CSPSV - compute the solution to a complex system of linear
equations A * X = B,
SYNOPSIS
SUBROUTINE CSPSV( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )
CHARACTER UPLO
INTEGER INFO, LDB, N, NRHS
INTEGER IPIV( * )
COMPLEX AP( * ), B( LDB, * )
PURPOSE
CSPSV 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 matri
ces.
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 fac
tored form of A is then used to solve the system of equa
tions 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 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.
tipliers 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 CSPTRF, 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 CSPTRF. If IPIV(k) >
0, then rows and columns k and IPIV(k) were inter
changed, 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 inter
changed 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 inter
changed and D(k:k+1,k:k+1) is a 2-by-2 diagonal
block.
B (input/output) COMPLEX 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 ]
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