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zspsvx


 NAME
      ZSPSVX - use the diagonal pivoting factorization A =
      U*D*U**T or A = L*D*L**T to compute the solution to a com-
      plex 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

 SYNOPSIS
      SUBROUTINE ZSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B,
                         LDB, X, LDX, RCOND, FERR, BERR, WORK,
                         RWORK, INFO )

          CHARACTER      FACT, UPLO

          INTEGER        INFO, LDB, LDX, N, NRHS

          DOUBLE         PRECISION RCOND

          INTEGER        IPIV( * )

          DOUBLE         PRECISION BERR( * ), FERR( * ), RWORK( *
                         )

          COMPLEX*16     AFP( * ), AP( * ), B( LDB, * ), WORK( *
                         ), 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 sym-
      metric 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. 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, steps 3 and 4 are skipped.

      3. The system of equations is solved for X using the fac-
      tored 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)*(2n-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 inter-
              changed 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 inter-
              changed 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, 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, and the
              solution and error bounds are not computed.

      FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
              The estimated forward error bounds for each solution
              vector X(j) (the j-th column of the solution matrix
              X).  If XTRUE is the true solution, FERR(j) bounds
              the magnitude of the largest entry in (X(j) - XTRUE)
              divided by the magnitude of the largest entry in

              X(j).  The quality of the error bound depends on the
              quality of the estimate of norm(inv(A)) computed in
              the code; if the estimate of norm(inv(A)) is accu-
              rate, the error bound is guaranteed.

      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 entry 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 and <= N: if INFO = i, D(i,i) is exactly zero.
              The factorization has been completed, but the block
              diagonal matrix D is exactly singular, so the solu-
              tion and error bounds could not be computed.  = N+1:
              the block diagonal matrix D is nonsingular, but
              RCOND is less than machine precision.  The factori-
              zation has been completed, but the matrix is singu-
              lar to working precision, so the solution and error
              bounds have not been computed.

 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 ]