# NAG CL Interfacef07bdc (dgbtrf)

Settings help

CL Name Style:

## 1Purpose

f07bdc computes the $LU$ factorization of a real $m×n$ band matrix.

## 2Specification

 #include
 void f07bdc (Nag_OrderType order, Integer m, Integer n, Integer kl, Integer ku, double ab[], Integer pdab, Integer ipiv[], NagError *fail)
The function may be called by the names: f07bdc, nag_lapacklin_dgbtrf or nag_dgbtrf.

## 3Description

f07bdc forms the $LU$ factorization of a real $m×n$ 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: $\mathbf{order}$Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{kl}$Integer Input
On entry: ${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
5: $\mathbf{ku}$Integer Input
On entry: ${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
6: $\mathbf{ab}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array ab must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdab}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pdab}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $m×n$ matrix $A$.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements ${A}_{ij}$, for row $i=1,\dots ,m$ and column $j=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,i-{k}_{l}\right),\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,i+{k}_{u}\right)$, depends on the order argument as follows:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${A}_{ij}$ is stored as ${\mathbf{ab}}\left[\left(j-1\right)×{\mathbf{pdab}}+{\mathbf{kl}}+{\mathbf{ku}}+i-j\right]$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${A}_{ij}$ is stored as ${\mathbf{ab}}\left[\left(i-1\right)×{\mathbf{pdab}}+{\mathbf{kl}}+j-i\right]$.
See Section 9 in f07bac for further details.
On exit: ab is overwritten by details of the factorization.
The elements, ${u}_{ij}$, of the upper triangular band factor $U$ with ${k}_{l}+{k}_{u}$ super-diagonals, and the multipliers, ${l}_{ij}$, used to form the lower triangular factor $L$ are stored. The elements ${u}_{ij}$, for $i=1,\dots ,m$ and $j=i,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,i+{k}_{l}+{k}_{u}\right)$, and ${l}_{ij}$, for $i=1,\dots ,m$ and $j=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,i-{k}_{l}\right),\dots ,i$, are stored where ${A}_{ij}$ is stored on entry.
7: $\mathbf{pdab}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array ab.
Constraint: ${\mathbf{pdab}}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
8: $\mathbf{ipiv}\left[\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right]$Integer Output
On exit: the pivot indices that define the permutation matrix. At the $\mathit{i}$th step, if ${\mathbf{ipiv}}\left[\mathit{i}-1\right]>\mathit{i}$ then row $\mathit{i}$ of the matrix $A$ was interchanged with row ${\mathbf{ipiv}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$. ${\mathbf{ipiv}}\left[i-1\right]\le i$ indicates that, at the $i$th step, a row interchange was not required.
9: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{kl}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{kl}}\ge 0$.
On entry, ${\mathbf{ku}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ku}}\ge 0$.
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pdab}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdab}}>0$.
NE_INT_3
On entry, ${\mathbf{pdab}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{kl}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{ku}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdab}}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR
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

f07bdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bdc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

## 9Further Comments

The total number of floating-point operations varies between approximately $2n{k}_{l}\left({k}_{u}+1\right)$ and $2n{k}_{l}\left({k}_{l}+{k}_{u}+1\right)$, depending on the interchanges, assuming $m=n\gg {k}_{l}$ and $n\gg {k}_{u}$.
A call to f07bdc may be followed by calls to the functions:
• f07bec to solve $AX=B$ or ${A}^{\mathrm{T}}X=B$;
• f07bgc to estimate the condition number of $A$.
The complex analogue of this function is f07brc.

## 10Example

This example computes the $LU$ factorization of the matrix $A$, where
 $A= ( -0.23 2.54 -3.66 0.00 -6.98 2.46 -2.73 -2.13 0.00 2.56 2.46 4.07 0.00 0.00 -4.78 -3.82 ) .$
Here $A$ is treated as a band matrix with one subdiagonal and two superdiagonals.

### 10.1Program Text

Program Text (f07bdce.c)

### 10.2Program Data

Program Data (f07bdce.d)

### 10.3Program Results

Program Results (f07bdce.r)