CXML
        	    
LAPACK version 3.0
cgtsvx(3)

PURPOSE
  CGTSVX - 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,

SYNTAX
  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, * )

DESCRIPTION
  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 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 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 contains
	  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 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 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 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
	  CGTTRF.

	  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 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 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 (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.