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NAME
dptsvx - use the factorization 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 positive definite tridiagonal
matrix and X and B are N-by-NRHS matrices
SYNOPSIS
SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X,
LDX, RCOND, FERR, BERR, WORK, INFO )
CHARACTER FACT
INTEGER INFO, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ), E(
* ), EF( * ), FERR( * ), WORK( * ), X( LDX, * )
#include <sunperf.h>
void dptsvx(char fact, int n, int nrhs, double *d, double
*e, double *df, double *ef, double *db, int ldb,
double *dx, int ldx, double *drcond, double *ferr,
double *berr, int *info) ;
PURPOSE
DPTSVX uses the factorization 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 positive definite tridiagonal
matrix 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 matrix A is factored as A = L*D*L**T,
where L is a unit lower bidiagonal matrix and D is diagonal.
The factorization can also be regarded as having the form
A = U**T*D*U.
2. The factored form of A is used to compute 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
factored 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, DF
and EF contain the factored form of A. D, E, DF,
and EF will not be modified. = 'N': The matrix A
will be copied to DF and EF and factored.
N (input) INTEGER
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.
D (input) DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the tridiagonal matrix
A.
E (input) DOUBLE PRECISION array, dimension (N-1)
The (n-1) subdiagonal elements of the tridiagonal
matrix A.
DF (input or output) DOUBLE PRECISION array, dimen-
sion (N)
If FACT = 'F', then DF is an input argument and on
entry contains the n diagonal elements of the
diagonal matrix D from the L*D*L**T factorization
of A. If FACT = 'N', then DF is an output argu-
ment and on exit contains the n diagonal elements
of the diagonal matrix D from the L*D*L**T factor-
ization of A.
EF (input or output) DOUBLE PRECISION array, dimen-
sion (N-1)
If FACT = 'F', then EF is an input argument and on
entry contains the (n-1) subdiagonal elements of
the unit bidiagonal factor L from the L*D*L**T
factorization of A. If FACT = 'N', then EF is an
output argument and on exit contains the (n-1)
subdiagonal elements of the unit bidiagonal factor
L from the L*D*L**T factorization of A.
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 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) DOUBLE PRECISION array, dimension (NRHS)
The forward error bound for each solution 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 element in (X(j) - XTRUE)
divided by the magnitude of the largest element in
X(j).
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
(2*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 the leading
minor of order i of A is not positive definite, so
the factorization could not be completed unless i
= N, and the solution and error bounds could not
be computed. = N+1 RCOND is less than machine
precision. The factorization has been completed,
but the matrix is singular to working precision,
and the solution and error bounds have not been
computed.
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