
DGBRFS(l) ) DGBRFS(l)
NAME
DGBRFS  improve the computed solution to a system of linear equations when the coeffi
cient matrix is banded, and provides error bounds and backward error estimates for the
solution
SYNOPSIS
SUBROUTINE DGBRFS( 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( * )
DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), BERR( * ), FERR(
* ), WORK( * ), X( LDX, * )
PURPOSE
DGBRFS improves the computed solution to a system of linear equations when the coefficient
matrix is banded, and provides 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 = Transpose)
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) DOUBLE PRECISION array, dimension (LDAB,N)
The original band matrix A, stored in rows 1 to KL+KU+1. The jth column of A is
stored in the jth column of the array AB as follows: AB(ku+1+ij,j) = A(i,j) for
max(1,jku)<=i<=min(n,j+kl).
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KL+KU+1.
AFB (input) DOUBLE PRECISION array, dimension (LDAFB,N)
Details of the LU factorization of the band matrix A, as computed by DGBTRF. 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 DGBTRF; for 1<=i<=N, row i of the matrix was interchanged
with row IPIV(i).
B (input) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by DGBTRS. On exit, the improved
solution matrix X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
FERR (output) DOUBLE PRECISION array, dimension (NRHS)
The estimated forward error bound for each solution vector X(j) (the jth 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 esti
mate is as reliable as the estimate for RCOND, and is almost always a slight over
estimate of the true error.
BERR (output) DOUBLE PRECISION 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 solu
tion).
WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
IWORK (workspace) INTEGER array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
PARAMETERS
ITMAX is the maximum number of steps of iterative refinement.
LAPACK version 3.0 15 June 2000 DGBRFS(l) 
