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DGTSVX(1)		 LAPACK routine (version 3.2)		     DGTSVX(1)

NAME
       DGTSVX  -  uses	the LU factorization to compute the solution to a real
       system of linear equations A * X = B or A**T * X = B,

SYNOPSIS
       SUBROUTINE DGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,  DU2,
			  IPIV,	 B,  LDB,  X,  LDX,  RCOND,  FERR, BERR, WORK,
			  IWORK, INFO )

	   CHARACTER	  FACT, TRANS

	   INTEGER	  INFO, LDB, LDX, N, NRHS

	   DOUBLE	  PRECISION RCOND

	   INTEGER	  IPIV( * ), IWORK( * )

	   DOUBLE	  PRECISION B( LDB, * ), BERR( * ), D( * ), DF(	 *  ),
			  DL(  *  ),  DLF(  *  ), DU( * ), DU2( * ), DUF( * ),
			  FERR( * ), WORK( * ), X( LDX, * )

PURPOSE
       DGTSVX uses the LU factorization to compute the solution to a real sys‐
       tem  of linear equations A * X = B or A**T * X = B, where A is a tridi‐
       agonal 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 = 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) DOUBLE PRECISION array, dimension (N-1)
	       The (n-1) subdiagonal elements of A.

       D       (input) DOUBLE PRECISION array, dimension (N)
	       The n diagonal elements of A.

       DU      (input) DOUBLE PRECISION array, dimension (N-1)
	       The (n-1) superdiagonal elements of A.

       DLF     (input or output) DOUBLE PRECISION 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 DGTTRF.	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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 DGTTRF.	 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 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) DOUBLE PRECISION array, dimension (3*N)

       IWORK   (workspace) INTEGER 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			     DGTSVX(1)
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