NAG Library Routine Document

F06UKF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     $\mathrm{NORM}$ – 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 ${\displaystyle \underset{i,j}{\max }}\left|a_{ij}\right|$ (not a norm).

Constraint: $\mathbf{NORM}=\text{'1'}$, $\text{'O'}$, $\text{'I'}$, $\text{'F'}$, $\text{'E'}$ or $\text{'M'}$.

2:     $\mathrm{UPLO}$ – 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:     $\mathrm{DIAG}$ – 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:     $\mathrm{N}$ – INTEGERInput

On entry: $n$, the order of the matrix $A$.

When $\mathbf{N}=0$, F06UKF returns zero.

Constraint: $\mathbf{N}\ge 0$.

5:     $\mathrm{AP}\left(*\right)$ – COMPLEX (KIND=nag_wp) arrayInput

Note: the dimension of the array AP must be at least $\mathbf{N}\times \left(\mathbf{N}+1\right)/2$.

On entry: the $n$ by $n$ triangular matrix $A$, packed by columns.

More precisely,

• if $\mathbf{UPLO}=\text{'U'}$, the upper triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{AP}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if $\mathbf{UPLO}=\text{'L'}$, the lower triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{AP}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.

If $\mathbf{DIAG}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced; the same storage scheme is used whether $\mathbf{DIAG}=\text{'N'}$ or ‘U’.

6:     $\mathrm{WORK}\left(*\right)$ – REAL (KIND=nag_wp) arrayWorkspace

Note: the dimension of the array WORK must be at least $\max \left(1,\mathbf{N}\right)$ if $\mathbf{NORM}=\text{'I'}$, and at least $1$ otherwise.

6  Error Indicators and Warnings

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example