.SH NAME DSGESV - the solution to a real system of linear equations A * X = B, .SH SYNOPSIS .TP 19 SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK, .TP 19 .ti +4 + SWORK, ITER, INFO) .TP 19 .ti +4 INTEGER INFO,ITER,LDA,LDB,LDX,N,NRHS .TP 19 .ti +4 INTEGER IPIV(*) .TP 19 .ti +4 REAL SWORK(*) .TP 19 .ti +4 DOUBLE PRECISION A(LDA,*),B(LDB,*),WORK(N,*),X(LDX,*) .SH PURPOSE DSGESV computes 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. DSGESV first attempts to factorize the matrix in SINGLE PRECISION and use this factorization within an iterative refinement procedure to produce a solution with DOUBLE PRECISION normwise backward error quality (see below). If the approach fails the method switches to a DOUBLE PRECISION factorization and solve. .br The iterative refinement is not going to be a winning strategy if the ratio SINGLE PRECISION performance over DOUBLE PRECISION performance is too small. A reasonable strategy should take the number of right-hand sides and the size of the matrix into account. This might be done with a call to ILAENV in the future. Up to now, we always try iterative refinement. The iterative refinement process is stopped if .br ITER > ITERMAX .br or for all the RHS we have: .br RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX .br where .br o ITER is the number of the current iteration in the iterative refinement process .br o RNRM is the infinity-norm of the residual .br o XNRM is the infinity-norm of the solution .br o ANRM is the infinity-operator-norm of the matrix A .br o EPS is the machine epsilon returned by DLAMCH(\(aqEpsilon\(aq) The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 respectively. .SH ARGUMENTS .TP 8 N (input) INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0. .TP 8 NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. .TP 8 A (input or input/ouptut) DOUBLE PRECISION array, dimension (LDA,N) On entry, the N-by-N coefficient matrix A. On exit, if iterative refinement has been successfully used (INFO.EQ.0 and ITER.GE.0, see description below), then A is unchanged, if double precision factorization has been used (INFO.EQ.0 and ITER.LT.0, see description below), then the array A contains the factors L and U from the factorization A = P*L*U; the unit diagonal elements of L are not stored. .TP 8 LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). .TP 8 IPIV (output) INTEGER array, dimension (N) The pivot indices that define the permutation matrix P; row i of the matrix was interchanged with row IPIV(i). Corresponds either to the single precision factorization (if INFO.EQ.0 and ITER.GE.0) or the double precision factorization (if INFO.EQ.0 and ITER.LT.0). .TP 8 B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) The N-by-NRHS matrix of right hand side matrix B. .TP 8 LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). .TP 8 X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) If INFO = 0, the N-by-NRHS solution matrix X. .TP 8 LDX (input) INTEGER The leading dimension of the array X. LDX >= max(1,N). .TP 8 WORK (workspace) DOUBLE PRECISION array, dimension (N*NRHS) This array is used to hold the residual vectors. .TP 8 SWORK (workspace) REAL array, dimension (N*(N+NRHS)) This array is used to use the single precision matrix and the right-hand sides or solutions in single precision. .TP 8 ITER (output) INTEGER < 0: iterative refinement has failed, double precision factorization has been performed -1 : taking into account machine parameters, N, NRHS, it is a priori not worth working in SINGLE PRECISION -2 : overflow of an entry when moving from double to SINGLE PRECISION -3 : failure of SGETRF .br -31: stop the iterative refinement after the 30th iterations > 0: iterative refinement has been sucessfully used. Returns the number of iterations .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: if INFO = i, U(i,i) computed in DOUBLE PRECISION is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed. =========