f16 Chapter Contents
f16 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_dgb_norm (f16rbc)

## 1  Purpose

nag_dgb_norm (f16rbc) calculates the value of the $1$-norm, the $\infty$-norm, the Frobenius norm or the maximum absolute value of the elements of a real $m$ by $n$ band matrix, stored in banded form.

## 2  Specification

 #include #include
 void nag_dgb_norm (Nag_OrderType order, Nag_NormType norm, Integer m, Integer n, Integer kl, Integer ku, const double ab[], Integer pdab, double *r, NagError *fail)

## 3  Description

Given a real $m$ by $n$ banded matrix, $A$, nag_dgb_norm (f16rbc) calculates one of the values given by
 ${‖A‖}_{1}=\underset{j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\sum _{i=1}^{m}\left|{a}_{ij}\right|$ (the $1$-norm of $A$), ${‖A‖}_{\infty }=\underset{i}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\sum _{j=1}^{n}\left|{a}_{ij}\right|$ (the $\infty$-norm of $A$), ${‖A‖}_{F}={\left(\sum _{i=1}^{m}\sum _{j=1}^{n}{\left|{a}_{ij}\right|}^{2}\right)}^{1/2}$ (the Frobenius norm of $A$),   or $\underset{i,j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\left|{a}_{ij}\right|$ (the maximum absolute element value of $A$).

## 4  References

The BLAS Technical Forum Standard (2001) http://www.netlib.org/blas/blast-forum

## 5  Arguments

1:     orderNag_OrderTypeInput
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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     normNag_NormTypeInput
On entry: specifies the value to be returned.
${\mathbf{norm}}=\mathrm{Nag_OneNorm}$
The $1$-norm.
${\mathbf{norm}}=\mathrm{Nag_InfNorm}$
The $\infty$-norm.
${\mathbf{norm}}=\mathrm{Nag_FrobeniusNorm}$
The Frobenius (or Euclidean) norm.
${\mathbf{norm}}=\mathrm{Nag_MaxNorm}$
The value $\underset{i,j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\left|{a}_{ij}\right|$ (not a norm).
Constraint: ${\mathbf{norm}}=\mathrm{Nag_OneNorm}$, $\mathrm{Nag_TwoNorm}$, $\mathrm{Nag_FrobeniusNorm}$, $\mathrm{Nag_InfNorm}$ or $\mathrm{Nag_MaxNorm}$.
3:     mIntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
4:     nIntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     klIntegerInput
On entry: ${k}_{l}$, the number of subdiagonals within the band of $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
6:     kuIntegerInput
On entry: ${k}_{u}$, the number of superdiagonals within the band of $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
7:     ab[$\mathit{dim}$]const doubleInput
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 $m$ by $n$ band 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 ${\mathbf{ab}}\left[\left(j-1\right)×{\mathbf{pdab}}+{\mathbf{ku}}+i-j\right]$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${A}_{ij}$ is stored as ${\mathbf{ab}}\left[\left(i-1\right)×{\mathbf{pdab}}+{\mathbf{kl}}+j-i\right]$.
8:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array ab.
Constraint: ${\mathbf{pdab}}\ge {\mathbf{kl}}+{\mathbf{ku}}+1$.
9:     rdouble *Output
On exit: the value of the norm specified by norm.
10:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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: ${\mathbf{n}}\ge 0$.
NE_INT_3
On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$, ${\mathbf{kl}}=〈\mathit{\text{value}}〉$, ${\mathbf{ku}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdab}}\ge {\mathbf{kl}}+{\mathbf{ku}}+1$.

## 7  Accuracy

The algorithms used take care to avoid unnecessary over/underflow in conformance with the BLAS standard (see Section 2.7 of The BLAS Technical Forum Standard (2001)).

None.

## 9  Example

Calculates the various norms of a $6$ by $4$ banded matrix with two subdiagonals and one superdiagonal.

### 9.1  Program Text

Program Text (f16rbce.c)

### 9.2  Program Data

Program Data (f16rbce.d)

### 9.3  Program Results

Program Results (f16rbce.r)