zspsv(3)

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, * )

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 sym

metric 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 de

tails.

On exit, the block diagonal matrix D and the mul

tipliers 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 struc

ture 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 IP

IV(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 ille

gal 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 ]

LAPACK version 3.0 15 June 2000

ZSPSV(3)