CGTSVX(1) LAPACK routine (version 3.2) CGTSVX(1)NAME
CGTSVX - uses the LU factorization to compute the solution to a complex
system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
SYNOPSIS
SUBROUTINE CGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
RWORK, INFO )
CHARACTER FACT, TRANS
INTEGER INFO, LDB, LDX, N, NRHS
REAL RCOND
INTEGER IPIV( * )
REAL BERR( * ), FERR( * ), RWORK( * )
COMPLEX B( LDB, * ), D( * ), DF( * ), DL( * ), DLF( * ), DU(
* ), DU2( * ), DUF( * ), WORK( * ), X( LDX, * )
PURPOSE
CGTSVX uses the LU factorization to compute the solution to a complex
system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
matrices.
Error bounds on the solution and a condition estimate are also pro‐
vided.
DESCRIPTION
The following steps are performed:
1. If FACT = 'N', the LU decomposition is used to factor the matrix A
as A = L * U, where L is a product of permutation and unit lower
bidiagonal matrices and U is upper triangular with nonzeros in
only the main diagonal and first two superdiagonals.
2. If some U(i,i)=0, so that U is exactly singular, 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': DLF, DF, DUF, DU2, and IPIV contain
the factored form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
will not be modified. = 'N': The matrix will be copied to
DLF, DF, and DUF and factored.
TRANS (input) CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose)
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 matrix B. NRHS >= 0.
DL (input) COMPLEX array, dimension (N-1)
The (n-1) subdiagonal elements of A.
D (input) COMPLEX array, dimension (N)
The n diagonal elements of A.
DU (input) COMPLEX array, dimension (N-1)
The (n-1) superdiagonal elements of A.
DLF (input or output) COMPLEX array, dimension (N-1)
If FACT = 'F', then DLF is an input argument and on entry con‐
tains the (n-1) multipliers that define the matrix L from the
LU factorization of A as computed by CGTTRF. If FACT = 'N',
then DLF is an output argument and on exit contains the (n-1)
multipliers that define the matrix L from the LU factorization
of A.
DF (input or output) COMPLEX array, dimension (N)
If FACT = 'F', then DF is an input argument and on entry con‐
tains the n diagonal elements of the upper triangular matrix U
from the LU factorization of A. If FACT = 'N', then DF is an
output argument and on exit contains the n diagonal elements of
the upper triangular matrix U from the LU factorization of A.
DUF (input or output) COMPLEX array, dimension (N-1)
If FACT = 'F', then DUF is an input argument and on entry con‐
tains the (n-1) elements of the first superdiagonal of U. If
FACT = 'N', then DUF is an output argument and on exit contains
the (n-1) elements of the first superdiagonal of U.
DU2 (input or output) COMPLEX array, dimension (N-2)
If FACT = 'F', then DU2 is an input argument and on entry con‐
tains the (n-2) elements of the second superdiagonal of U. If
FACT = 'N', then DU2 is an output argument and on exit contains
the (n-2) elements of the second superdiagonal of U.
IPIV (input or output) INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry con‐
tains the pivot indices from the LU factorization of A as com‐
puted by CGTTRF. If FACT = 'N', then IPIV is an output argu‐
ment and on exit contains the pivot indices from the LU factor‐
ization of A; row i of the matrix was interchanged with row
IPIV(i). IPIV(i) will always be either i or i+1; IPIV(i) = i
indicates a row interchange was not required.
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 or 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 estimate of 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 estimated 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).
The estimate is as reliable as the estimate for RCOND, and is
almost always a slight overestimate of the true error.
BERR (output) REAL 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) COMPLEX array, dimension (2*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: U(i,i) is exactly zero. The factorization has not been
completed unless i = N, but the factor U is exactly singular,
so the solution and error bounds could not be 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.
LAPACK routine (version 3.2) November 2008 CGTSVX(1)
