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NAME
sgtsvx - use the LU factorization to compute the solution to
a real system of linear equations A * X = B or A**T * X = B,
SYNOPSIS
SUBROUTINE SGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF,
DUF, DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
WORK, IWORK, INFO )
CHARACTER FACT, TRANS
INTEGER INFO, LDB, LDX, N, NRHS
REAL RCOND
INTEGER IPIV( * ), IWORK( * )
REAL B( LDB, * ), BERR( * ), D( * ), DF( * ), DL( * ), DLF(
* ), DU( * ), DU2( * ), DUF( * ), FERR( * ), WORK(
* ), X( LDX, * )
#include <sunperf.h>
void sgtsvx(char fact, char trans, int n, int nrhs, float
*dl, float *d, float *du, float *dlf, float *df,
float *duf, float *du2, int *ipivot, float *sb,
int ldb, float *sx, int ldx, float *srcond, float
*ferr, float *berr, int *info);
PURPOSE
SGTSVX uses the LU factorization to compute the solution to
a real system of linear equations A * X = B or A**T * X = B,
where A is a tridiagonal matrix of order N 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 LU decomposition is used to factor the
matrix A as A = L * U, where L is a product of permutation
and unit lower bidiagonal matrices and U is upper triangular
with nonzeros in only the main diagonal and first two super-
diagonals.
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': DLF, DF, DUF,
DU2, and IPIV contain the factored form of A; DL,
D, DU, DLF, DF, DUF, DU2 and IPIV will not be
modified. = 'N': The matrix will be copied to
DLF, DF, and DUF and factored.
TRANS (input) CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose = Tran-
spose)
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 matrix B. NRHS >= 0.
DL (input) REAL array, dimension (N-1)
The (n-1) subdiagonal elements of A.
D (input) REAL array, dimension (N)
The n diagonal elements of A.
DU (input) REAL array, dimension (N-1)
The (n-1) superdiagonal elements of A.
DLF (input or output) REAL array, dimension (N-1)
If FACT = 'F', then DLF is an input argument and
on entry contains the (n-1) multipliers that
define the matrix L from the LU factorization of A
as computed by SGTTRF.
If FACT = 'N', then DLF is an output argument and
on exit contains the (n-1) multipliers that define
the matrix L from the LU factorization of A.
DF (input or output) REAL array, dimension (N)
If FACT = 'F', then DF is an input argument and on
entry contains the n diagonal elements of the
upper triangular matrix U from the LU factoriza-
tion of A.
If FACT = 'N', then DF is an output argument and
on exit contains the n diagonal elements of the
upper triangular matrix U from the LU factoriza-
tion of A.
DUF (input or output) REAL array, dimension (N-1)
If FACT = 'F', then DUF is an input argument and
on entry contains the (n-1) elements of the first
superdiagonal of U.
If FACT = 'N', then DUF is an output argument and
on exit contains the (n-1) elements of the first
superdiagonal of U.
DU2 (input or output) REAL array, dimension (N-2)
If FACT = 'F', then DU2 is an input argument and
on entry contains the (n-2) elements of the second
superdiagonal of U.
If FACT = 'N', then DU2 is an output argument and
on exit contains the (n-2) elements of the second
superdiagonal of U.
IPIV (input or output) INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and
on entry contains the pivot indices from the LU
factorization of A as computed by SGTTRF.
If FACT = 'N', then IPIV is an output argument and
on exit contains the pivot indices from the LU
factorization of A; row i of the matrix was inter-
changed with row IPIV(i). IPIV(i) will always be
either i or i+1; IPIV(i) = i indicates a row
interchange was not required.
B (input) REAL 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) REAL 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 com-
puted.
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
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) 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) REAL 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: if INFO = i, and i is
<= N: U(i,i) is exactly zero. The factorization
has not been completed unless i = N, but the fac-
tor U is exactly singular, so 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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