dptsvx man page on Scientific

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DPTSVX(1)		 LAPACK routine (version 3.2)		     DPTSVX(1)

NAME
       DPTSVX - uses the factorization A = L*D*L**T to compute the solution to
       a real system of linear equations A*X = B, where A is an N-by-N symmet‐
       ric  positive  definite	tridiagonal  matrix  and X and B are N-by-NRHS
       matrices

SYNOPSIS
       SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,	RCOND,
			  FERR, BERR, WORK, INFO )

	   CHARACTER	  FACT

	   INTEGER	  INFO, LDB, LDX, N, NRHS

	   DOUBLE	  PRECISION RCOND

	   DOUBLE	  PRECISION  B(	 LDB, * ), BERR( * ), D( * ), DF( * ),
			  E( * ), EF( * ), FERR( * ), WORK( * ), X( LDX, * )

PURPOSE
       DPTSVX uses the factorization A = L*D*L**T to compute the solution to a
       real system of linear equations A*X = B, where A is an N-by-N symmetric
       positive definite tridiagonal matrix and X and B are  N-by-NRHS	matri‐
       ces.   Error  bounds  on the solution and a condition estimate are also
       provided.

DESCRIPTION
       The following steps are performed:
       1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L
	  is a unit lower bidiagonal matrix and D is diagonal.	The
	  factorization can also be regarded as having the form
	  A = U**T*D*U.
       2. If the leading i-by-i principal minor is not positive definite,
	  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':  On entry, DF and EF contain the fac‐
	       tored form of A.	 D, E, DF, and EF will	not  be	 modified.   =
	       'N':  The matrix A will be copied to DF and EF and factored.

       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 matrices B and X.  NRHS >= 0.

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

       E       (input) DOUBLE PRECISION array, dimension (N-1)
	       The (n-1) subdiagonal elements of the tridiagonal matrix 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 diagonal matrix D from the
	       L*D*L**T factorization of A.  If FACT = 'N', then DF is an out‐
	       put  argument  and  on exit contains the n diagonal elements of
	       the diagonal matrix D from the L*D*L**T factorization of A.

       EF      (input or output) DOUBLE PRECISION array, dimension (N-1)
	       If FACT = 'F', then EF is an input argument and on  entry  con‐
	       tains  the  (n-1)  subdiagonal  elements of the unit bidiagonal
	       factor L from the L*D*L**T factorization of A.  If FACT =  'N',
	       then  EF	 is  an output argument and on exit contains the (n-1)
	       subdiagonal elements of the unit bidiagonal factor L  from  the
	       L*D*L**T factorization of A.

       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 of 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  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 forward error bound for each solution vector X(j) (the j-th
	       column  of  the solution matrix X).  If XTRUE is the true solu‐
	       tion 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).

       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 (2*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:  the leading minor of order i of A is not positive	 defi‐
	       nite,  so  the  factorization  could  not be completed, and the
	       solution has not been 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.  Nev‐
	       ertheless,  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			     DPTSVX(1)
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