sptsvx(3)

NAME

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

SYNOPSIS

SUBROUTINE SPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X,
LDX, RCOND, FERR, BERR, WORK, INFO )
    CHARACTER      FACT
    INTEGER        INFO, LDB, LDX, N, NRHS
    REAL           RCOND
    REAL           B( LDB, * ), BERR( * ), D( * ),  DF(  *
), E( * ), EF( * ), FERR( * ), WORK( * ), X( LDX, * )

PURPOSE

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.

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. 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.

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) REAL array, dimension (N)
The n diagonal elements of the tridiagonal matrix

A.

E (input) REAL array, dimension (N-1)
The (n-1) subdiagonal elements of the tridiagonal

matrix 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 diagonal matrix D

from the L*D*L**T factorization of A. If FACT = 'N', then DF is

an output argument and on exit contains the n diagonal elements

of the diagonal matrix D from the L*D*L**T factorization of A.

EF (input or output) REAL array, dimension (N-1)
If FACT = 'F', then EF is an input argument and on

entry contains the (n-1) subdiagonal elements of the unit bidiag

onal 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) 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 of 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 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 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 up

per 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) 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 (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,

and the solution has not been 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 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

SPTSVX(3)