
SPTSVX(l) ) SPTSVX(l)
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 NbyN symmetric positive definite tridiagonal
matrix and X and B are NbyNRHS 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 NbyN symmetric positive definite tridiagonal
matrix and X and B are NbyNRHS 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 ibyi 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 factored 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 (N1)
The (n1) 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 (N1)
If FACT = 'F', then EF is an input argument and on entry contains the (n1) subdi
agonal 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 (n1)
subdiagonal elements of the unit bidiagonal factor L from the L*D*L**T factoriza
tion of A.
B (input) REAL array, dimension (LDB,NRHS)
The NbyNRHS 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 NbyNRHS 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 jth 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) 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 solu
tion).
WORK (workspace) REAL array, dimension (2*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
> 0: if INFO = i, and i is
<= N: the leading minor of order i of A is not positive definite, so the factor
ization 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. Nevertheless, the solu
tion 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(l) 
