*> \brief <b> DSGESV computes the solution to system of linear equations A * X = B for GE matrices</b> (mixed precision with iterative refinement)
*
* =========== DOCUMENTATION ===========
*
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*
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*
* 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 = 0 and ITER >= 0, see description below), then A is
*> unchanged, if double precision factorization has been used
*> (INFO = 0 and ITER < 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 = 0 and ITER >= 0) or the double precision
*> factorization (if INFO = 0 and ITER < 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.
*
*> \ingroup gesv_mixed
*
* =====================================================================
SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK,
$ SWORK, ITER, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. 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, DGETRF,
$ DGETRS,
$ SGETRF, SGETRS, SLAG2D, 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 satisfied 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