*> \brief DSGESV computes the solution to system of linear equations A * X = B for GE matrices (mixed precision with iterative refinement) * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DSGESV + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK, * SWORK, ITER, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS * .. * .. Array Arguments .. * INTEGER IPIV( * ) * REAL SWORK( * ) * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( N, * ), * \$ X( LDX, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> 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. *> \endverbatim * * Arguments: * ========== * *> \param[in] N *> \verbatim *> N is INTEGER *> The number of linear equations, i.e., the order of the *> matrix A. N >= 0. *> \endverbatim *> *> \param[in] NRHS *> \verbatim *> NRHS is INTEGER *> The number of right hand sides, i.e., the number of columns *> of the matrix B. NRHS >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is 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. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[out] IPIV *> \verbatim *> IPIV is 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). *> \endverbatim *> *> \param[in] B *> \verbatim *> B is DOUBLE PRECISION array, dimension (LDB,NRHS) *> The N-by-NRHS right hand side matrix B. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= max(1,N). *> \endverbatim *> *> \param[out] X *> \verbatim *> X is DOUBLE PRECISION array, dimension (LDX,NRHS) *> If INFO = 0, the N-by-NRHS solution matrix X. *> \endverbatim *> *> \param[in] LDX *> \verbatim *> LDX is INTEGER *> The leading dimension of the array X. LDX >= max(1,N). *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (N,NRHS) *> This array is used to hold the residual vectors. *> \endverbatim *> *> \param[out] SWORK *> \verbatim *> SWORK is 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. *> \endverbatim *> *> \param[out] ITER *> \verbatim *> ITER is 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 successfully used. *> Returns the number of iterations *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is 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. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date June 2016 * *> \ingroup doubleGEsolve * * ===================================================================== SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK, \$ SWORK, ITER, INFO ) * * -- LAPACK driver routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * June 2016 * * .. Scalar Arguments .. INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS * .. * .. Array Arguments .. INTEGER IPIV( * ) REAL SWORK( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( N, * ), \$ X( LDX, * ) * .. * * ===================================================================== * * .. Parameters .. LOGICAL DOITREF PARAMETER ( DOITREF = .TRUE. ) * INTEGER ITERMAX PARAMETER ( ITERMAX = 30 ) * DOUBLE PRECISION BWDMAX PARAMETER ( BWDMAX = 1.0E+00 ) * DOUBLE PRECISION NEGONE, ONE PARAMETER ( NEGONE = -1.0D+0, ONE = 1.0D+0 ) * * .. Local Scalars .. INTEGER I, IITER, PTSA, PTSX DOUBLE PRECISION ANRM, CTE, EPS, RNRM, XNRM * * .. External Subroutines .. EXTERNAL DAXPY, DGEMM, DLACPY, DLAG2S, SLAG2D, SGETRF, \$ SGETRS, XERBLA * .. * .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL IDAMAX, DLAMCH, DLANGE * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX, SQRT * .. * .. Executable Statements .. * INFO = 0 ITER = 0 * * Test the input parameters. * IF( N.LT.0 ) THEN INFO = -1 ELSE IF( NRHS.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -9 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSGESV', -INFO ) RETURN END IF * * Quick return if (N.EQ.0). * IF( N.EQ.0 ) \$ RETURN * * Skip single precision iterative refinement if a priori slower * than double precision factorization. * IF( .NOT.DOITREF ) THEN ITER = -1 GO TO 40 END IF * * Compute some constants. * ANRM = DLANGE( 'I', N, N, A, LDA, WORK ) EPS = DLAMCH( 'Epsilon' ) CTE = ANRM*EPS*SQRT( DBLE( N ) )*BWDMAX * * Set the indices PTSA, PTSX for referencing SA and SX in SWORK. * PTSA = 1 PTSX = PTSA + N*N * * Convert B from double precision to single precision and store the * result in SX. * CALL DLAG2S( N, NRHS, B, LDB, SWORK( PTSX ), N, INFO ) * IF( INFO.NE.0 ) THEN ITER = -2 GO TO 40 END IF * * Convert A from double precision to single precision and store the * result in SA. * CALL DLAG2S( N, N, A, LDA, SWORK( PTSA ), N, INFO ) * IF( INFO.NE.0 ) THEN ITER = -2 GO TO 40 END IF * * Compute the LU factorization of SA. * CALL SGETRF( N, N, SWORK( PTSA ), N, IPIV, INFO ) * IF( INFO.NE.0 ) THEN ITER = -3 GO TO 40 END IF * * Solve the system SA*SX = SB. * CALL SGETRS( 'No transpose', N, NRHS, SWORK( PTSA ), N, IPIV, \$ SWORK( PTSX ), N, INFO ) * * Convert SX back to double precision * CALL SLAG2D( N, NRHS, SWORK( PTSX ), N, X, LDX, INFO ) * * Compute R = B - AX (R is WORK). * CALL DLACPY( 'All', N, NRHS, B, LDB, WORK, N ) * CALL DGEMM( 'No Transpose', 'No Transpose', N, NRHS, N, NEGONE, A, \$ LDA, X, LDX, ONE, WORK, N ) * * Check whether the NRHS normwise backward errors satisfy the * stopping criterion. If yes, set ITER=0 and return. * DO I = 1, NRHS XNRM = ABS( X( IDAMAX( N, X( 1, I ), 1 ), I ) ) RNRM = ABS( WORK( IDAMAX( N, WORK( 1, I ), 1 ), I ) ) IF( RNRM.GT.XNRM*CTE ) \$ GO TO 10 END DO * * If we are here, the NRHS normwise backward errors satisfy the * stopping criterion. We are good to exit. * ITER = 0 RETURN * 10 CONTINUE * DO 30 IITER = 1, ITERMAX * * Convert R (in WORK) from double precision to single precision * and store the result in SX. * CALL DLAG2S( N, NRHS, WORK, N, SWORK( PTSX ), N, INFO ) * IF( INFO.NE.0 ) THEN ITER = -2 GO TO 40 END IF * * Solve the system SA*SX = SR. * CALL SGETRS( 'No transpose', N, NRHS, SWORK( PTSA ), N, IPIV, \$ SWORK( PTSX ), N, INFO ) * * Convert SX back to double precision and update the current * iterate. * CALL SLAG2D( N, NRHS, SWORK( PTSX ), N, WORK, N, INFO ) * DO I = 1, NRHS CALL DAXPY( N, ONE, WORK( 1, I ), 1, X( 1, I ), 1 ) END DO * * Compute R = B - AX (R is WORK). * CALL DLACPY( 'All', N, NRHS, B, LDB, WORK, N ) * CALL DGEMM( 'No Transpose', 'No Transpose', N, NRHS, N, NEGONE, \$ A, LDA, X, LDX, ONE, WORK, N ) * * Check whether the NRHS normwise backward errors satisfy the * stopping criterion. If yes, set ITER=IITER>0 and return. * DO I = 1, NRHS XNRM = ABS( X( IDAMAX( N, X( 1, I ), 1 ), I ) ) RNRM = ABS( WORK( IDAMAX( N, WORK( 1, I ), 1 ), I ) ) IF( RNRM.GT.XNRM*CTE ) \$ GO TO 20 END DO * * If we are here, the NRHS normwise backward errors satisfy the * stopping criterion, we are good to exit. * ITER = IITER * RETURN * 20 CONTINUE * 30 CONTINUE * * If we are at this place of the code, this is because we have * performed ITER=ITERMAX iterations and never satisified the * stopping criterion, set up the ITER flag accordingly and follow up * on double precision routine. * ITER = -ITERMAX - 1 * 40 CONTINUE * * Single-precision iterative refinement failed to converge to a * satisfactory solution, so we resort to double precision. * CALL DGETRF( N, N, A, LDA, IPIV, INFO ) * IF( INFO.NE.0 ) \$ RETURN * CALL DLACPY( 'All', N, NRHS, B, LDB, X, LDX ) CALL DGETRS( 'No transpose', N, NRHS, A, LDA, IPIV, X, LDX, \$ INFO ) * RETURN * * End of DSGESV. * END