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NAME
CHPSVX - use the diagonal pivoting factorization A =
U*D*U**H or A = L*D*L**H to compute the solution to a com-
plex system of linear equations A * X = B, where A is an N-
by-N Hermitian matrix stored in packed format and X and B
are N-by-NRHS matrices
SYNOPSIS
SUBROUTINE CHPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B,
LDB, X, LDX, RCOND, FERR, BERR, WORK,
RWORK, INFO )
CHARACTER FACT, UPLO
INTEGER INFO, LDB, LDX, N, NRHS
REAL RCOND
INTEGER IPIV( * )
REAL BERR( * ), FERR( * ), RWORK( * )
COMPLEX AFP( * ), AP( * ), B( LDB, * ), WORK( *
), X( LDX, * )
PURPOSE
CHPSVX uses the diagonal pivoting factorization A = U*D*U**H
or A = L*D*L**H to compute the solution to a complex system
of linear equations A * X = B, where A is an N-by-N Hermi-
tian matrix stored in packed format and X and B are N-by-
NRHS matrices.
Error bounds on the solution and a condition estimate are
also provided.
DESCRIPTION
The following steps are performed:
1. If FACT = 'N', the diagonal pivoting method is used to
factor A as
A = U * D * U**H, if UPLO = 'U', or
A = L * D * L**H, if UPLO = 'L',
where U (or L) is a product of permutation and unit upper
(lower)
triangular matrices and D is Hermitian and block diagonal
with
1-by-1 and 2-by-2 diagonal blocks.
2. The factored form of A is used to estimate the condition
number
of the matrix A. If the reciprocal of the condition
number is
less than machine precision, steps 3 and 4 are skipped.
3. The system of equations is solved for X using the fac-
tored form
of A.
4. Iterative refinement is applied to improve the computed
solution
matrix and calculate error bounds and backward error
estimates
for it.
ARGUMENTS
FACT (input) CHARACTER*1
Specifies whether or not the factored form of A has
been supplied on entry. = 'F': On entry, AFP and
IPIV contain the factored form of A. AFP and IPIV
will not be modified. = 'N': The matrix A will be
copied to AFP and factored.
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 matrices B and X. NRHS >= 0.
AP (input) COMPLEX array, dimension (N*(N+1)/2)
The upper or lower triangle of the Hermitian 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.
AFP (input or output) COMPLEX array, dimension (N*(N+1)/2)
If FACT = 'F', then AFP is an input argument and on
entry contains the block diagonal matrix D and the
multipliers used to obtain the factor U or L from
the factorization A = U*D*U**H or A = L*D*L**H as
computed by CHPTRF, stored as a packed triangular
matrix in the same storage format as A.
If FACT = 'N', then AFP is an output argument and on
exit contains the block diagonal matrix D and the
multipliers used to obtain the factor U or L from
the factorization A = U*D*U**H or A = L*D*L**H as
computed by CHPTRF, stored as a packed triangular
matrix in the same storage format as A.
IPIV (input or output) INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on
entry contains details of the interchanges and the
block structure of D, as determined by CHPTRF. 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 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.
If FACT = 'N', then IPIV is an output argument and
on exit contains details of the interchanges and the
block structure of D, as determined by CHPTRF.
B (input) COMPLEX array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >=
max(1,N).
X (output) COMPLEX array, dimension (LDX,NRHS)
If INFO = 0, the N-by-NRHS solution matrix X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >=
max(1,N).
RCOND (output) REAL
The estimate of the reciprocal condition number of
the matrix A. If RCOND is less than the machine
precision (in particular, if RCOND = 0), the matrix
is singular to working precision. This condition is
indicated by a return code of INFO > 0, and the
solution and error bounds are not computed.
FERR (output) REAL array, dimension (NRHS)
The estimated forward error bounds for each solution
vector X(j) (the j-th column of the solution matrix
X). If XTRUE is the true solution, FERR(j) bounds
the magnitude of the largest entry in (X(j) - XTRUE)
divided by the magnitude of the largest entry in
X(j). The quality of the error bound depends on the
quality of the estimate of norm(inv(A)) computed in
the code; if the estimate of norm(inv(A)) is accu-
rate, the error bound is guaranteed.
BERR (output) REAL array, dimension (NRHS)
The componentwise relative backward error of each
solution vector X(j) (i.e., the smallest relative
change in any entry of A or B that makes X(j) an
exact solution).
WORK (workspace) COMPLEX array, dimension (2*N)
RWORK (workspace) REAL array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal
value
> 0 and <= N: 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 solu-
tion and error bounds could not be computed. = N+1:
the block diagonal matrix D is nonsingular, but
RCOND is less than machine precision. The factori-
zation has been completed, but the matrix is singu-
lar to working precision, so the solution and error
bounds have not been computed.
FURTHER DETAILS
The packed storage scheme is illustrated by the following
example when N = 4, UPLO = 'U':
Two-dimensional storage of the Hermitian matrix A:
a11 a12 a13 a14
a22 a23 a24
a33 a34 (aij = conjg(aji))
a44
Packed storage of the upper triangle of A:
AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]