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NAME
dspsvx - use the diagonal pivoting factorization A =
U*D*U**T or A = L*D*L**T to compute the solution to a real
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 DSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B,
LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO
)
CHARACTER FACT, UPLO
INTEGER INFO, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
FERR( * ), WORK( * ), X( LDX, * )
#include <sunperf.h>
void dspsvx(char fact, char uplo, int n, int nrhs, double
*dap, double *afp, int *ipivot, double *db, int
ldb, double *x, int ldx, double *drcond, double
*ferr, double *berr, int *info) ;
PURPOSE
DSPSVX uses the diagonal pivoting factorization A = U*D*U**T
or A = L*D*L**T to compute the solution to a real 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
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. 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimen-
sion
(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 DSPTRF, stored as a
packed triangular matrix in the same storage for-
mat 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**T or A =
L*D*L**T as computed by DSPTRF, 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 DSPTRF.
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 DSPTRF.
B (input) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION
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 com-
puted.
FERR (output) DOUBLE PRECISION 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
corresponding to X(j), FERR(j) is an estimated
upper bound for the magnitude of the largest ele-
ment in (X(j) - XTRUE) divided by the magnitude of
the largest element in X(j). The estimate is as
reliable as the estimate for RCOND, and is almost
always a slight overestimate of the true error.
BERR (output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension
(3*N)
IWORK (workspace) INTEGER array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille-
gal 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 solution and error bounds could not be com-
puted. = N+1: the block diagonal matrix D is non-
singular, but RCOND is less than machine preci-
sion. The factorization has been completed, but
the matrix is singular to working precision, so
the solution and error bounds have not been com-
puted.
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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