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NAME
     sgbrfs - improve the computed solution to a system of linear
     equations  when  the  coefficient matrix is banded, and pro-
     vides error bounds and  backward  error  estimates  for  the
     solution

SYNOPSIS
     SUBROUTINE SGBRFS( TRANS, N, KL, KU, NRHS,  AB,  LDAB,  AFB,
               LDAFB,  IPIV,  B,  LDB,  X, LDX, FERR, BERR, WORK,
               IWORK, INFO )

     CHARACTER TRANS

     INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS

     INTEGER IPIV( * ), IWORK( * )

     REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), BERR( * ),
               FERR( * ), WORK( * ), X( LDX, * )



     #include <sunperf.h>

     void sgbrfs(char trans, int n, int kl,  int  ku,  int  nrhs,
               float  *sab,  int ldab, float *afb, int ldafb, int
               *ipivot, float *sb, int ldb, float *sx,  int  ldx,
               float *ferr, float *berr, int *info) ;

PURPOSE
     SGBRFS improves the computed solution to a system of  linear
     equations  when  the  coefficient matrix is banded, and pro-
     vides 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 = Tran-
               spose)

     N         (input) INTEGER
               The order of the matrix A.  N >= 0.

     KL        (input) INTEGER
               The number of subdiagonals within the band  of  A.
               KL >= 0.

     KU        (input) INTEGER
               The number of superdiagonals within the band of A.
               KU >= 0.

     NRHS      (input) INTEGER
               The number of right hand sides, i.e.,  the  number
               of columns of the matrices B and X.  NRHS >= 0.

     AB        (input) REAL array, dimension (LDAB,N)
               The original band matrix A, stored in  rows  1  to
               KL+KU+1.   The  j-th  column of A is stored in the
               j-th  column  of  the   array   AB   as   follows:
               AB(ku+1+i-j,j)     =     A(i,j)    for    max(1,j-
               ku)<=i<=min(n,j+kl).

     LDAB      (input) INTEGER
               The leading dimension of the array  AB.   LDAB  >=
               KL+KU+1.

     AFB       (input) REAL array, dimension (LDAFB,N)
               Details of the LU factorization of the band matrix
               A, as computed by SGBTRF.  U is stored as an upper
               triangular band matrix with  KL+KU  superdiagonals
               in  rows  1  to  KL+KU+1, and the multipliers used
               during  the  factorization  are  stored  in   rows
               KL+KU+2 to 2*KL+KU+1.

     LDAFB     (input) INTEGER
               The leading dimension of the array AFB.  LDAFB  >=
               2*KL*KU+1.

     IPIV      (input) INTEGER array, dimension (N)
               The pivot indices from SGBTRF; for 1<=i<=N, row  i
               of the matrix was interchanged with row IPIV(i).

     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
               SGBTRS.  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
               corresponding  to  X(j),  FERR(j)  is an estimated
               upper bound for the magnitude of the largest  ele-
               ment in (X(j) - XTRUE) divided by the magnitude of
               the largest element in X(j).  The estimate  is  as
               reliable  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).

     WORK      (workspace) REAL array, dimension (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

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