dsgesv.f

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      SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK,
     $                   SWORK, ITER, INFO )
*
*  -- LAPACK PROTOTYPE driver routine (version 3.3.1) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*  -- April 2011                                                      --
*
*     ..
*     .. 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, * )
*     ..
*
*  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.
*
*  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.
*
*  Arguments
*  =========
*
*  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.
*
*  =====================================================================
*
*     .. 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