ZSPSV(3f)

NAME
ZSPSV - 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, * )

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 ]