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# sgbrfs

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

SYNOPSIS
SUBROUTINE SGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
LDAFB, IPIV, B, LDB, X, LDX, FERR, BERR,
WORK, IWORK, INFO )

CHARACTER      TRANS

INTEGER        INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N,
NRHS

INTEGER        IPIV( * ), IWORK( * )

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

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

ARGUMENTS
TRANS   (input) CHARACTER*1
Specifies the form of the system of equations:
= 'N':  A * X = B     (No transpose)
= 'T':  A**T * X = B  (Transpose)
= 'C':  A**H * X = B  (Conjugate transpose = Tran-
spose)

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

KL      (input) INTEGER
The number of subdiagonals within the band of A.  KL
>= 0.

KU      (input) INTEGER
The number of superdiagonals within the band of A.
KU >= 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 original band matrix A, stored in rows 1 to
KL+KU+1.  The j-th column of A is stored in the j-th
column of the array AB as follows: AB(ku+1+i-j,j) =
A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

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

AFB     (input) REAL array, dimension (LDAFB,N)
Details of the LU factorization of the band matrix
A, as computed by SGBTRF.  U is stored as an upper
triangular band matrix with KL+KU superdiagonals in
rows 1 to KL+KU+1, and the multipliers used during
the factorization are stored in rows KL+KU+2 to
2*KL+KU+1.

LDAFB   (input) INTEGER
The leading dimension of the array AFB.  LDAFB >=
2*KL*KU+1.

IPIV    (input) INTEGER array, dimension (N)
The pivot indices from SGBTRF; for 1<=i<=N, row i of
the matrix was interchanged with row IPIV(i).

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
SGBTRS.  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.
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