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 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. 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  fac-
	       tored 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  fac-
	       torization 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 fac-
	       torization 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 superdiagonal 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 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 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 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) 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 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) 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 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.  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 version 3.0						   15 June 2000 							 ZGTSVX(l)