NAME

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

SYNOPSIS

  SUBROUTINE ZSPSV( UPLO, N, NRHS, A, IPIVOT, B, LDB, INFO)
  CHARACTER * 1 UPLO
  DOUBLE COMPLEX A(*), B(LDB,*)
  INTEGER N, NRHS, LDB, INFO
  INTEGER IPIVOT(*)

  SUBROUTINE ZSPSV_64( UPLO, N, NRHS, A, IPIVOT, B, LDB, INFO)
  CHARACTER * 1 UPLO
  DOUBLE COMPLEX A(*), B(LDB,*)
  INTEGER*8 N, NRHS, LDB, INFO
  INTEGER*8 IPIVOT(*)

F95 INTERFACE

  SUBROUTINE SPSV( UPLO, N, NRHS, A, IPIVOT, B, [LDB], [INFO])
  CHARACTER(LEN=1) :: UPLO
  COMPLEX(8), DIMENSION(:) :: A
  COMPLEX(8), DIMENSION(:,:) :: B
  INTEGER :: N, NRHS, LDB, INFO
  INTEGER, DIMENSION(:) :: IPIVOT

  SUBROUTINE SPSV_64( UPLO, N, NRHS, A, IPIVOT, B, [LDB], [INFO])
  CHARACTER(LEN=1) :: UPLO
  COMPLEX(8), DIMENSION(:) :: A
  COMPLEX(8), DIMENSION(:,:) :: B
  INTEGER(8) :: N, NRHS, LDB, INFO
  INTEGER(8), DIMENSION(:) :: IPIVOT

C INTERFACE

#include <sunperf.h>

void zspsv(char uplo, int n, int nrhs, doublecomplex *a, int *ipivot, doublecomplex *b, int ldb, int *info);

void zspsv_64(char uplo, long n, long nrhs, doublecomplex *a, long *ipivot, doublecomplex *b, long ldb, long *info);

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)

 = 'U':  Upper triangle of A is stored;

 = 'L':  Lower triangle of A is stored.

N (input)
The number of linear equations, i.e., the order of the matrix A. N > = 0.
NRHS (input)
The number of right hand sides, i.e., the number of columns of the matrix B. NRHS > = 0.
A (input/output)
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 A as follows: if UPLO = 'U', A(i + (j-1)*j/2) = A(i,j) for 1 < =i < =j; if UPLO = 'L', A(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 CSPTRF, stored as a packed triangular matrix in the same storage format as A.
IPIVOT (output)
Details of the interchanges and the block structure of D, as determined by CSPTRF. If IPIVOT(k) > 0, then rows and columns k and IPIVOT(k) were interchanged, and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and IPIVOT(k) = IPIVOT(k-1) < 0, then rows and columns k-1 and -IPIVOT(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIVOT(k) = IPIVOT(k+1) < 0, then rows and columns k+1 and -IPIVOT(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
B (input/output)
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)
The leading dimension of the array B. LDB > = max(1,N).

INFO (output)

 = 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:

A = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]