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spbrfs

 NAME
      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

 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 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 fac-
              torization 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 accu-
              rate, 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 refine-
      ment.