CPTSVX - use the factorization A = L*D*L**H to compute the
       solution to a complex system of linear equations A*X =  B,
       where A is an N-by-N Hermitian positive definite tridiago­
       nal matrix and X and B are N-by-NRHS matrices


SYNOPSIS

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

           CHARACTER      FACT

           INTEGER        INFO, LDB, LDX, N, NRHS

           REAL           RCOND

           REAL           BERR( * ), D( * ), DF( * ), FERR( *  ),
                          RWORK( * )

           COMPLEX        B(  LDB,  * ), E( * ), EF( * ), WORK( *
                          ), X( LDX, * )


PURPOSE

       CPTSVX uses the factorization A = L*D*L**H to compute  the
       solution  to a complex system of linear equations A*X = B,
       where A is an N-by-N Hermitian 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**H, where L
          is a unit lower bidiagonal matrix and  D  is  diagonal.
       The
          factorization can also be regarded as having the form
          A = U**H*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


       3.  The system of equations is solved for X using the fac­
       tored 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  the
               matrix  A is 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) COMPLEX 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**H  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**H factor­
               ization of A.

       EF      (input or output) COMPLEX 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**H
               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**H factorization of A.

               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) COMPLEX 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  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) COMPLEX array, dimension (N)

       RWORK   (workspace) REAL array, dimension (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

               accurate than the value of RCOND would suggest.


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