# NAG FL Interfacef07brf (zgbtrf)

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## 1Purpose

f07brf computes the $LU$ factorization of a complex $m×n$ band matrix.

## 2Specification

Fortran Interface
 Subroutine f07brf ( m, n, kl, ku, ab, ldab, ipiv, info)
 Integer, Intent (In) :: m, n, kl, ku, ldab Integer, Intent (Out) :: ipiv(min(m,n)), info Complex (Kind=nag_wp), Intent (Inout) :: ab(ldab,*)
#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)
The routine may be called by the names f07brf, nagf_lapacklin_zgbtrf or its LAPACK name zgbtrf.

## 3Description

f07brf forms the $LU$ factorization of a complex $m×n$ 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.

## 4References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

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 $m×n$ 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
 $ab(kl+ku+1+i-j,j)=Aij for ​max(1,j-ku)≤i≤min(m,j+kl).$
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: ${\mathbf{ldab}}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
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).

## 6Error 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 $⟨\mathit{\text{value}}⟩$ 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.

## 7Accuracy

The computed factors $L$ and $U$ are the exact factors of a perturbed matrix $A+E$, where
 $|E|≤c(k)εP|L||U| ,$
$c\left(k\right)$ is a modest linear function of $k={k}_{l}+{k}_{u}+1$, and $\epsilon$ is the machine precision. This assumes $k\ll \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$.

## 8Parallelism 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 implementation-specific information.

The total number of real floating-point 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.

## 10Example

This example computes the $LU$ factorization of the matrix $A$, where
 $A= ( -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00+0.00i 0.00+6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00+0.00i -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00+0.00i 0.00+0.00i 4.48-1.09i -0.46-1.72i ) .$
Here $A$ is treated as a band matrix with one subdiagonal and two superdiagonals.

### 10.1Program Text

Program Text (f07brfe.f90)

### 10.2Program Data

Program Data (f07brfe.d)

### 10.3Program Results

Program Results (f07brfe.r)