SGTRFS - 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 SGTRFS( 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( * )

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


PURPOSE

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

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

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

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

               TRF.

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

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

       DU2     (input) REAL 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) REAL 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) REAL array, dimension (LDX,NRHS)
               On  entry,  the  solution matrix X, as computed by
               SGTTRS.  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).


       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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