CXML
        	    
LAPACK version 3.0
sptsvx(3)

PURPOSE
  SPTSVX - use 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 matrices

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

      CHARACTER	     FACT

      INTEGER	     INFO, LDB, LDX, N, NRHS

      REAL	     RCOND

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

DESCRIPTION
  SPTSVX 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 matrices.
  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 supplied
	  on entry.  = 'F':  On entry, DF and EF contain the factored 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) REAL array, dimension (N)
	  The n diagonal elements of the tridiagonal matrix A.

  E	  (input) REAL array, dimension (N-1)
	  The (n-1) subdiagonal elements of the tridiagonal matrix 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 diagonal matrix D from the L*D*L**T
	  factorization of A.  If FACT = 'N', then DF is an output 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) REAL array, dimension (N-1)
	  If FACT = 'F', then EF is an input argument and on entry contains
	  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) 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 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) REAL
	  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 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).

  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) REAL 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 definite,
	  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.  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.