CPTSVX(3F)CPTSVX(3F)NAME
CPTSVX - use the factorization A = L*D*L**H to compute the solution to a
complex system of linear equations A*X = B, where A is an N-by-N
Hermitian positive definite tridiagonal matrix and X and B are N-by-NRHS
matrices
SYNOPSIS
SUBROUTINE CPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND,
FERR, BERR, WORK, RWORK, INFO )
CHARACTER FACT
INTEGER INFO, LDB, LDX, N, NRHS
REAL RCOND
REAL BERR( * ), D( * ), DF( * ), FERR( * ), RWORK( * )
COMPLEX B( LDB, * ), E( * ), EF( * ), WORK( * ), X( LDX, * )
PURPOSE
CPTSVX uses the factorization A = L*D*L**H to compute the solution to a
complex system of linear equations A*X = B, where A is an N-by-N
Hermitian 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**H, where L
is a unit lower bidiagonal matrix and D is diagonal. The
factorization can also be regarded as having the form
A = U**H*D*U.
2. The factored form of A is used to compute the condition number
of the matrix A. If the reciprocal of the condition number is
less than machine precision, steps 3 and 4 are skipped.
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.
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CPTSVX(3F)CPTSVX(3F)ARGUMENTS
FACT (input) CHARACTER*1
Specifies whether or not the factored form of the matrix A is
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) COMPLEX 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**H 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**H factorization of A.
EF (input or output) COMPLEX 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**H 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**H
factorization of A.
B (input) COMPLEX 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) COMPLEX array, dimension (LDX,NRHS)
If INFO = 0, 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, and the solution and
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CPTSVX(3F)CPTSVX(3F)
error bounds are not computed.
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) COMPLEX array, dimension (N)
RWORK (workspace) REAL array, dimension (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 definite, so the factorization could not be
completed unless i = N, and the solution and error bounds could
not be computed. = N+1 RCOND is less than machine precision.
The factorization has been completed, but the matrix is singular
to working precision, and the solution and error bounds have not
been computed.
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