.TH SPTSVX 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) "
.SH NAME
SPTSVX - 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
.SH SYNOPSIS
.TP 19
SUBROUTINE SPTSVX(
FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
RCOND, FERR, BERR, WORK, INFO )
.TP 19
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CHARACTER
FACT
.TP 19
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INTEGER
INFO, LDB, LDX, N, NRHS
.TP 19
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REAL
RCOND
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REAL
B( LDB, * ), BERR( * ), D( * ), DF( * ),
E( * ), EF( * ), FERR( * ), WORK( * ),
X( LDX, * )
.SH 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.
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.SH DESCRIPTION
The following steps are performed:
.br
1. If FACT = \(aqN\(aq, 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
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A = U**T*D*U.
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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.
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3. The system of equations is solved for X using the factored form
of A.
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4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
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.SH ARGUMENTS
.TP 8
FACT (input) CHARACTER*1
Specifies whether or not the factored form of A has been
supplied on entry.
= \(aqF\(aq: On entry, DF and EF contain the factored form of A.
D, E, DF, and EF will not be modified.
= \(aqN\(aq: The matrix A will be copied to DF and EF and
factored.
.TP 8
N (input) INTEGER
The order of the matrix A. N >= 0.
.TP 8
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
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D (input) REAL array, dimension (N)
The n diagonal elements of the tridiagonal matrix A.
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E (input) REAL array, dimension (N-1)
The (n-1) subdiagonal elements of the tridiagonal matrix A.
.TP 8
DF (input or output) REAL array, dimension (N)
If FACT = \(aqF\(aq, 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 = \(aqN\(aq, 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.
.TP 8
EF (input or output) REAL array, dimension (N-1)
If FACT = \(aqF\(aq, then EF is an input argument and on entry
contains the (n-1) subdiagonal elements of the unit
bidiagonal factor L from the L*D*L**T factorization of A.
If FACT = \(aqN\(aq, 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.
.TP 8
B (input) REAL array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.
.TP 8
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
.TP 8
X (output) REAL array, dimension (LDX,NRHS)
If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X.
.TP 8
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
.TP 8
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.
.TP 8
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 upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j).
.TP 8
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).
.TP 8
WORK (workspace) REAL array, dimension (2*N)
.TP 8
INFO (output) INTEGER
= 0: successful exit
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< 0: if INFO = -i, the i-th argument had an illegal value
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> 0: if INFO = i, and i is
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<= 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. 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.