NAG FL Interface
f07brf (zgbtrf)
1
Purpose
f07brf computes the $LU$ factorization of a complex $m$ by $n$ band matrix.
2
Specification
Fortran Interface
Integer, Intent (In) 
:: 
m, n, kl, ku, ldab 
Integer, Intent (Out) 
:: 
ipiv(min(m,n)), info 
Complex (Kind=nag_wp), Intent (Inout) 
:: 
ab(ldab,*) 

C Header Interface
#include <nag.h>
void 
f07brf_ (const Integer *m, const Integer *n, const Integer *kl, const Integer *ku, Complex ab[], const Integer *ldab, Integer ipiv[], Integer *info) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f07brf_ (const Integer &m, const Integer &n, const Integer &kl, const Integer &ku, Complex ab[], const Integer &ldab, Integer ipiv[], Integer &info) 
}

The routine may be called by the names f07brf, nagf_lapacklin_zgbtrf or its LAPACK name zgbtrf.
3
Description
f07brf forms the $LU$ factorization of a complex $m$ by $n$ band matrix $A$ using partial pivoting, with row interchanges. Usually $m=n$, and then, if $A$ has ${k}_{l}$ nonzero subdiagonals and ${k}_{u}$ nonzero superdiagonals, the factorization has the form $A=PLU$, where $P$ is a permutation matrix, $L$ is a lower triangular matrix with unit diagonal elements and at most ${k}_{l}$ nonzero elements in each column, and $U$ is an upper triangular band matrix with ${k}_{l}+{k}_{u}$ superdiagonals.
Note that $L$ is not a band matrix, but the nonzero elements of $L$ can be stored in the same space as the subdiagonal elements of $A$. $U$ is a band matrix but with ${k}_{l}$ additional superdiagonals compared with $A$. These additional superdiagonals are created by the row interchanges.
4
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments

1:
$\mathbf{m}$ – Integer
Input

On entry: $m$, the number of rows of the matrix $A$.
Constraint:
${\mathbf{m}}\ge 0$.

2:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of columns of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

3:
$\mathbf{kl}$ – Integer
Input

On entry: ${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint:
${\mathbf{kl}}\ge 0$.

4:
$\mathbf{ku}$ – Integer
Input

On entry: ${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint:
${\mathbf{ku}}\ge 0$.

5:
$\mathbf{ab}\left({\mathbf{ldab}},*\right)$ – Complex (Kind=nag_wp) array
Input/Output

Note: the second dimension of the array
ab
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the
$m$ by
$n$ matrix
$A$.
The matrix is stored in rows
${k}_{l}+1$ to
$2{k}_{l}+{k}_{u}+1$; the first
${k}_{l}$ rows need not be set, more precisely, the element
${A}_{ij}$ must be stored in
See
Section 9 in
f07bnf for further details.
On exit: if
${\mathbf{info}}\ge {\mathbf{0}}$,
ab is overwritten by details of the factorization.
The upper triangular band matrix $U$, with ${k}_{l}+{k}_{u}$ superdiagonals, is stored in rows $1$ to ${k}_{l}+{k}_{u}+1$ of the array, and the multipliers used to form the matrix $L$ are stored in rows ${k}_{l}+{k}_{u}+2$ to $2{k}_{l}+{k}_{u}+1$.

6:
$\mathbf{ldab}$ – Integer
Input

On entry: the first dimension of the array
ab as declared in the (sub)program from which
f07brf is called.
Constraint:
${\mathbf{ldab}}\ge 2\times {\mathbf{kl}}+{\mathbf{ku}}+1$.

7:
$\mathbf{ipiv}\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$ – Integer array
Output

On exit: the pivot indices that define the permutation matrix. At the
$\mathit{i}$th step, if ${\mathbf{ipiv}}\left(\mathit{i}\right)>\mathit{i}$ then row $\mathit{i}$ of the matrix $A$ was interchanged with row ${\mathbf{ipiv}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$. ${\mathbf{ipiv}}\left(i\right)\le i$ indicates that, at the $i$th step, a row interchange was not required.

8:
$\mathbf{info}$ – Integer
Output
On exit:
${\mathbf{info}}=0$ unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
 ${\mathbf{info}}<0$

If ${\mathbf{info}}=i$, argument $i$ had an illegal value.
If
${\mathbf{info}}=999$, dynamic memory allocation failed. See
Section 9 in the Introduction to the NAG Library FL Interface for further information. An explanatory message is output, and execution of the program is terminated.
 ${\mathbf{info}}>0$

Element $\u2329\mathit{\text{value}}\u232a$ of the diagonal is exactly zero.
The factorization has been completed, but the factor $U$
is exactly singular, and division by zero will occur if it is used to solve
a system of equations.
7
Accuracy
The computed factors
$L$ and
$U$ are the exact factors of a perturbed matrix
$A+E$, where
$c\left(k\right)$ is a modest linear function of
$k={k}_{l}+{k}_{u}+1$, and
$\epsilon $ is the
machine precision. This assumes
$k\ll \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$.
8
Parallelism and Performance
f07brf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07brf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The total number of real floatingpoint operations varies between approximately $8n{k}_{l}\left({k}_{u}+1\right)$ and $8n{k}_{l}\left({k}_{l}+{k}_{u}+1\right)$, depending on the interchanges, assuming $m=n\gg {k}_{l}$ and $n\gg {k}_{u}$.
A call to
f07brf may be followed by calls to the routines:
 f07bsf to solve $AX=B$, ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$;
 f07buf to estimate the condition number of $A$.
The real analogue of this routine is
f07bdf.
10
Example
This example computes the
$LU$ factorization of the matrix
$A$, where
Here
$A$ is treated as a band matrix with one subdiagonal and two superdiagonals.
10.1
Program Text
10.2
Program Data
10.3
Program Results