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# zgtsvx

```
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 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 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. 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, steps 3 and 4 are skipped.

3. The system of equations is solved for X using the fac-
tored 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 (N-1)
The (n-1) subdiagonal elements of A.

D       (input) COMPLEX*16 array, dimension (N)
The n diagonal elements of A.

DU      (input) COMPLEX*16 array, dimension (N-1)
The (n-1) superdiagonal elements of A.

DLF     (input or output) COMPLEX*16 array, dimension (N-1)
If FACT = 'F', then DLF is an input argument and on
entry contains the (n-1) multipliers 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 (n-1) multipliers 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 (N-1)
If FACT = 'F', then DUF is an input argument and on
entry contains 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 super-
diagonal of U.

DU2     (input or output) COMPLEX*16 array, dimension (N-2)
If FACT = 'F', then DU2 is an input argument and on
entry contains 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) INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on
entry contains the pivot indices from the LU factor-
ization 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 fac-
torization of A; row i of the matrix was inter-
changed with row IPIV(i).  IPIV(i) will always be
either i or i+1; IPIV(i) = i indicates a row inter-
change was not required.

B       (input) COMPLEX*16 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*16 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) 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, and the
solution and error bounds are not computed.

FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
The estimated forward error bounds for each solution
vector X(j) (the j-th column of the solution matrix
X).  If XTRUE is the true solution, FERR(j) bounds
the magnitude of the largest entry in (X(j) - XTRUE)
divided by the magnitude of the largest entry in
X(j).  The quality of the error bound depends on the
quality of the estimate of norm(inv(A)) computed in
the code; if the estimate of norm(inv(A)) is accu-
rate, the error bound is guaranteed.

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 entry of A or B that makes X(j) an
exact solution).

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 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.  = 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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