DGTRFS - 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 DGTRFS( TRANS, N, NRHS, DL,  D,  DU,  DLF,  DF,
                          DUF,  DU2,  IPIV, B, LDB, X, LDX, FERR,
                          BERR, WORK, IWORK, INFO )

           CHARACTER      TRANS

           INTEGER        INFO, LDB, LDX, N, NRHS

           INTEGER        IPIV( * ), IWORK( * )

           DOUBLE         PRECISION B( LDB, * ), BERR( * ), D(  *
                          ), DF( * ), DL( * ), DLF( * ), DU( * ),
                          DU2( * ), DUF( * ), FERR( * ), WORK(  *
                          ), X( LDX, * )


PURPOSE

       DGTRFS  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  =
               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) DOUBLE PRECISION array, dimension (N-1)
               The (n-1) subdiagonal elements of A.

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

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

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

               TRF.

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

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

       DU2     (input) DOUBLE PRECISION 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) DOUBLE PRECISION 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)   DOUBLE   PRECISION   array,
               dimension (LDX,NRHS)
               On entry, the solution matrix X,  as  computed  by
               DGTTRS.   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
               exact solution).

               (3*N)

       IWORK   (workspace) INTEGER 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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