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, * )
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 superdiagonals.
2. If some U(i,i)=0, so that U 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
precision,
INFO = N+1 is returned as a warning, but the routine
still goes on
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': DLF, DF, DUF,
DU2, and IPIV contain the factored form of A; DL,
D, DU, DLF, DF, DUF, DU2 and IPIV will not be mod
ified. = '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 =
Transpose)
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
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 or INFO = N+1, the N-by-NRHS solution
matrix X.
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
(X(j) - XTRUE) divided by the magnitude of the
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) 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. RCOND = 0 is
returned. = N+1: U is nonsingular, but RCOND is
less than machine precision, meaning that the
matrix is singular to working precision. Never
theless, the solution and error bounds are com
puted because there are a number of situations
where the computed solution can be more accurate
than the value of RCOND would suggest.
Man(1) output converted with
man2html