# NAG Library Routine Document

## 1Purpose

f06ulf 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 complex $n$ by $n$ triangular band matrix.

## 2Specification

Fortran Interface
 Function f06ulf ( norm, uplo, diag, n, k, ab, ldab, work)
 Real (Kind=nag_wp) :: f06ulf Integer, Intent (In) :: n, k, ldab Real (Kind=nag_wp), Intent (Inout) :: work(*) Complex (Kind=nag_wp), Intent (In) :: ab(ldab,*) Character (1), Intent (In) :: norm, uplo, diag
#include <nagmk26.h>
 double f06ulf_ (const char *norm, const char *uplo, const char *diag, const Integer *n, const Integer *k, const Complex ab[], const Integer *ldab, 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 triangular.
${\mathbf{uplo}}=\text{'U'}$
$A$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A$ is lower triangular.
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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
When ${\mathbf{n}}=0$, f06ulf returns zero.
Constraint: ${\mathbf{n}}\ge 0$.
5:     $\mathbf{k}$ – IntegerInput
On entry: $k$, the number of subdiagonals or superdiagonals of the matrix $A$.
Constraint: ${\mathbf{k}}\ge 0$.
6:     $\mathbf{ab}\left({\mathbf{ldab}},*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab must be at least ${\mathbf{n}}$.
On entry: the $n$ by $n$ triangular band matrix $A$
The matrix is stored in rows $1$ to $k+1$, more precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left(k+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-k\right)\le i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+k\right)\text{.}$
If ${\mathbf{diag}}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced.
7:     $\mathbf{ldab}$ – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f06ulf is called.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{k}}+1$.
8:     $\mathbf{work}\left(*\right)$ – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{norm}}=\text{'I'}$, and at least $1$ otherwise.

None.

Not applicable.

## 8Parallelism and Performance

f06ulf is not threaded in any implementation.