SGBRFS(l)		LAPACK routine (version	1.1)		    SGBRFS(l)

NAME
  SGBRFS - improve the computed	solution to a system of	linear equations when
  the coefficient matrix is banded, and	provides 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, * )

PURPOSE
  SGBRFS improves the computed solution	to a system of linear equations	when
  the coefficient matrix is banded, 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.

  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 fol-
	  lows:	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 bounds for each solution vector X(j)
	  (the j-th column of the solution matrix X).  If XTRUE	is the true
	  solution, FERR(j) bounds the magnitude of the	largest	entry in
	  (X(j)	- XTRUE) divided by the	magnitude of the largest entry in
	  X(j).	 The quality of	the error bound	depends	on the quality of the
	  estimate of norm(inv(A)) computed in the code; if the	estimate of
	  norm(inv(A)) is accurate, the	error bound is guaranteed.

  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 entry	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 illegal value

PARAMETERS

  ITMAX	is the maximum number of steps of iterative refinement.