CGTRFS - improve the computed solution to a system of lin­
       ear equations when the coefficient matrix is  tridiagonal,
       and provides error bounds and backward error estimates for
       the solution


SYNOPSIS

       SUBROUTINE CGTRFS( TRANS, N, NRHS, DL,  D,  DU,  DLF,  DF,
                          DUF,  DU2,  IPIV, B, LDB, X, LDX, FERR,
                          BERR, WORK, RWORK, INFO )

           CHARACTER      TRANS

           INTEGER        INFO, LDB, LDX, N, NRHS

           INTEGER        IPIV( * )

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

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


PURPOSE

       CGTRFS improves the computed solution to a system of  lin­
       ear  equations when the coefficient matrix is tridiagonal,
       and provides error bounds and backward error estimates for
       the solution.


ARGUMENTS

       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)

       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) COMPLEX array, dimension (N-1)
               The (n-1) subdiagonal elements of A.

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

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

       DLF     (input) COMPLEX array, dimension (N-1)
               The (n-1) multipliers that  define  the  matrix  L

               TRF.

       DF      (input) COMPLEX array, dimension (N)
               The n diagonal elements of  the  upper  triangular
               matrix U from the LU factorization of A.

       DUF     (input) COMPLEX array, dimension (N-1)
               The  (n-1)  elements of the first superdiagonal of
               U.

       DU2     (input) COMPLEX array, dimension (N-2)
               The (n-2) elements of the second superdiagonal  of
               U.

       IPIV    (input) INTEGER array, dimension (N)
               The  pivot  indices; for 1 <= i <= n, row i of the
               matrix was interchanged with row IPIV(i).  IPIV(i)
               will  always be either i or i+1; IPIV(i) = i indi­
               cates a row interchange was not required.

       B       (input) COMPLEX array, dimension (LDB,NRHS)
               The right hand side matrix B.

       LDB     (input) INTEGER
               The leading dimension of  the  array  B.   LDB  >=
               max(1,N).

       X       (input/output) COMPLEX array, dimension (LDX,NRHS)
               On entry, the solution matrix X,  as  computed  by
               CGTTRS.   On exit, the improved solution matrix X.

       LDX     (input) INTEGER
               The leading dimension of  the  array  X.   LDX  >=
               max(1,N).

       FERR    (output) REAL array, dimension (NRHS)
               The  estimated  forward error bound for each solu­
               tion vector X(j) (the j-th column of the  solution
               matrix  X).   If XTRUE is the true solution corre­
               sponding 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 estimate is as reli­
               able as the estimate  for  RCOND,  and  is  almost
               always a slight overestimate 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 solution).


       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


PARAMETERS

       ITMAX  is the maximum number of steps of iterative refine­
       ment.


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