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SGTSVX(l)					)					SGTSVX(l)

NAME
       SGTSVX - use the LU factorization to compute the solution to a real system of linear equa-
       tions A * X = B or A**T * X = B,

SYNOPSIS
       SUBROUTINE SGTSVX( 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

	   REAL 	  RCOND

	   INTEGER	  IPIV( * ), IWORK( * )

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

PURPOSE
       SGTSVX uses the LU factorization to compute the solution to a real system of linear  equa-
       tions  A  *  X = B or A**T * 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 = 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) REAL array, dimension (N-1)
	       The (n-1) subdiagonal elements of A.

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

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

       DLF     (input or output) REAL array, dimension (N-1)
	       If FACT = 'F', then DLF is an input argument and on entry contains the (n-1)  mul-
	       tipliers  that  define  the matrix L from the LU factorization of A as computed by
	       SGTTRF.

	       If FACT = 'N', then DLF is an output argument and on exit contains the (n-1)  mul-
	       tipliers that define the matrix L from the LU factorization of A.

       DF      (input or output) REAL 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) REAL array, dimension (N-1)
	       If  FACT = 'F', then DUF is an input argument and on entry contains the (n-1) ele-
	       ments of the first superdiagonal of U.

	       If FACT = 'N', then DUF is an output argument and on exit contains the (n-1)  ele-
	       ments of the first superdiagonal of U.

       DU2     (input or output) REAL array, dimension (N-2)
	       If  FACT = 'F', then DU2 is an input argument and on entry contains the (n-2) ele-
	       ments of the second superdiagonal of U.

	       If FACT = 'N', then DU2 is an output argument and on exit contains the (n-2)  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 SGTTRF.

	       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) REAL 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) REAL 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 esti-
	       mate is as reliable as the estimate for RCOND, and is almost always a slight over-
	       estimate 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 solu-
	       tion).

       WORK    (workspace) REAL 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 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 				SGTSVX(l)
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