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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.
```