zspsv(3S)

NAME
     ZSPSV - compute the solution to a complex system of linear equations A *
     X = B,

SYNOPSIS
     SUBROUTINE ZSPSV( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )

         CHARACTER     UPLO

         INTEGER       INFO, LDB, N, NRHS

         INTEGER       IPIV( * )

         COMPLEX*16    AP( * ), B( LDB, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     ZSPSV computes 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.

     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, D is symmetric and block diagonal with 1-by-1 and
     2-by-2 diagonal blocks.  The factored form of A is then used to solve the
     system of equations A * X = B.


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.

     AP      (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
             On entry, 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.

             On exit, 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    (output) INTEGER array, dimension (N)
             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.

     B       (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
             On entry, the N-by-NRHS right hand side matrix B.  On exit, if
             INFO = 0, the N-by-NRHS solution matrix X.

     LDB     (input) INTEGER
             The leading dimension of the array B.  LDB >= max(1,N).

     INFO    (output) INTEGER
             = 0:  successful exit
             < 0:  if INFO = -i, the i-th argument had an illegal value
             > 0:  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 could not be 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 ]


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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