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version 1.6, 2010/08/13 21:04:14 version 1.11, 2012/07/31 11:06:39
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   *> \brief <b> ZSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices</b>
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download ZSPSVX + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zspsvx.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zspsvx.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zspsvx.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
   *                          LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          FACT, UPLO
   *       INTEGER            INFO, LDB, LDX, N, NRHS
   *       DOUBLE PRECISION   RCOND
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IPIV( * )
   *       DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
   *       COMPLEX*16         AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
   *      $                   X( LDX, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
   *> A = L*D*L**T to compute the solution to a complex system of linear
   *> equations A * X = B, where A is an N-by-N symmetric matrix stored
   *> in packed format and X and B are N-by-NRHS matrices.
   *>
   *> Error bounds on the solution and a condition estimate are also
   *> provided.
   *> \endverbatim
   *
   *> \par Description:
   *  =================
   *>
   *> \verbatim
   *>
   *> The following steps are performed:
   *>
   *> 1. If FACT = 'N', the diagonal pivoting method is used to factor A as
   *>       A = U * D * U**T,  if UPLO = 'U', or
   *>       A = L * D * L**T,  if UPLO = 'L',
   *>    where U (or L) is a product of permutation and unit upper (lower)
   *>    triangular matrices and D is symmetric and block diagonal with
   *>    1-by-1 and 2-by-2 diagonal blocks.
   *>
   *> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
   *>    returns with INFO = i. Otherwise, the factored form of A is used
   *>    to estimate the condition number of the matrix A.  If the
   *>    reciprocal of the condition number is less than machine precision,
   *>    INFO = N+1 is returned as a warning, but the routine still goes on
   *>    to solve for X and compute error bounds as described below.
   *>
   *> 3. The system of equations is solved for X using the factored form
   *>    of A.
   *>
   *> 4. Iterative refinement is applied to improve the computed solution
   *>    matrix and calculate error bounds and backward error estimates
   *>    for it.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] FACT
   *> \verbatim
   *>          FACT is CHARACTER*1
   *>          Specifies whether or not the factored form of A has been
   *>          supplied on entry.
   *>          = 'F':  On entry, AFP and IPIV contain the factored form
   *>                  of A.  AP, AFP and IPIV will not be modified.
   *>          = 'N':  The matrix A will be copied to AFP and factored.
   *> \endverbatim
   *>
   *> \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 matrices B and X.  NRHS >= 0.
   *> \endverbatim
   *>
   *> \param[in] AP
   *> \verbatim
   *>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
   *>          The upper or lower triangle of the symmetric matrix A, packed
   *>          columnwise in a linear array.  The j-th column of A is stored
   *>          in the array AP as follows:
   *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
   *>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
   *>          See below for further details.
   *> \endverbatim
   *>
   *> \param[in,out] AFP
   *> \verbatim
   *>          AFP is COMPLEX*16 array, dimension (N*(N+1)/2)
   *>          If FACT = 'F', then AFP is an input argument and on entry
   *>          contains the block diagonal matrix D and the multipliers used
   *>          to obtain the factor U or L from the factorization
   *>          A = U*D*U**T or A = L*D*L**T as computed by ZSPTRF, stored as
   *>          a packed triangular matrix in the same storage format as A.
   *>
   *>          If FACT = 'N', then AFP is an output argument and on exit
   *>          contains the block diagonal matrix D and the multipliers used
   *>          to obtain the factor U or L from the factorization
   *>          A = U*D*U**T or A = L*D*L**T as computed by ZSPTRF, stored as
   *>          a packed triangular matrix in the same storage format as A.
   *> \endverbatim
   *>
   *> \param[in,out] IPIV
   *> \verbatim
   *>          IPIV is INTEGER array, dimension (N)
   *>          If FACT = 'F', then IPIV is an input argument and on entry
   *>          contains details of the interchanges and the block structure
   *>          of D, as determined by ZSPTRF.
   *>          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
   *>          interchanged and D(k,k) is a 1-by-1 diagonal block.
   *>          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
   *>          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
   *>          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
   *>          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
   *>          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
   *>
   *>          If FACT = 'N', then IPIV is an output argument and on exit
   *>          contains details of the interchanges and the block structure
   *>          of D, as determined by ZSPTRF.
   *> \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 or INFO = N+1, 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] RCOND
   *> \verbatim
   *>          RCOND is DOUBLE PRECISION
   *>          The estimate of the reciprocal condition number of the matrix
   *>          A.  If RCOND is less than the machine precision (in
   *>          particular, if RCOND = 0), the matrix is singular to working
   *>          precision.  This condition is indicated by a return code of
   *>          INFO > 0.
   *> \endverbatim
   *>
   *> \param[out] FERR
   *> \verbatim
   *>          FERR is DOUBLE PRECISION array, dimension (NRHS)
   *>          The estimated forward error bound for each solution vector
   *>          X(j) (the j-th column of the solution matrix X).
   *>          If XTRUE is the true solution corresponding to X(j), FERR(j)
   *>          is an estimated upper bound for the magnitude of the largest
   *>          element in (X(j) - XTRUE) divided by the magnitude of the
   *>          largest element in X(j).  The estimate is as reliable as
   *>          the estimate for RCOND, and is almost always a slight
   *>          overestimate of the true error.
   *> \endverbatim
   *>
   *> \param[out] BERR
   *> \verbatim
   *>          BERR is DOUBLE PRECISION array, dimension (NRHS)
   *>          The componentwise relative backward error of each solution
   *>          vector X(j) (i.e., the smallest relative change in
   *>          any element of A or B that makes X(j) an exact solution).
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is COMPLEX*16 array, dimension (2*N)
   *> \endverbatim
   *>
   *> \param[out] RWORK
   *> \verbatim
   *>          RWORK is DOUBLE PRECISION array, dimension (N)
   *> \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, and i is
   *>                <= N:  D(i,i) is exactly zero.  The factorization
   *>                       has been completed but the factor D is exactly
   *>                       singular, so the solution and error bounds could
   *>                       not be computed. RCOND = 0 is returned.
   *>                = N+1: D is nonsingular, but RCOND is less than machine
   *>                       precision, meaning that the matrix is singular
   *>                       to working precision.  Nevertheless, the
   *>                       solution and error bounds are computed because
   *>                       there are a number of situations where the
   *>                       computed solution can be more accurate than the
   *>                       value of RCOND would suggest.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date April 2012
   *
   *> \ingroup complex16OTHERsolve
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  The packed storage scheme is illustrated by the following example
   *>  when N = 4, UPLO = 'U':
   *>
   *>  Two-dimensional storage of the symmetric matrix A:
   *>
   *>     a11 a12 a13 a14
   *>         a22 a23 a24
   *>             a33 a34     (aij = aji)
   *>                 a44
   *>
   *>  Packed storage of the upper triangle of A:
   *>
   *>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE ZSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,        SUBROUTINE ZSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
      $                   LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )       $                   LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.2) --  *  -- LAPACK driver routine (version 3.4.1) --
 *  -- 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..--
 *     November 2006  *     April 2012
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          FACT, UPLO        CHARACTER          FACT, UPLO
Line 18 Line 294
      $                   X( LDX, * )       $                   X( LDX, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZSPSVX uses the diagonal pivoting factorization A = U*D*U**T or  
 *  A = L*D*L**T to compute the solution to a complex system of linear  
 *  equations A * X = B, where A is an N-by-N symmetric matrix stored  
 *  in packed format and X and B are N-by-NRHS matrices.  
 *  
 *  Error bounds on the solution and a condition estimate are also  
 *  provided.  
 *  
 *  Description  
 *  ===========  
 *  
 *  The following steps are performed:  
 *  
 *  1. If FACT = 'N', the diagonal pivoting method is used to factor A as  
 *        A = U * D * U**T,  if UPLO = 'U', or  
 *        A = L * D * L**T,  if UPLO = 'L',  
 *     where U (or L) is a product of permutation and unit upper (lower)  
 *     triangular matrices and D is symmetric and block diagonal with  
 *     1-by-1 and 2-by-2 diagonal blocks.  
 *  
 *  2. If some D(i,i)=0, so that D is exactly singular, then the routine  
 *     returns with INFO = i. Otherwise, the factored form of A is used  
 *     to estimate the condition number of the matrix A.  If the  
 *     reciprocal of the condition number is less than machine precision,  
 *     INFO = N+1 is returned as a warning, but the routine still goes on  
 *     to solve for X and compute error bounds as described below.  
 *  
 *  3. The system of equations is solved for X using the factored form  
 *     of A.  
 *  
 *  4. Iterative refinement is applied to improve the computed solution  
 *     matrix and calculate error bounds and backward error estimates  
 *     for it.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  FACT    (input) CHARACTER*1  
 *          Specifies whether or not the factored form of A has been  
 *          supplied on entry.  
 *          = 'F':  On entry, AFP and IPIV contain the factored form  
 *                  of A.  AP, AFP and IPIV will not be modified.  
 *          = 'N':  The matrix A will be copied to AFP and factored.  
 *  
 *  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 matrices B and X.  NRHS >= 0.  
 *  
 *  AP      (input) COMPLEX*16 array, dimension (N*(N+1)/2)  
 *          The upper or lower triangle of the symmetric matrix A, packed  
 *          columnwise in a linear array.  The j-th column of A is stored  
 *          in the array AP as follows:  
 *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;  
 *          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.  
 *          See below for further details.  
 *  
 *  AFP     (input or output) COMPLEX*16 array, dimension (N*(N+1)/2)  
 *          If FACT = 'F', then AFP is an input argument and on entry  
 *          contains the block diagonal matrix D and the multipliers used  
 *          to obtain the factor U or L from the factorization  
 *          A = U*D*U**T or A = L*D*L**T as computed by ZSPTRF, stored as  
 *          a packed triangular matrix in the same storage format as A.  
 *  
 *          If FACT = 'N', then AFP is an output argument and on exit  
 *          contains the block diagonal matrix D and the multipliers used  
 *          to obtain the factor U or L from the factorization  
 *          A = U*D*U**T or A = L*D*L**T as computed by ZSPTRF, stored as  
 *          a packed triangular matrix in the same storage format as A.  
 *  
 *  IPIV    (input or output) INTEGER array, dimension (N)  
 *          If FACT = 'F', then IPIV is an input argument and on entry  
 *          contains details of the interchanges and the block structure  
 *          of D, as determined by ZSPTRF.  
 *          If IPIV(k) > 0, then rows and columns k and IPIV(k) were  
 *          interchanged and D(k,k) is a 1-by-1 diagonal block.  
 *          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and  
 *          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)  
 *          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =  
 *          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were  
 *          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.  
 *  
 *          If FACT = 'N', then IPIV is an output argument and on exit  
 *          contains details of the interchanges and the block structure  
 *          of D, as determined by ZSPTRF.  
 *  
 *  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 or INFO = N+1, the N-by-NRHS solution matrix X.  
 *  
 *  LDX     (input) INTEGER  
 *          The leading dimension of the array X.  LDX >= max(1,N).  
 *  
 *  RCOND   (output) DOUBLE PRECISION  
 *          The estimate of the reciprocal condition number of the matrix  
 *          A.  If RCOND is less than the machine precision (in  
 *          particular, if RCOND = 0), the matrix is singular to working  
 *          precision.  This condition is indicated by a return code of  
 *          INFO > 0.  
 *  
 *  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)  
 *          The estimated forward error bound for each solution vector  
 *          X(j) (the j-th column of the solution matrix X).  
 *          If XTRUE is the true solution corresponding to X(j), FERR(j)  
 *          is an estimated upper bound for the magnitude of the largest  
 *          element in (X(j) - XTRUE) divided by the magnitude of the  
 *          largest element in X(j).  The estimate is as reliable as  
 *          the estimate for RCOND, and is almost always a slight  
 *          overestimate of the true error.  
 *  
 *  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)  
 *          The componentwise relative backward error of each solution  
 *          vector X(j) (i.e., the smallest relative change in  
 *          any element of A or B that makes X(j) an exact solution).  
 *  
 *  WORK    (workspace) COMPLEX*16 array, dimension (2*N)  
 *  
 *  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)  
 *  
 *  INFO    (output) INTEGER  
 *          = 0: successful exit  
 *          < 0: if INFO = -i, the i-th argument had an illegal value  
 *          > 0:  if INFO = i, and i is  
 *                <= N:  D(i,i) is exactly zero.  The factorization  
 *                       has been completed but the factor D is exactly  
 *                       singular, so the solution and error bounds could  
 *                       not be computed. RCOND = 0 is returned.  
 *                = N+1: D is nonsingular, but RCOND is less than machine  
 *                       precision, meaning that the matrix is singular  
 *                       to working precision.  Nevertheless, the  
 *                       solution and error bounds are computed because  
 *                       there are a number of situations where the  
 *                       computed solution can be more accurate than the  
 *                       value of RCOND would suggest.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  The packed storage scheme is illustrated by the following example  
 *  when N = 4, UPLO = 'U':  
 *  
 *  Two-dimensional storage of the symmetric matrix A:  
 *  
 *     a11 a12 a13 a14  
 *         a22 a23 a24  
 *             a33 a34     (aij = aji)  
 *                 a44  
 *  
 *  Packed storage of the upper triangle of A:  
 *  
 *  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Line 234 Line 343
 *  *
       IF( NOFACT ) THEN        IF( NOFACT ) THEN
 *  *
 *        Compute the factorization A = U*D*U' or A = L*D*L'.  *        Compute the factorization A = U*D*U**T or A = L*D*L**T.
 *  *
          CALL ZCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )           CALL ZCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
          CALL ZSPTRF( UPLO, N, AFP, IPIV, INFO )           CALL ZSPTRF( UPLO, N, AFP, IPIV, INFO )
Removed from v.1.6  
changed lines
  Added in v.1.11

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