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version 1.5, 2010/12/21 13:53:42 version 1.18, 2020/05/21 21:46:03
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       SUBROUTINE ZCPOSV( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK,  *> \brief <b> ZCPOSV computes the solution to system of linear equations A * X = B for PO matrices</b>
      +                   SWORK, RWORK, ITER, INFO )  *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download ZCPOSV + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zcposv.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zcposv.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zcposv.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZCPOSV( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK,
   *                          SWORK, RWORK, ITER, INFO )
   *
   *       .. Scalar Arguments ..
   *       CHARACTER          UPLO
   *       INTEGER            INFO, ITER, LDA, LDB, LDX, N, NRHS
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   RWORK( * )
   *       COMPLEX            SWORK( * )
   *       COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( N, * ),
   *      $                   X( LDX, * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZCPOSV computes the solution to a complex system of linear equations
   *>    A * X = B,
   *> where A is an N-by-N Hermitian positive definite matrix and X and B
   *> are N-by-NRHS matrices.
   *>
   *> ZCPOSV first attempts to factorize the matrix in COMPLEX and use this
   *> factorization within an iterative refinement procedure to produce a
   *> solution with COMPLEX*16 normwise backward error quality (see below).
   *> If the approach fails the method switches to a COMPLEX*16
   *> factorization and solve.
   *>
   *> The iterative refinement is not going to be a winning strategy if
   *> the ratio COMPLEX performance over COMPLEX*16 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] UPLO
   *> \verbatim
   *>          UPLO is CHARACTER*1
   *>          = 'U':  Upper triangle of A is stored;
   *>          = 'L':  Lower triangle of A is stored.
   *> \endverbatim
   *>
   *> \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 COMPLEX*16 array,
   *>          dimension (LDA,N)
   *>          On entry, the Hermitian matrix A. If UPLO = 'U', the leading
   *>          N-by-N upper triangular part of A contains the upper
   *>          triangular part of the matrix A, and the strictly lower
   *>          triangular part of A is not referenced.  If UPLO = 'L', the
   *>          leading N-by-N lower triangular part of A contains the lower
   *>          triangular part of the matrix A, and the strictly upper
   *>          triangular part of A is not referenced.
   *>
   *>          Note that the imaginary parts of the diagonal
   *>          elements need not be set and are assumed to be zero.
   *>
   *>          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 factor U or L from the Cholesky
   *>          factorization A = U**H*U or A = L*L**H.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in] B
   *> \verbatim
   *>          B is COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 array, dimension (N,NRHS)
   *>          This array is used to hold the residual vectors.
   *> \endverbatim
   *>
   *> \param[out] SWORK
   *> \verbatim
   *>          SWORK is COMPLEX 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] RWORK
   *> \verbatim
   *>          RWORK is DOUBLE PRECISION array, dimension (N)
   *> \endverbatim
   *>
   *> \param[out] ITER
   *> \verbatim
   *>          ITER is INTEGER
   *>          < 0: iterative refinement has failed, COMPLEX*16
   *>               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 CPOTRF
   *>               -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, the leading minor of order i of
   *>                (COMPLEX*16) A is not positive definite, so the
   *>                factorization could not be completed, and the solution
   *>                has not been computed.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee
   *> \author Univ. of California Berkeley
   *> \author Univ. of Colorado Denver
   *> \author NAG Ltd.
   *
   *> \date June 2016
 *  *
 *  -- LAPACK PROTOTYPE driver routine (version 3.3.0)                 --  *> \ingroup complex16POsolve
 *  *
 *     November 2010  *  =====================================================================
         SUBROUTINE ZCPOSV( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK,
        $                   SWORK, RWORK, ITER, INFO )
 *  *
   *  -- LAPACK driver routine (version 3.8.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     ..  *     June 2016
   *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          UPLO        CHARACTER          UPLO
       INTEGER            INFO, ITER, LDA, LDB, LDX, N, NRHS        INTEGER            INFO, ITER, LDA, LDB, LDX, N, NRHS
Line 16 Line 222
       DOUBLE PRECISION   RWORK( * )        DOUBLE PRECISION   RWORK( * )
       COMPLEX            SWORK( * )        COMPLEX            SWORK( * )
       COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( N, * ),        COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( N, * ),
      +                   X( LDX, * )       $                   X( LDX, * )
 *     ..  *     ..
 *  *
 *  Purpose  *  =====================================================================
 *  =======  
 *  
 *  ZCPOSV computes the solution to a complex system of linear equations  
 *     A * X = B,  
 *  where A is an N-by-N Hermitian positive definite matrix and X and B  
 *  are N-by-NRHS matrices.  
 *  
 *  ZCPOSV first attempts to factorize the matrix in COMPLEX and use this  
 *  factorization within an iterative refinement procedure to produce a  
 *  solution with COMPLEX*16 normwise backward error quality (see below).  
 *  If the approach fails the method switches to a COMPLEX*16  
 *  factorization and solve.  
 *  
 *  The iterative refinement is not going to be a winning strategy if  
 *  the ratio COMPLEX performance over COMPLEX*16 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  
 *  =========  
 *  
 *  UPLO    (input) CHARACTER*1  
 *          = 'U':  Upper triangle of A is stored;  
 *          = 'L':  Lower triangle of A is stored.  
 *  
 *  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) COMPLEX*16 array,  
 *          dimension (LDA,N)  
 *          On entry, the Hermitian matrix A. If UPLO = 'U', the leading  
 *          N-by-N upper triangular part of A contains the upper  
 *          triangular part of the matrix A, and the strictly lower  
 *          triangular part of A is not referenced.  If UPLO = 'L', the  
 *          leading N-by-N lower triangular part of A contains the lower  
 *          triangular part of the matrix A, and the strictly upper  
 *          triangular part of A is not referenced.  
 *  
 *          Note that the imaginary parts of the diagonal  
 *          elements need not be set and are assumed to be zero.  
 *  
 *          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 factor U or L from the Cholesky  
 *          factorization A = U**H*U or A = L*L**H.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,N).  
 *  
 *  B       (input) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 array, dimension (N*NRHS)  
 *          This array is used to hold the residual vectors.  
 *  
 *  SWORK   (workspace) COMPLEX 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.  
 *  
 *  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)  
 *  
 *  ITER    (output) INTEGER  
 *          < 0: iterative refinement has failed, COMPLEX*16  
 *               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 CPOTRF  
 *               -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, the leading minor of order i of  
 *                (COMPLEX*16) A is not positive definite, so the  
 *                factorization could not be completed, and the solution  
 *                has not been computed.  
 *  
 *  =========  
 *  *
 *     .. Parameters ..  *     .. Parameters ..
       LOGICAL            DOITREF        LOGICAL            DOITREF
Line 148 Line 239
 *  *
       COMPLEX*16         NEGONE, ONE        COMPLEX*16         NEGONE, ONE
       PARAMETER          ( NEGONE = ( -1.0D+00, 0.0D+00 ),        PARAMETER          ( NEGONE = ( -1.0D+00, 0.0D+00 ),
      +                   ONE = ( 1.0D+00, 0.0D+00 ) )       $                   ONE = ( 1.0D+00, 0.0D+00 ) )
 *  *
 *     .. Local Scalars ..  *     .. Local Scalars ..
       INTEGER            I, IITER, PTSA, PTSX        INTEGER            I, IITER, PTSA, PTSX
Line 157 Line 248
 *  *
 *     .. External Subroutines ..  *     .. External Subroutines ..
       EXTERNAL           ZAXPY, ZHEMM, ZLACPY, ZLAT2C, ZLAG2C, CLAG2Z,        EXTERNAL           ZAXPY, ZHEMM, ZLACPY, ZLAT2C, ZLAG2C, CLAG2Z,
      +                   CPOTRF, CPOTRS, XERBLA       $                   CPOTRF, CPOTRS, XERBLA, ZPOTRF, ZPOTRS
 *     ..  *     ..
 *     .. External Functions ..  *     .. External Functions ..
       INTEGER            IZAMAX        INTEGER            IZAMAX
Line 201 Line 292
 *     Quick return if (N.EQ.0).  *     Quick return if (N.EQ.0).
 *  *
       IF( N.EQ.0 )        IF( N.EQ.0 )
      +   RETURN       $   RETURN
 *  *
 *     Skip single precision iterative refinement if a priori slower  *     Skip single precision iterative refinement if a priori slower
 *     than double precision factorization.  *     than double precision factorization.
Line 254 Line 345
 *     Solve the system SA*SX = SB.  *     Solve the system SA*SX = SB.
 *  *
       CALL CPOTRS( UPLO, N, NRHS, SWORK( PTSA ), N, SWORK( PTSX ), N,        CALL CPOTRS( UPLO, N, NRHS, SWORK( PTSA ), N, SWORK( PTSX ), N,
      +             INFO )       $             INFO )
 *  *
 *     Convert SX back to COMPLEX*16  *     Convert SX back to COMPLEX*16
 *  *
Line 265 Line 356
       CALL ZLACPY( 'All', N, NRHS, B, LDB, WORK, N )        CALL ZLACPY( 'All', N, NRHS, B, LDB, WORK, N )
 *  *
       CALL ZHEMM( 'Left', UPLO, N, NRHS, NEGONE, A, LDA, X, LDX, ONE,        CALL ZHEMM( 'Left', UPLO, N, NRHS, NEGONE, A, LDA, X, LDX, ONE,
      +            WORK, N )       $            WORK, N )
 *  *
 *     Check whether the NRHS normwise backward errors satisfy the  *     Check whether the NRHS normwise backward errors satisfy the
 *     stopping criterion. If yes, set ITER=0 and return.  *     stopping criterion. If yes, set ITER=0 and return.
Line 274 Line 365
          XNRM = CABS1( X( IZAMAX( N, X( 1, I ), 1 ), I ) )           XNRM = CABS1( X( IZAMAX( N, X( 1, I ), 1 ), I ) )
          RNRM = CABS1( WORK( IZAMAX( N, WORK( 1, I ), 1 ), I ) )           RNRM = CABS1( WORK( IZAMAX( N, WORK( 1, I ), 1 ), I ) )
          IF( RNRM.GT.XNRM*CTE )           IF( RNRM.GT.XNRM*CTE )
      +      GO TO 10       $      GO TO 10
       END DO        END DO
 *  *
 *     If we are here, the NRHS normwise backward errors satisfy the  *     If we are here, the NRHS normwise backward errors satisfy the
Line 300 Line 391
 *        Solve the system SA*SX = SR.  *        Solve the system SA*SX = SR.
 *  *
          CALL CPOTRS( UPLO, N, NRHS, SWORK( PTSA ), N, SWORK( PTSX ), N,           CALL CPOTRS( UPLO, N, NRHS, SWORK( PTSA ), N, SWORK( PTSX ), N,
      +                INFO )       $                INFO )
 *  *
 *        Convert SX back to double precision and update the current  *        Convert SX back to double precision and update the current
 *        iterate.  *        iterate.
Line 316 Line 407
          CALL ZLACPY( 'All', N, NRHS, B, LDB, WORK, N )           CALL ZLACPY( 'All', N, NRHS, B, LDB, WORK, N )
 *  *
          CALL ZHEMM( 'L', UPLO, N, NRHS, NEGONE, A, LDA, X, LDX, ONE,           CALL ZHEMM( 'L', UPLO, N, NRHS, NEGONE, A, LDA, X, LDX, ONE,
      +               WORK, N )       $               WORK, N )
 *  *
 *        Check whether the NRHS normwise backward errors satisfy the  *        Check whether the NRHS normwise backward errors satisfy the
 *        stopping criterion. If yes, set ITER=IITER>0 and return.  *        stopping criterion. If yes, set ITER=IITER>0 and return.
Line 325 Line 416
             XNRM = CABS1( X( IZAMAX( N, X( 1, I ), 1 ), I ) )              XNRM = CABS1( X( IZAMAX( N, X( 1, I ), 1 ), I ) )
             RNRM = CABS1( WORK( IZAMAX( N, WORK( 1, I ), 1 ), I ) )              RNRM = CABS1( WORK( IZAMAX( N, WORK( 1, I ), 1 ), I ) )
             IF( RNRM.GT.XNRM*CTE )              IF( RNRM.GT.XNRM*CTE )
      +         GO TO 20       $         GO TO 20
          END DO           END DO
 *  *
 *        If we are here, the NRHS normwise backward errors satisfy the  *        If we are here, the NRHS normwise backward errors satisfy the
Line 340 Line 431
    30 CONTINUE     30 CONTINUE
 *  *
 *     If we are at this place of the code, this is because we have  *     If we are at this place of the code, this is because we have
 *     performed ITER=ITERMAX iterations and never satisified the  *     performed ITER=ITERMAX iterations and never satisfied the
 *     stopping criterion, set up the ITER flag accordingly and follow  *     stopping criterion, set up the ITER flag accordingly and follow
 *     up on double precision routine.  *     up on double precision routine.
 *  *
Line 354 Line 445
       CALL ZPOTRF( UPLO, N, A, LDA, INFO )        CALL ZPOTRF( UPLO, N, A, LDA, INFO )
 *  *
       IF( INFO.NE.0 )        IF( INFO.NE.0 )
      +   RETURN       $   RETURN
 *  *
       CALL ZLACPY( 'All', N, NRHS, B, LDB, X, LDX )        CALL ZLACPY( 'All', N, NRHS, B, LDB, X, LDX )
       CALL ZPOTRS( UPLO, N, NRHS, A, LDA, X, LDX, INFO )        CALL ZPOTRS( UPLO, N, NRHS, A, LDA, X, LDX, INFO )
Removed from v.1.5  
changed lines
  Added in v.1.18

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