F06 Chapter Contents
F06 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF06ULF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

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.

## 2  Specification

 FUNCTION F06ULF ( NORM, UPLO, DIAG, N, K, AB, LDAB, WORK)
 REAL (KIND=nag_wp) F06ULF
 INTEGER N, K, LDAB REAL (KIND=nag_wp) WORK(*) COMPLEX (KIND=nag_wp) AB(LDAB,*) CHARACTER(1) NORM, UPLO, DIAG

None.

None.

## 5  Parameters

1:     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:     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:     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:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
When ${\mathbf{N}}=0$, F06ULF returns zero.
Constraint: ${\mathbf{N}}\ge 0$.
5:     K – INTEGERInput
On entry: $k$, the number of subdiagonals or superdiagonals of the matrix $A$.
Constraint: ${\mathbf{K}}\ge 0$.
6:     AB(LDAB,$*$) – 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:     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:     WORK($*$) – 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.