ZGTRFS - 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 ZGTRFS( 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( * )

           DOUBLE         PRECISION BERR( * ), FERR( * ),  RWORK(
                          * )

           COMPLEX*16     B(  LDB, * ), D( * ), DF( * ), DL( * ),
                          DLF( * ), DU( * ), DU2( * ), DUF( *  ),
                          WORK( * ), X( LDX, * )


PURPOSE

       ZGTRFS  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*16 array, dimension (N-1)
               The (n-1) subdiagonal elements of A.

       D       (input) COMPLEX*16 array, dimension (N)
               The diagonal elements of A.

       DU      (input) COMPLEX*16 array, dimension (N-1)
               The (n-1) superdiagonal elements of A.

               The  (n-1)  multipliers  that  define the matrix L
               from the LU factorization of A as computed by ZGT­
               TRF.

       DF      (input) COMPLEX*16 array, dimension (N)
               The  n  diagonal  elements of the upper triangular
               matrix U from the LU factorization of A.

       DUF     (input) COMPLEX*16 array, dimension (N-1)
               The (n-1) elements of the first  superdiagonal  of
               U.

       DU2     (input) COMPLEX*16 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*16 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*16 array, dimension
               (LDX,NRHS)
               On  entry,  the  solution matrix X, as computed by
               ZGTTRS.  On exit, the improved solution matrix  X.

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

       FERR    (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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


       WORK    (workspace) COMPLEX*16 array, dimension (2*N)

       RWORK   (workspace) DOUBLE PRECISION 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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