# NAG Library Routine Document

## 1Purpose

f06rjf returns, via the function name, 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 $n$ trapezoidal matrix (triangular if $m=n$).

## 2Specification

Fortran Interface
 Function f06rjf ( norm, uplo, diag, m, n, a, lda, work)
 Real (Kind=nag_wp) :: f06rjf Integer, Intent (In) :: m, n, lda Real (Kind=nag_wp), Intent (In) :: a(lda,*) Real (Kind=nag_wp), Intent (Inout) :: work(*) Character (1), Intent (In) :: norm, uplo, diag
#include nagmk26.h
 double f06rjf_ ( const char *norm, const char *uplo, const char *diag, const Integer *m, const Integer *n, const double a[], const Integer *lda, double work[], const Charlen length_norm, const Charlen length_uplo, const Charlen length_diag)

None.
None.

## 5Arguments

1:     $\mathbf{norm}$ – Character(1)Input
On entry: specifies the value to be returned.
${\mathbf{norm}}=\text{'1'}$ or $\text{'O'}$
The $1$-norm.
${\mathbf{norm}}=\text{'I'}$
The $\infty$-norm.
${\mathbf{norm}}=\text{'F'}$ or $\text{'E'}$
The Frobenius (or Euclidean) norm.
${\mathbf{norm}}=\text{'M'}$
The value $\underset{i,j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\left|{a}_{ij}\right|$ (not a norm).
Constraint: ${\mathbf{norm}}=\text{'1'}$, $\text{'O'}$, $\text{'I'}$, $\text{'F'}$, $\text{'E'}$ or $\text{'M'}$.
2:     $\mathbf{uplo}$ – Character(1)Input
On entry: specifies whether $A$ is upper or lower trapezoidal.
${\mathbf{uplo}}=\text{'U'}$
$A$ is upper trapezoidal.
${\mathbf{uplo}}=\text{'L'}$
$A$ is lower trapezoidal.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3:     $\mathbf{diag}$ – Character(1)Input
On entry: specifies whether $A$ has nonunit or unit diagonal elements.
${\mathbf{diag}}=\text{'N'}$
The diagonal elements are stored explicitly.
${\mathbf{diag}}=\text{'U'}$
The diagonal elements are assumed to be $1$, and are not referenced.
Constraint: ${\mathbf{diag}}=\text{'N'}$ or $\text{'U'}$.
4:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
When ${\mathbf{m}}=0$, f06rjf is set to zero.
Constraint: ${\mathbf{m}}\ge 0$.
5:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrix $A$.
When ${\mathbf{n}}=0$, f06rjf is set to zero.
Constraint: ${\mathbf{n}}\ge 0$.
6:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array a must be at least ${\mathbf{n}}$.
On entry: the $m$ by $n$ trapezoidal matrix $A$.
• If ${\mathbf{uplo}}=\text{'U'}$, $A$ is upper trapezoidal and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{uplo}}=\text{'L'}$, $A$ is lower trapezoidal and the elements of the array above the diagonal are not referenced.
• If ${\mathbf{diag}}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced.
7:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f06rjf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
8:     $\mathbf{work}\left(*\right)$ – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}}\right)$ if ${\mathbf{norm}}=\text{'I'}$, and at least $1$ otherwise.

None.

Not applicable.

## 8Parallelism and Performance

f06rjf is not threaded in any implementation.