sgtsvx(3)

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, * )

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
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.

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 =

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 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 factorization 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 factorization 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 interchanged with row IPIV(i). IP

IV(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.

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 pre

cision. 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 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). 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 factor 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. 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.

LAPACK version 3.0 15 June 2000

SGTSVX(3)