zspsvx(3F)

NAME
     ZSPSVX - use 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

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 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. 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 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, 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 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 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 solution and error bounds could not
             be computed.  = N+1: the block diagonal matrix D is nonsingular,
             but RCOND is less than machine precision.  The factorization has
             been completed, but the matrix is singular 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 ]