cgbrfs man page on Scientific

```CGBRFS(1)		 LAPACK routine (version 3.2)		     CGBRFS(1)

NAME
CGBRFS - 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

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

CHARACTER	  TRANS

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

INTEGER	  IPIV( * )

REAL		  BERR( * ), FERR( * ), RWORK( * )

COMPLEX	  AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), WORK( *
), X( LDX, * )

PURPOSE
CGBRFS  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)

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) COMPLEX 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) COMPLEX array, dimension (LDAFB,N)
Details	of  the LU factorization of the band matrix A, as com‐
puted by CGBTRF.	 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 CGBTRF; for 1<=i<=N, row i of the matrix
was interchanged with row IPIV(i).

B       (input) COMPLEX 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) COMPLEX array, dimension (LDX,NRHS)
On entry, the solution matrix X, as  computed  by  CGBTRS.   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 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.

BERR    (output) REAL array, dimension (NRHS)
The componentwise relative backward error of each solution vec‐
tor X(j) (i.e., the smallest relative change in any element  of
A or B that makes X(j) an exact solution).

WORK    (workspace) COMPLEX array, dimension (2*N)

RWORK   (workspace) REAL 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 refinement.

LAPACK routine (version 3.2)	 November 2008			     CGBRFS(1)
```
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