Contents
sptsvx - 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
SUBROUTINE SPTSVX(FACT, N, NRHS, D, E, DF, EF, B, LDB, X,
LDX, RCOND, FERR, BERR, WORK, INFO)
CHARACTER * 1 FACT
INTEGER N, NRHS, LDB, LDX, INFO
REAL RCOND
REAL D(*), E(*), DF(*), EF(*), B(LDB,*), X(LDX,*), FERR(*),
BERR(*), WORK(*)
SUBROUTINE SPTSVX_64(FACT, N, NRHS, D, E, DF, EF, B, LDB,
X, LDX, RCOND, FERR, BERR, WORK, INFO)
CHARACTER * 1 FACT
INTEGER*8 N, NRHS, LDB, LDX, INFO
REAL RCOND
REAL D(*), E(*), DF(*), EF(*), B(LDB,*), X(LDX,*), FERR(*),
BERR(*), WORK(*)
F95 INTERFACE
SUBROUTINE PTSVX(FACT, [N], [NRHS], D, E, DF, EF, B, [LDB],
X, [LDX], RCOND, FERR, BERR, [WORK], [INFO])
CHARACTER(LEN=1) :: FACT
INTEGER :: N, NRHS, LDB, LDX, INFO
REAL :: RCOND
REAL, DIMENSION(:) :: D, E, DF, EF, FERR, BERR, WORK
REAL, DIMENSION(:,:) :: B, X
SUBROUTINE PTSVX_64(FACT, [N], [NRHS], D, E, DF, EF, B,
[LDB], X, [LDX], RCOND, FERR, BERR, [WORK], [INFO])
CHARACTER(LEN=1) :: FACT
INTEGER(8) :: N, NRHS, LDB, LDX, INFO
REAL :: RCOND
REAL, DIMENSION(:) :: D, E, DF, EF, FERR, BERR, WORK
REAL, DIMENSION(:,:) :: B, X
C INTERFACE
#include <sunperf.h>
void sptsvx(char fact, int n, int nrhs, float *d, float *e,
float *df, float *ef, float *b, int ldb, float *x,
int ldx, float *rcond, float *ferr, float *berr,
int *info);
void sptsvx_64(char fact, long n, long nrhs, float *d, float
*e, float *df, float *ef, float *b, long ldb,
float *x, long ldx, float *rcond, float *ferr,
float *berr, long *info);
sptsvx 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.
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. If the leading i-by-i principal minor is not positive
definite,
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
precision, 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.
FACT (input)
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) The order of the matrix A. N >= 0.
NRHS (input)
The number of right hand sides, i.e., the number
of columns of the matrices B and X. NRHS >= 0.
D (input) The n diagonal elements of the tridiagonal matrix
A.
E (input) The (n-1) subdiagonal elements of the tridiagonal
matrix A.
DF (input or output)
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)
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) The N-by-NRHS right hand side matrix B.
LDB (input)
The leading dimension of the array B. LDB >=
max(1,N).
X (output)
If INFO = 0 of INFO = N+1, the N-by-NRHS solution
matrix X.
LDX (input)
The leading dimension of the array X. LDX >=
max(1,N).
RCOND (output)
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)
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)
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)
dimension(2*N)
INFO (output)
= 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, and the solution has not been com-
puted. RCOND = 0 is returned. = N+1: U is non-
singular, but RCOND is less than machine preci-
sion, meaning that the matrix is singular to work-
ing precision. Nevertheless, the solution and
error bounds are computed because there are a
number of situations where the computed solution
can be more accurate than the value of RCOND would
suggest.