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# zspsv

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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 diago-
nal 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 details.

On exit, the block diagonal matrix D and the multi-
pliers used to obtain the factor U or L from the
factorization A = U*D*U**T or A = L*D*L**T as com-
puted 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 fac-
torization has been completed, but the block diago-
nal 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 ]
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