.TH  CPBRFS 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) " 
.SH NAME
CPBRFS - the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and banded, and provides error bounds and backward error estimates for the solution
.SH SYNOPSIS
.TP 19
SUBROUTINE CPBRFS(
UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
.TP 19
.ti +4
CHARACTER
UPLO
.TP 19
.ti +4
INTEGER
INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
.TP 19
.ti +4
REAL
BERR( * ), FERR( * ), RWORK( * )
.TP 19
.ti +4
COMPLEX
AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
WORK( * ), X( LDX, * )
.SH PURPOSE
CPBRFS improves the computed solution to a system of linear
equations when the coefficient matrix is Hermitian positive definite
and banded, and provides error bounds and backward error estimates
for the solution.

.SH ARGUMENTS
.TP 8
UPLO    (input) CHARACTER*1
= \(aqU\(aq:  Upper triangle of A is stored;
.br
= \(aqL\(aq:  Lower triangle of A is stored.
.TP 8
N       (input) INTEGER
The order of the matrix A.  N >= 0.
.TP 8
KD      (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = \(aqU\(aq,
or the number of subdiagonals if UPLO = \(aqL\(aq.  KD >= 0.
.TP 8
NRHS    (input) INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X.  NRHS >= 0.
.TP 8
AB      (input) COMPLEX array, dimension (LDAB,N)
The upper or lower triangle of the Hermitian 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 = \(aqU\(aq, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = \(aqL\(aq, AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
.TP 8
LDAB    (input) INTEGER
The leading dimension of the array AB.  LDAB >= KD+1.
.TP 8
AFB     (input) COMPLEX array, dimension (LDAFB,N)
The triangular factor U or L from the Cholesky factorization
A = U**H*U or A = L*L**H of the band matrix A as computed by
CPBTRF, in the same storage format as A (see AB).
.TP 8
LDAFB   (input) INTEGER
The leading dimension of the array AFB.  LDAFB >= KD+1.
.TP 8
B       (input) COMPLEX array, dimension (LDB,NRHS)
The right hand side matrix B.
.TP 8
LDB     (input) INTEGER
The leading dimension of the array B.  LDB >= max(1,N).
.TP 8
X       (input/output) COMPLEX array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by CPBTRS.
On exit, the improved solution matrix X.
.TP 8
LDX     (input) INTEGER
The leading dimension of the array X.  LDX >= max(1,N).
.TP 8
FERR    (output) 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.
.TP 8
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 element of A or B that makes X(j) an exact solution).
.TP 8
WORK    (workspace) COMPLEX array, dimension (2*N)
.TP 8
RWORK   (workspace) REAL array, dimension (N)
.TP 8
INFO    (output) INTEGER
= 0:  successful exit
.br
< 0:  if INFO = -i, the i-th argument had an illegal value
.SH PARAMETERS

ITMAX is the maximum number of steps of iterative refinement.
