434 lines
13 KiB
Fortran
434 lines
13 KiB
Fortran
*> \brief <b> DSGESV computes the solution to system of linear equations A * X = B for GE matrices</b> (mixed precision with iterative refinement)
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
*> \htmlonly
|
|
*> Download DSGESV + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsgesv.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsgesv.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsgesv.f">
|
|
*> [TXT]</a>
|
|
*> \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 = 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.
|
|
*
|
|
*> \date June 2016
|
|
*
|
|
*> \ingroup doubleGEsolve
|
|
*
|
|
* =====================================================================
|
|
SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK,
|
|
$ SWORK, ITER, INFO )
|
|
*
|
|
* -- LAPACK driver routine (version 3.8.0) --
|
|
* -- 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, 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
|