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 tridiago­
       nal matrix and X and B are N-by-NRHS matrices


SYNOPSIS

       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, * )


PURPOSE

       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 tridiago­
       nal 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

          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 argu­
               ment and on exit contains the n diagonal  elements
               of the diagonal matrix D from the L*D*L**T factor­
               ization 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.

               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 ille­
               gal 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 com­
               puted. RCOND = 0 is returned.  = N+1: U is nonsin­
               gular,  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.


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