CXML
        	    
LAPACK version 3.0
dgbrfs(3)

PURPOSE
  DGBRFS - 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

SYNTAX
  SUBROUTINE DGBRFS( 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( * )

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

DESCRIPTION
  DGBRFS 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDAFB,N)
	  Details of the LU factorization of the band matrix A, as computed
	  by DGBTRF.  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 DGBTRF; for 1<=i<=N, row i of the matrix was
	  interchanged with row IPIV(i).

  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 DGBTRS.  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 solution 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 element 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) 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).

  WORK	  (workspace) DOUBLE PRECISION 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.