SPBRFS(3)      LAPACK routine of NEC Numeric Library Collection      SPBRFS(3)



NAME
       SPBRFS

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



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

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

           KD        (input)
                     KD is INTEGER
                     The number of superdiagonals of the matrix A if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

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

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

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

           B         (input)
                     B is REAL array, dimension (LDB,NRHS)
                     The right hand side matrix B.

           LDB       (input)
                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,N).

           X         (input/output)
                     X is REAL array, dimension (LDX,NRHS)
                     On entry, the solution matrix X, as computed by SPBTRS.
                     On exit, the improved solution matrix X.

           LDX       (input)
                     LDX is INTEGER
                     The leading dimension of the array X.  LDX >= max(1,N).

           FERR      (output)
                     FERR is 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)
                     BERR is 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      (output)
                     WORK is REAL array, dimension (3*N)

           IWORK     (output)
                     IWORK is INTEGER array, dimension (N)

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value



       Internal Parameters:


             ITMAX is the maximum number of steps of iterative refinement.



LAPACK routine                  31 October 2017                      SPBRFS(3)