CSPSVX - use the diagonal pivoting factorization A =
U*D*U**T or A = L*D*L**T to compute the solution to a com
plex 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
SYNOPSIS
SUBROUTINE CSPSVX( 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
CSPSVX uses the diagonal pivoting factorization A =
U*D*U**T or A = L*D*L**T to compute the solution to a com
plex 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. 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**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 and D is symmetric and block diago
nal with
1-by-1 and 2-by-2 diagonal blocks.
2. If some D(i,i)=0, so that D is exactly singular, then
the routine
returns with INFO = i. Otherwise, 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
INFO = N+1 is returned as a warning, but the routine
still goes on
to solve for X and compute error bounds as described
below.
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. AP, 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 symmetric
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)*(2*n-
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**T or A =
L*D*L**T as computed by CSPTRF, stored as a packed
triangular matrix in the same storage format as A.
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**T or A =
L*D*L**T as computed by CSPTRF, 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 CSPTRF.
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.
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 CSPTRF.
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 or INFO = N+1, 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.
FERR (output) REAL array, dimension (NRHS)
The estimated forward error bound for each solu
tion vector X(j) (the j-th column of the solution
matrix X). If XTRUE is the true solution corre
sponding to X(j), FERR(j) is an estimated upper
bound for the magnitude of the largest element in
largest element in X(j). The estimate is as reli
able as the estimate for RCOND, and is almost
always a slight overestimate of the true error.
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 element 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 ille
gal value
> 0: if INFO = i, and i is
<= N: D(i,i) is exactly zero. The factorization
has been completed but the factor D is exactly
singular, so the solution and error bounds could
not be computed. RCOND = 0 is returned. = N+1: D
is nonsingular, but RCOND is less than machine
precision, meaning that the matrix is singular to
working precision. Nevertheless, the solution and
error bounds are computed because there are a num
ber of situations where the computed solution can
be more accurate than the value of RCOND would
suggest.
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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