SPBRFS(l)		LAPACK routine (version	1.1)		    SPBRFS(l)

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

SYNOPSIS

  SUBROUTINE SPBRFS( UPLO, N, KD, NRHS,	AB, LDAB, AFB, LDAFB, B, LDB, X, LDX,
		     FERR, BERR, WORK, IWORK, INFO )

      CHARACTER	     UPLO

      INTEGER	     INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS

      INTEGER	     IWORK( * )

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

PURPOSE
  SPBRFS improves the computed solution	to a system of linear equations	when
  the coefficient matrix is symmetric positive definite	and banded, and	pro-
  vides	error bounds and backward error	estimates for the solution.

ARGUMENTS

  UPLO	  (input) CHARACTER*1
	  = 'U':  Upper	triangle of A is stored;
	  = 'L':  Lower	triangle of A is stored.

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

  KD	  (input) INTEGER
	  The number of	superdiagonals of the matrix A if UPLO = 'U', or the
	  number of subdiagonals if UPLO = 'L'.	 KD >= 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 upper or lower triangle of the symmetric band matrix A, stored
	  in the first KD+1 rows of the	array.	The j-th column	of A is
	  stored in the	j-th column of the array AB as follows:	if UPLO	=
	  'U', AB(kd+1+i-j,j) =	A(i,j) for max(1,j-kd)<=i<=j; if UPLO =	'L',
	  AB(1+i-j,j)	 = A(i,j) for j<=i<=min(n,j+kd).

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

  AFB	  (input) REAL array, dimension	(LDAFB,N)
	  The triangular factor	U or L from the	Cholesky factorization A =
	  U**T*U or A =	L*L**T of the band matrix A as computed	by SPBTRF, in
	  the same storage format as A (see AB).

  LDAFB	  (input) INTEGER
	  The leading dimension	of the array AFB.  LDAFB >= KD+1.

  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 SPBTRS.  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.