CXML
        	    
LAPACK version 3.0
spbrfs(3)

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

SYNTAX
  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, * )

DESCRIPTION
  SPBRFS improves the computed solution to a system of linear equations when
  the coefficient matrix is symmetric positive definite and banded, and
  provides 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 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) 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 illegal value

PARAMETERS
  ITMAX is the maximum number of steps of iterative refinement.