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
// $ WORK( * ), X( LDX, * )
//* ..
//*
//* Purpose
//* =======
//*
//* ZGESVX uses the LU factorization to compute the solution to a real
//* system of linear equations
//* A * X = B,
//* where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
//*
//* Error bounds on the solution and a condition estimate are also
//* provided.
//*
//* Description
//* ===========
//*
//* The following steps are performed:
//*
//* 1. If FACT = 'E', real scaling factors are computed to equilibrate
//* the system:
//* TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
//* TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
//* TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
//* Whether or not the system will be equilibrated depends on the
//* scaling of the matrix A, but if equilibration is used, A is
//* overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
//* or diag(C)*B (if TRANS = 'T' or 'C').
//*
//* 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
//* matrix A (after equilibration if FACT = 'E') as
//* A = P * L * U,
//* where P is a permutation matrix, L is a unit lower triangular
//* matrix, and U is upper triangular.
//*
//* 3. If some U(i,i)=0, so that U is exactly singular, then the routine
//* returns with INFO = i. Otherwise, the factored form of A is used
//* to estimate the condition number of the matrix A. If the
//* reciprocal of the condition number is less than machine precision,
//* INFO = N+1 is returned as a warning, but the routine still goes on
//* to solve for X and compute error bounds as described below.
//*
//* 4. The system of equations is solved for X using the factored form
//* of A.
//*
//* 5. Iterative refinement is applied to improve the computed solution
//* matrix and calculate error bounds and backward error estimates
//* for it.
//*
//* 6. If equilibration was used, the matrix X is premultiplied by
//* diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
//* that it solves the original system before equilibration.
//*
//* Arguments
//* =========
//*
//* FACT (input) CHARACTER*1
//* Specifies whether or not the factored form of the matrix A is
//* supplied on entry, and if not, whether the matrix A should be
//* equilibrated before it is factored.
//* = 'F': On entry, AF and IPIV contain the factored form of A.
//* If EQUED is not 'N', the matrix A has been
//* equilibrated with scaling factors given by R and C.
//* A, AF, and IPIV are not modified.
//* = 'N': The matrix A will be copied to AF and factored.
//* = 'E': The matrix A will be equilibrated if necessary, then
//* copied to AF and factored.
char FACT = 'E';
//*
//* TRANS (input) CHARACTER*1
//* Specifies the form of the system of equations:
//* = 'N': A * X = B (No transpose)
//* = 'T': A**T * X = B (Transpose)
//* = 'C': A**H * X = B (Transpose)
char TRANS = 'N';
//*
//* N (input) INTEGER
//* The number of linear equations, i.e., the order of the
//* matrix A. N >= 0.
//*
int N = m;
//* NRHS (input) INTEGER
//* The number of right hand sides, i.e., the number of columns
//* of the matrices B and X. NRHS >= 0.
int NRHS = n;
//* A (input/output) COMPLEX*16 array, dimension (LDA,N)
//* On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
//* not 'N', then A must have been equilibrated by the scaling
//* factors in R and/or C. A is not modified if FACT = 'F' or
//* 'N', or if FACT = 'E' and EQUED = 'N' on exit.
//*
//* On exit, if EQUED .ne. 'N', A is scaled as follows:
//* EQUED = 'R': A := diag(R) * A
//* EQUED = 'C': A := A * diag(C)
//* EQUED = 'B': A := diag(R) * A * diag(C).
double* A = a;
//* LDA (input) INTEGER
//* The leading dimension of the array A. LDA >= max(1,N).
//*
int LDA = m;
//* AF (input or output) COMPLEX*16 array, dimension (LDAF,N)
//* If FACT = 'F', then AF is an input argument and on entry
//* contains the factors L and U from the factorization
//* A = P*L*U as computed by DGETRF. If EQUED .ne. 'N', then
//* AF is the factored form of the equilibrated matrix A.
//*
//* If FACT = 'N', then AF is an output argument and on exit
//* returns the factors L and U from the factorization A = P*L*U
//* of the original matrix A.
//*
//* If FACT = 'E', then AF is an output argument and on exit
//* returns the factors L and U from the factorization A = P*L*U
//* of the equilibrated matrix A (see the description of A for
//* the form of the equilibrated matrix).
double *AF = (double*) Malloc(2*sizeof(double)*LDA*N);
//* LDAF (input) INTEGER
//* The leading dimension of the array AF. LDAF >= max(1,N).
//*
int LDAF = m;
//* IPIV (input or output) INTEGER array, dimension (N)
//* If FACT = 'F', then IPIV is an input argument and on entry
//* contains the pivot indices from the factorization A = P*L*U
//* as computed by DGETRF; row i of the matrix was interchanged
//* with row IPIV(i).
//*
//* If FACT = 'N', then IPIV is an output argument and on exit
//* contains the pivot indices from the factorization A = P*L*U
//* of the original matrix A.
//*
//* If FACT = 'E', then IPIV is an output argument and on exit
//* contains the pivot indices from the factorization A = P*L*U
//* of the equilibrated matrix A.
int *IPIV = (int*) Malloc(sizeof(int)*N);
//* EQUED (input or output) CHARACTER*1
//* Specifies the form of equilibration that was done.
//* = 'N': No equilibration (always true if FACT = 'N').
//* = 'R': Row equilibration, i.e., A has been premultiplied by
//* diag(R).
//* = 'C': Column equilibration, i.e., A has been postmultiplied
//* by diag(C).
//* = 'B': Both row and column equilibration, i.e., A has been
//* replaced by diag(R) * A * diag(C).
//* EQUED is an input argument if FACT = 'F'; otherwise, it is an
//* output argument.
char EQUED;
//* R (input or output) DOUBLE PRECISION array, dimension (N)
//* The row scale factors for A. If EQUED = 'R' or 'B', A is
//* multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
//* is not accessed. R is an input argument if FACT = 'F';
//* otherwise, R is an output argument. If FACT = 'F' and
//* EQUED = 'R' or 'B', each element of R must be positive.
double *R = (double*) Malloc(sizeof(double)*N);
//* C (input or output) DOUBLE PRECISION array, dimension (N)
//* The column scale factors for A. If EQUED = 'C' or 'B', A is
//* multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
//* is not accessed. C is an input argument if FACT = 'F';
//* otherwise, C is an output argument. If FACT = 'F' and
//* EQUED = 'C' or 'B', each element of C must be positive.
double *C = (double*) Malloc(sizeof(double)*N);
//* B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
//* On entry, the N-by-NRHS right hand side matrix B.
//* On exit,
//* if EQUED = 'N', B is not modified;
//* if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
//* diag(R)*B;
//* if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
//* overwritten by diag(C)*B.
double *B = b;
//* LDB (input) INTEGER
//* The leading dimension of the array B. LDB >= max(1,N).
int LDB = m;
//* X (output) COMPLEX*16 array, dimension (LDX,NRHS)
//* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
//* to the original system of equations. Note that A and B are
//* modified on exit if EQUED .ne. 'N', and the solution to the
//* equilibrated system is inv(diag(C))*X if TRANS = 'N' and
//* EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
//* and EQUED = 'R' or 'B'.
double *X = c;
//* LDX (input) INTEGER
//* The leading dimension of the array X. LDX >= max(1,N).
int LDX = m;
//* RCOND (output) DOUBLE PRECISION
//* The estimate of the reciprocal condition number of the matrix
//* A after equilibration (if done). If RCOND is less than the
//* machine precision (in particular, if RCOND = 0), the matrix
//* is singular to working precision. This condition is
//* indicated by a return code of INFO > 0.
double RCOND;
//* FERR (output) DOUBLE PRECISION array, dimension (NRHS)
//* The estimated forward error bound for each solution vector
//* X(j) (the j-th column of the solution matrix X).
//* If XTRUE is the true solution corresponding to X(j), FERR(j)
//* is an estimated upper bound for the magnitude of the largest
//* element in (X(j) - XTRUE) divided by the magnitude of the
//* largest element in X(j). The estimate is as reliable as
//* the estimate for RCOND, and is almost always a slight
//* overestimate of the true error.
double *FERR = (double*) Malloc(n*sizeof(double));
//* BERR (output) DOUBLE PRECISION array, dimension (NRHS)
//* The componentwise relative backward error of each solution
//* vector X(j) (i.e., the smallest relative change in
//* any element of A or B that makes X(j) an exact solution).
double *BERR = (double*) Malloc(n*sizeof(double));
//* WORK (workspace) COMPLEX*16 array, dimension (2*N)
double *WORK = (double*) Malloc(2*(2*N)*sizeof(double));
//* RWORK (workspace/output) DOUBLE PRECISION array, dimension (2*N)
//* On exit, RWORK(1) contains the reciprocal pivot growth
//* factor norm(A)/norm(U). The "max absolute element" norm is
//* used. If RWORK(1) is much less than 1, then the stability
//* of the LU factorization of the (equilibrated) matrix A
//* could be poor. This also means that the solution X, condition
//* estimator RCOND, and forward error bound FERR could be
//* unreliable. If factorization fails with 0<INFO<=N, then
//* RWORK(1) contains the reciprocal pivot growth factor for the
//* leading INFO columns of A.
double *RWORK = (double*) Malloc(2*N*sizeof(double));
//* INFO (output) INTEGER
//* = 0: successful exit
//* < 0: if INFO = -i, the i-th argument had an illegal value
//* > 0: if INFO = i, and i is
//* <= N: U(i,i) is exactly zero. The factorization has
//* been completed, but the factor U is exactly
//* singular, so the solution and error bounds
//* could not be computed. RCOND = 0 is returned.
//* = N+1: U is nonsingular, but RCOND is less than machine
//* precision, meaning that the matrix is singular
//* to working precision. Nevertheless, the
//* solution and error bounds are computed because
//* there are a number of situations where the
//* computed solution can be more accurate than the
//* value of RCOND would suggest.
//*
//* =====================================================================
int INFO;
zgesvx_(&FACT, &TRANS, &N, &NRHS, A, &LDA, AF, &LDAF, IPIV, &EQUED, R, C, B,
&LDB, X, &LDX, &RCOND, FERR, BERR, WORK, RWORK, &INFO);
if ((INFO == N) || (INFO == N+1) || (RCOND < getEPS())) {
snprintf(msgBuffer,MSGBUFLEN,"Matrix is singular to working precision. RCOND = %e\n",RCOND);
eval->warningMessage(msgBuffer);
}
// Free the allocated arrays...
Free(AF);
Free(IPIV);
Free(R);
Free(C);
Free(WORK);
Free(RWORK);
Free(FERR);
Free(BERR);
}
/***************************************************************************
* Linear equation solver for float matrices
***************************************************************************/
// Solve A*C = B, where A is m x m, and B is m x n, all quantities are real.
void floatSolveLinEq(Interpreter* eval, int m, int n, float *c, float* a, float *b) {
if ((m == 0) || (n == 0)) return;
char msgBuffer[MSGBUFLEN];
// Here are the comments from the LAPACK routine used:
//SUBROUTINE SGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
//$ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
//$ WORK, IWORK, INFO )
//*
//* -- LAPACK driver routine (version 3.0) --
//* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
//* Courant Institute, Argonne National Lab, and Rice University
//* June 30, 1999
//*
//* .. Scalar Arguments ..
// CHARACTER EQUED, FACT, TRANS
// INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
// REAL RCOND
//* ..
//* .. Array Arguments ..
// INTEGER IPIV( * ), IWORK( * )
// REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
// $ BERR( * ), C( * ), FERR( * ), R( * ),
// $ WORK( * ), X( LDX, * )
//* ..
//*
//* Purpose
//* =======
//*
//* SGESVX uses the LU factorization to compute the solution to a real
//* system of linear equations
//* A * X = B,
//* where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
//*
//* Error bounds on the solution and a condition estimate are also
//* provided.
//*
//* Description
//* ===========
//*
//* The following steps are performed:
//*
//* 1. If FACT = 'E', real scaling factors are computed to equilibrate
//* the system:
//* TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
//* TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
//* TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
//* Whether or not the system will be equilibrated depends on the
//* scaling of the matrix A, but if equilibration is used, A is
//* overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
//* or diag(C)*B (if TRANS = 'T' or 'C').
//*
//* 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
//* matrix A (after equilibration if FACT = 'E') as
//* A = P * L * U,
//* where P is a permutation matrix, L is a unit lower triangular
//* matrix, and U is upper triangular.
//*
//* 3. If some U(i,i)=0, so that U is exactly singular, then the routine
//* returns with INFO = i. Otherwise, the factored form of A is used
//* to estimate the condition number of the matrix A. If the
//* reciprocal of the condition number is less than machine precision,
//* INFO = N+1 is returned as a warning, but the routine still goes on
//* to solve for X and compute error bounds as described below.
//*
//* 4. The system of equations is solved for X using the factored form
//* of A.
//*
//* 5. Iterative refinement is applied to improve the computed solution
//* matrix and calculate error bounds and backward error estimates
//* for it.
//*
//* 6. If equilibration was used, the matrix X is premultiplied by
//* diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
//* that it solves the original system before equilibration.
//*
//* Arguments
//* =========
//*
//* FACT (input) CHARACTER*1
//* Specifies whether or not the factored form of the matrix A is
//* supplied on entry, and if not, whether the matrix A should be
//* equilibrated before it is factored.
//* = 'F': On entry, AF and IPIV contain the factored form of A.
//* If EQUED is not 'N', the matrix A has been
//* equilibrated with scaling factors given by R and C.
//* A, AF, and IPIV are not modified.
//* = 'N': The matrix A will be copied to AF and factored.
//* = 'E': The matrix A will be equilibrated if necessary, then
//* copied to AF and factored.
char FACT = 'E';
//*
//* TRANS (input) CHARACTER*1
//* Specifies the form of the system of equations:
//* = 'N': A * X = B (No transpose)
//* = 'T': A**T * X = B (Transpose)
//* = 'C': A**H * X = B (Transpose)
char TRANS = 'N';
//*
//* N (input) INTEGER
//* The number of linear equations, i.e., the order of the
//* matrix A. N >= 0.
//*
int N = m;
//* NRHS (input) INTEGER
//* The number of right hand sides, i.e., the number of columns
//* of the matrices B and X. NRHS >= 0.
int NRHS = n;
//*
//* A (input/output) REAL array, dimension (LDA,N)
//* On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
//* not 'N', then A must have been equilibrated by the scaling
//* factors in R and/or C. A is not modified if FACT = 'F' or
//* 'N', or if FACT = 'E' and EQUED = 'N' on exit.
//*
//* On exit, if EQUED .ne. 'N', A is scaled as follows:
//* EQUED = 'R': A := diag(R) * A
//* EQUED = 'C': A := A * diag(C)
//* EQUED = 'B': A := diag(R) * A * diag(C).
float* A = a;
//* LDA (input) INTEGER
//* The leading dimension of the array A. LDA >= max(1,N).
//*
int LDA = m;
//* AF (input or output) REAL array, dimension (LDAF,N)
//* If FACT = 'F', then AF is an input argument and on entry
//* contains the factors L and U from the factorization
//* A = P*L*U as computed by DGETRF. If EQUED .ne. 'N', then
//* AF is the factored form of the equilibrated matrix A.
//*
//* If FACT = 'N', then AF is an output argument and on exit
//* returns the factors L and U from the factorization A = P*L*U
//* of the original matrix A.
//*
//* If FACT = 'E', then AF is an output argument and on exit
//* returns the factors L and U from the factorization A = P*L*U
//* of the equilibrated matrix A (see the description of A for
//* the form of the equilibrated matrix).
float *AF = (float*) Malloc(sizeof(float)*LDA*N);
//* LDAF (input) INTEGER
//* The leading dimension of the array AF. LDAF >= max(1,N).
//*
int LDAF = m;
//* IPIV (input or output) INTEGER array, dimension (N)
//* If FACT = 'F', then IPIV is an input argument and on entry
//* contains the pivot indices from the factorization A = P*L*U
//* as computed by DGETRF; row i of the matrix was interchanged
//* with row IPIV(i).
//*
//* If FACT = 'N', then IPIV is an output argument and on exit
//* contains the pivot indices from the factorization A = P*L*U
//* of the original matrix A.
//*
//* If FACT = 'E', then IPIV is an output argument and on exit
//* contains the pivot indices from the factorization A = P*L*U
//* of the equilibrated matrix A.
int *IPIV = (int*) Malloc(sizeof(int)*N);
//* EQUED (input or output) CHARACTER*1
//* Specifies the form of equilibration that was done.
//* = 'N': No equilibration (always true if FACT = 'N').
//* = 'R': Row equilibration, i.e., A has been premultiplied by
//* diag(R).
//* = 'C': Column equilibration, i.e., A has been postmultiplied
//* by diag(C).
//* = 'B': Both row and column equilibration, i.e., A has been
//* replaced by diag(R) * A * diag(C).
//* EQUED is an input argument if FACT = 'F'; otherwise, it is an
//* output argument.
char EQUED;
//* R (input or output) REAL array, dimension (N)
//* The row scale factors for A. If EQUED = 'R' or 'B', A is
//* multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
//* is not accessed. R is an input argument if FACT = 'F';
//* otherwise, R is an output argument. If FACT = 'F' and
//* EQUED = 'R' or 'B', each element of R must be positive.
float *R = (float*) Malloc(sizeof(float)*N);
//* C (input or output) REAL array, dimension (N)
//* The column scale factors for A. If EQUED = 'C' or 'B', A is
//* multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
//* is not accessed. C is an input argument if FACT = 'F';
//* otherwise, C is an output argument. If FACT = 'F' and
//* EQUED = 'C' or 'B', each element of C must be positive.
float *C = (float*) Malloc(sizeof(float)*N);
//* B (input/output) REAL array, dimension (LDB,NRHS)
//* On entry, the N-by-NRHS right hand side matrix B.
//* On exit,
//* if EQUED = 'N', B is not modified;
//* if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
//* diag(R)*B;
//* if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
//* overwritten by diag(C)*B.
float *B = b;
//* LDB (input) INTEGER
//* The leading dimension of the array B. LDB >= max(1,N).
int LDB = m;
//* X (output) REAL array, dimension (LDX,NRHS)
//* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
//* to the original system of equations. Note that A and B are
//* modified on exit if EQUED .ne. 'N', and the solution to the
//* equilibrated system is inv(diag(C))*X if TRANS = 'N' and
//* EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
//* and EQUED = 'R' or 'B'.
float *X = c;
//* LDX (input) INTEGER
//* The leading dimension of the array X. LDX >= max(1,N).
int LDX = m;
//* RCOND (output) REAL
//* The estimate of the reciprocal condition number of the matrix
//* A after equilibration (if done). If RCOND is less than the
//* machine precision (in particular, if RCOND = 0), the matrix
//* is singular to working precision. This condition is
//* indicated by a return code of INFO > 0.
float RCOND;
//* FERR (output) REAL array, dimension (NRHS)
//* The estimated forward error bound for each solution vector
//* X(j) (the j-th column of the solution matrix X).
//* If XTRUE is the true solution corresponding to X(j), FERR(j)
//* is an estimated upper bound for the magnitude of the largest
//* element in (X(j) - XTRUE) divided by the magnitude of the
//* largest element in X(j). The estimate is as reliable as
//* the estimate for RCOND, and is almost always a slight
//* overestimate of the true error.
float *FERR = (float*) Malloc(n*sizeof(float));
//* BERR (output) REAL array, dimension (NRHS)
//* The componentwise relative backward error of each solution
//* vector X(j) (i.e., the smallest relative change in
//* any element of A or B that makes X(j) an exact solution).
float *BERR = (float*) Malloc(n*sizeof(float));
//* WORK (workspace/output) REAL array, dimension (4*N)
//* On exit, WORK(1) contains the reciprocal pivot growth
//* factor norm(A)/norm(U). The "max absolute element" norm is
//* used. If WORK(1) is much less than 1, then the stability
//* of the LU factorization of the (equilibrated) matrix A
//* could be poor. This also means that the solution X, condition
//* estimator RCOND, and forward error bound FERR could be
//* unreliable. If factorization fails with 0<INFO<=N, then
//* WORK(1) contains the reciprocal pivot growth factor for the
//* leading INFO columns of A.
float *WORK = (float*) Malloc(4*N*sizeof(float));
//* IWORK (workspace) INTEGER array, dimension (N)
int *IWORK = (int*) Malloc(4*N*sizeof(int));
//* INFO (output) INTEGER
//* = 0: successful exit
//* < 0: if INFO = -i, the i-th argument had an illegal value
//* > 0: if INFO = i, and i is
//* <= N: U(i,i) is exactly zero. The factorization has
//* been completed, but the factor U is exactly
//* singular, so the solution and error bounds
//* could not be computed. RCOND = 0 is returned.
//* = N+1: U is nonsingular, but RCOND is less than machine
//* precision, meaning that the matrix is singular
//* to working precision. Nevertheless, the
//* solution and error bounds are computed because
//* there are a number of situations where the
//* computed solution can be more accurate than the
//* value of RCOND would suggest.
//*
//* =====================================================================
int INFO;
sgesvx_(&FACT, &TRANS, &N, &NRHS, A, &LDA, AF, &LDAF, IPIV, &EQUED, R, C, B,
&LDB, X, &LDX, &RCOND, FERR, BERR, WORK, IWORK, &INFO);
if ((INFO == N) || (INFO == N+1) || (RCOND < getFloatEPS())) {
snprintf(msgBuffer,MSGBUFLEN,"Matrix is singular to working (single) precision. RCOND = %e\n",RCOND);
eval->warningMessage(msgBuffer);
}
// Free the allocated arrays...
Free(AF);
Free(IPIV);
Free(R);
Free(C);
Free(WORK);
Free(IWORK);
}
// Solve A*C = B, where A is m x m, and B is m x n, all quantities are real.
void complexSolveLinEq(Interpreter* eval, int m, int n, float *c, float* a, float *b) {
if ((m == 0) || (n == 0)) return;
char msgBuffer[MSGBUFLEN];
// Here are the comments from the LAPACK routine used:
//SUBROUTINE CGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
//$ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
//$ WORK, IWORK, INFO )
//*
//* -- LAPACK driver routine (version 3.0) --
//* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
//* Courant Institute, Argonne National Lab, and Rice University
//* June 30, 1999
//*
//* .. Scalar Arguments ..
// CHARACTER EQUED, FACT, TRANS
// INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
// REAL RCOND
//* ..
//* .. Array Arguments ..
// INTEGER IPIV( * )
// REAL BERR( * ), C( * ), FERR( * ), R( * ),
// $ RWORK( * )
// COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
// $ WORK( * ), X( LDX, * )
//* ..
//*
//* Purpose
//* =======
//*
//* ZGESVX uses the LU factorization to compute the solution to a real
//* system of linear equations
//* A * X = B,
//* where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
//*
//* Error bounds on the solution and a condition estimate are also
//* provided.
//*
//* Description
//* ===========
//*
//* The following steps are performed:
//*
//* 1. If FACT = 'E', real scaling factors are computed to equilibrate
//* the system:
//* TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
//* TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
//* TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
//* Whether or not the system will be equilibrated depends on the
//* scaling of the matrix A, but if equilibration is used, A is
//* overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
//* or diag(C)*B (if TRANS = 'T' or 'C').
//*
//* 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
//* matrix A (after equilibration if FACT = 'E') as
//* A = P * L * U,
//* where P is a permutation matrix, L is a unit lower triangular
//* matrix, and U is upper triangular.
//*
//* 3. If some U(i,i)=0, so that U is exactly singular, then the routine
//* returns with INFO = i. Otherwise, the factored form of A is used
//* to estimate the condition number of the matrix A. If the
//* reciprocal of the condition number is less than machine precision,
//* INFO = N+1 is returned as a warning, but the routine still goes on
//* to solve for X and compute error bounds as described below.
//*
//* 4. The system of equations is solved for X using the factored form
//* of A.
//*
//* 5. Iterative refinement is applied to improve the computed solution
//* matrix and calculate error bounds and backward error estimates
//* for it.
//*
//* 6. If equilibration was used, the matrix X is premultiplied by
//* diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
//* that it solves the original system before equilibration.
//*
//* Arguments
//* =========
//*
//* FACT (input) CHARACTER*1
//* Specifies whether or not the factored form of the matrix A is
//* supplied on entry, and if not, whether the matrix A should be
//* equilibrated before it is factored.
//* = 'F': On entry, AF and IPIV contain the factored form of A.
//* If EQUED is not 'N', the matrix A has been
//* equilibrated with scaling factors given by R and C.
//* A, AF, and IPIV are not modified.
//* = 'N': The matrix A will be copied to AF and factored.
//* = 'E': The matrix A will be equilibrated if necessary, then
//* copied to AF and factored.
char FACT = 'E';
//*
//* TRANS (input) CHARACTER*1
//* Specifies the form of the system of equations:
//* = 'N': A * X = B (No transpose)
//* = 'T': A**T * X = B (Transpose)
//* = 'C': A**H * X = B (Transpose)
char TRANS = 'N';
//*
//* N (input) INTEGER
//* The number of linear equations, i.e., the order of the
//* matrix A. N >= 0.
//*
int N = m;
//* NRHS (input) INTEGER
//* The number of right hand sides, i.e., the number of columns
//* of the matrices B and X. NRHS >= 0.
int NRHS = n;
//* A (input/output) COMPLEX array, dimension (LDA,N)
//* On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
//* not 'N', then A must have been equilibrated by the scaling
//* factors in R and/or C. A is not modified if FACT = 'F' or
//* 'N', or if FACT = 'E' and EQUED = 'N' on exit.
//*
//* On exit, if EQUED .ne. 'N', A is scaled as follows:
//* EQUED = 'R': A := diag(R) * A
//* EQUED = 'C': A := A * diag(C)
//* EQUED = 'B': A := diag(R) * A * diag(C).
float* A = a;
//* LDA (input) INTEGER
//* The leading dimension of the array A. LDA >= max(1,N).
//*
int LDA = m;
//* AF (input or output) COMPLEX array, dimension (LDAF,N)
//* If FACT = 'F', then AF is an input argument and on entry
//* contains the factors L and U from the factorization
//* A = P*L*U as computed by DGETRF. If EQUED .ne. 'N', then
//* AF is the factored form of the equilibrated matrix A.
//*
//* If FACT = 'N', then AF is an output argument and on exit
//* returns the factors L and U from the factorization A = P*L*U
//* of the original matrix A.
//*
//* If FACT = 'E', then AF is an output argument and on exit
//* returns the factors L and U from the factorization A = P*L*U
//* of the equilibrated matrix A (see the description of A for
//* the form of the equilibrated matrix).
float *AF = (float*) Malloc(2*sizeof(float)*LDA*N);
//* LDAF (input) INTEGER
//* The leading dimension of the array AF. LDAF >= max(1,N).
//*
int LDAF = m;
//* IPIV (input or output) INTEGER array, dimension (N)
//* If FACT = 'F', then IPIV is an input argument and on entry
//* contains the pivot indices from the factorization A = P*L*U
//* as computed by DGETRF; row i of the matrix was interchanged
//* with row IPIV(i).
//*
//* If FACT = 'N', then IPIV is an output argument and on exit
//* contains the pivot indices from the factorization A = P*L*U
//* of the original matrix A.
//*
//* If FACT = 'E', then IPIV is an output argument and on exit
//* contains the pivot indices from the factorization A = P*L*U
//* of the equilibrated matrix A.
int *IPIV = (int*) Malloc(sizeof(int)*N);
//* EQUED (input or output) CHARACTER*1
//* Specifies the form of equilibration that was done.
//* = 'N': No equilibration (always true if FACT = 'N').
//* = 'R': Row equilibration, i.e., A has been premultiplied by
//* diag(R).
//* = 'C': Column equilibration, i.e., A has been postmultiplied
//* by diag(C).
//* = 'B': Both row and column equilibration, i.e., A has been
//* replaced by diag(R) * A * diag(C).
//* EQUED is an input argument if FACT = 'F'; otherwise, it is an
//* output argument.
char EQUED;
//* R (input or output) REAL array, dimension (N)
//* The row scale factors for A. If EQUED = 'R' or 'B', A is
//* multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
//* is not accessed. R is an input argument if FACT = 'F';
//* otherwise, R is an output argument. If FACT = 'F' and
//* EQUED = 'R' or 'B', each element of R must be positive.
float *R = (float*) Malloc(sizeof(float)*N);
//* C (input or output) REAL array, dimension (N)
//* The column scale factors for A. If EQUED = 'C' or 'B', A is
//* multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
//* is not accessed. C is an input argument if FACT = 'F';
//* otherwise, C is an output argument. If FACT = 'F' and
//* EQUED = 'C' or 'B', each element of C must be positive.
float *C = (float*) Malloc(sizeof(float)*N);
//* B (input/output) COMPLEX array, dimension (LDB,NRHS)
//* On entry, the N-by-NRHS right hand side matrix B.
//* On exit,
//* if EQUED = 'N', B is not modified;
//* if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
//* diag(R)*B;
//* if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
//* overwritten by diag(C)*B.
float *B = b;
//* LDB (input) INTEGER
//* The leading dimension of the array B. LDB >= max(1,N).
int LDB = m;
//* X (output) COMPLEX array, dimension (LDX,NRHS)
//* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
//* to the original system of equations. Note that A and B are
//* modified on exit if EQUED .ne. 'N', and the solution to the
//* equilibrated system is inv(diag(C))*X if TRANS = 'N' and
//* EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
//* and EQUED = 'R' or 'B'.
float *X = c;
//* LDX (input) INTEGER
//* The leading dimension of the array X. LDX >= max(1,N).
int LDX = m;
//* RCOND (output) REAL
//* The estimate of the reciprocal condition number of the matrix
//* A after equilibration (if done). If RCOND is less than the
//* machine precision (in particular, if RCOND = 0), the matrix
//* is singular to working precision. This condition is
//* indicated by a return code of INFO > 0.
float RCOND;
//* FERR (output) REAL array, dimension (NRHS)
//* The estimated forward error bound for each solution vector
//* X(j) (the j-th column of the solution matrix X).
//* If XTRUE is the true solution corresponding to X(j), FERR(j)
//* is an estimated upper bound for the magnitude of the largest
//* element in (X(j) - XTRUE) divided by the magnitude of the
//* largest element in X(j). The estimate is as reliable as
//* the estimate for RCOND, and is almost always a slight
//* overestimate of the true error.
float *FERR = (float*) Malloc(n*sizeof(float));
//* BERR (output) REAL array, dimension (NRHS)
//* The componentwise relative backward error of each solution
//* vector X(j) (i.e., the smallest relative change in
//* any element of A or B that makes X(j) an exact solution).
float *BERR = (float*) Malloc(n*sizeof(float));
//* WORK (workspace) COMPLEX array, dimension (2*N)
float *WORK = (float*) Malloc(2*(2*N)*sizeof(float));
//* RWORK (workspace/output) REAL array, dimension (2*N)
//* On exit, RWORK(1) contains the reciprocal pivot growth
//* factor norm(A)/norm(U). The "max absolute element" norm is
//* used. If RWORK(1) is much less than 1, then the stability
//* of the LU factorization of the (equilibrated) matrix A
//* could be poor. This also means that the solution X, condition
//* estimator RCOND, and forward error bound FERR could be
//* unreliable. If factorization fails with 0<INFO<=N, then
//* RWORK(1) contains the reciprocal pivot growth factor for the
//* leading INFO columns of A.
float *RWORK = (float*) Malloc(2*N*sizeof(float));
//* INFO (output) INTEGER
//* = 0: successful exit
//* < 0: if INFO = -i, the i-th argument had an illegal value
//* > 0: if INFO = i, and i is
//* <= N: U(i,i) is exactly zero. The factorization has
//* been completed, but the factor U is exactly
//* singular, so the solution and error bounds
//* could not be computed. RCOND = 0 is returned.
//* = N+1: U is nonsingular, but RCOND is less than machine
//* precision, meaning that the matrix is singular
//* to working precision. Nevertheless, the
//* solution and error bounds are computed because
//* there are a number of situations where the
//* computed solution can be more accurate than the
//* value of RCOND would suggest.
//*
//* =====================================================================
int INFO;
cgesvx_(&FACT, &TRANS, &N, &NRHS, A, &LDA, AF, &LDAF, IPIV, &EQUED, R, C, B,
&LDB, X, &LDX, &RCOND, FERR, BERR, WORK, RWORK, &INFO);
if ((INFO == N) || (INFO == N+1) || (RCOND < getFloatEPS())) {
snprintf(msgBuffer,MSGBUFLEN,"Matrix is singular to working (single) precision. RCOND = %e\n",RCOND);
eval->warningMessage(msgBuffer);
}
// Free the allocated arrays...
Free(IPIV);
Free(R);
Free(C);
Free(WORK);
Free(RWORK);
Free(FERR);
Free(BERR);
}
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