DPTSVX(l) ) DPTSVX(l)NAMEDPTSVX - 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 symmet‐
ric positive definite tridiagonal matrix and X and B are N-by-NRHS
matrices
SYNOPSIS
SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND,
FERR, BERR, WORK, INFO )
CHARACTER FACT
INTEGER INFO, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
E( * ), EF( * ), FERR( * ), WORK( * ), X( LDX, * )
PURPOSEDPTSVX 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 matri‐
ces. 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 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 sup‐
plied on entry. = 'F': On entry, DF and EF contain the fac‐
tored 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) DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the tridiagonal matrix A.
E (input) DOUBLE PRECISION array, dimension (N-1)
The (n-1) subdiagonal elements of the tridiagonal matrix A.
DF (input or output) DOUBLE PRECISION array, dimension (N)
If FACT = 'F', then DF is an input argument and on entry con‐
tains 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 out‐
put 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) DOUBLE PRECISION array, dimension (N-1)
If FACT = 'F', then EF is an input argument and on entry con‐
tains 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION
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) DOUBLE PRECISION 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 solu‐
tion 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) DOUBLE PRECISION array, dimension (NRHS)
The componentwise relative backward error of each solution vec‐
tor X(j) (i.e., the smallest relative change in any element of
A or B that makes X(j) an exact solution).
WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= N: the leading minor of order i of A is not positive defi‐
nite, 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. Nev‐
ertheless, 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 DPTSVX(l)