
ZSPSV(l) ) ZSPSV(l)
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 NbyN symmetric matrix stored in packed format and X and B are NbyNRHS 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 1by1 and 2by2 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 jth column of A is stored in the array AP as follows: if
UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
(j1)*(2nj)/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 1by1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k1) < 0, then rows
and columns k1 and IPIV(k) were interchanged and D(k1:k,k1:k) is a 2by2
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 2by2 diagonal block.
B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
On entry, the NbyNRHS right hand side matrix B. On exit, if INFO = 0, the Nby
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 ith 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':
Twodimensional 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(l) 
