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 tridiago
nal 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 tridiago
nal 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
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 argu
ment and on exit contains the n diagonal elements
of the diagonal matrix D from the L*D*L**T factor
ization 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 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 (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.
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 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 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 com
puted. RCOND = 0 is returned. = N+1: U is nonsin
gular, 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.
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