
ZGTSVX(l) ) ZGTSVX(l)
NAME
ZGTSVX  use 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 ZGTSVX( 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
DOUBLE PRECISION RCOND
INTEGER IPIV( * )
DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
COMPLEX*16 B( LDB, * ), D( * ), DF( * ), DL( * ), DLF( * ), DU( * ), DU2( * ),
DUF( * ), WORK( * ), X( LDX, * )
PURPOSE
ZGTSVX 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 NbyNRHS matrices.
Error bounds on the solution and a condition estimate are also provided.
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 supplied 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*16 array, dimension (N1)
The (n1) subdiagonal elements of A.
D (input) COMPLEX*16 array, dimension (N)
The n diagonal elements of A.
DU (input) COMPLEX*16 array, dimension (N1)
The (n1) superdiagonal elements of A.
DLF (input or output) COMPLEX*16 array, dimension (N1)
If FACT = 'F', then DLF is an input argument and on entry contains the (n1) mul
tipliers that define the matrix L from the LU factorization of A as computed by
ZGTTRF.
If FACT = 'N', then DLF is an output argument and on exit contains the (n1) mul
tipliers that define the matrix L from the LU factorization of A.
DF (input or output) COMPLEX*16 array, dimension (N)
If FACT = 'F', then DF is an input argument and on entry contains 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*16 array, dimension (N1)
If FACT = 'F', then DUF is an input argument and on entry contains the (n1) ele
ments of the first superdiagonal of U.
If FACT = 'N', then DUF is an output argument and on exit contains the (n1) ele
ments of the first superdiagonal of U.
DU2 (input or output) COMPLEX*16 array, dimension (N2)
If FACT = 'F', then DU2 is an input argument and on entry contains the (n2) ele
ments of the second superdiagonal of U.
If FACT = 'N', then DU2 is an output argument and on exit contains the (n2) ele
ments 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 contains the pivot
indices from the LU factorization of A as computed by ZGTTRF.
If FACT = 'N', then IPIV is an output argument and on exit contains the pivot
indices from the LU factorization 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*16 array, dimension (LDB,NRHS)
The NbyNRHS right hand side matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (output) COMPLEX*16 array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the NbyNRHS solution matrix X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
RCOND (output) DOUBLE PRECISION
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) DOUBLE PRECISION array, dimension (NRHS)
The estimated forward error bound for each solution vector X(j) (the jth 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 esti
mate is as reliable as the estimate for RCOND, and is almost always a slight over
estimate of the true error.
BERR (output) DOUBLE PRECISION 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 solu
tion).
WORK (workspace) COMPLEX*16 array, dimension (2*N)
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith 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 preci
sion. 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 version 3.0 15 June 2000 ZGTSVX(l) 
