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LAPACK-3 - computes the solution to a real system of linear equations A * X = B,
SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK, SWORK, ITER, INFO ) INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS INTEGER IPIV( * ) REAL SWORK( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( N, * ), X( LDX, * )
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. 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 ITER > ITERMAX or for all the RHS we have: RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX where o ITER is the number of the current iteration in the iterative refinement process o RNRM is the infinity-norm of the residual o XNRM is the infinity-norm of the solution o ANRM is the infinity-operator-norm of the matrix A o EPS is the machine epsilon returned by DLAMCH('Epsilon') The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 respectively.
N (input) INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0. NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. A (input/output) 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. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). 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). B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) The N-by-NRHS right hand side matrix B. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) If INFO = 0, the N-by-NRHS solution matrix X. LDX (input) INTEGER The leading dimension of the array X. LDX >= max(1,N). WORK (workspace) DOUBLE PRECISION array, dimension (N,NRHS) This array is used to hold the residual vectors. 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. ITER (output) INTEGER < 0: iterative refinement has failed, double precision factorization has been performed -1 : the routine fell back to full precision for implementation- or machine-specific reasons -2 : narrowing the precision induced an overflow, the routine fell back to full precision -3 : failure of SGETRF -31: stop the iterative refinement after the 30th iterations > 0: iterative refinement has been sucessfully used. Returns the number of iterations INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 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. LAPACK PROTOTYPE driver routine (version 3.April 2011 DSGESV(3lapack)