NAG Library Function Document

1Purpose

nag_dsp_norm (f16rdc) calculates the value of the $1$-norm, the $\infty$-norm, the Frobenius norm or the maximum absolute value of the elements of a real $n$ by $n$ symmetric matrix, stored in packed form.

2Specification

 #include #include
 void nag_dsp_norm (Nag_OrderType order, Nag_NormType norm, Nag_UploType uplo, Integer n, const double ap[], double *r, NagError *fail)

3Description

Given a real $n$ by $n$ symmetric matrix, $A$, in packed storage, nag_dsp_norm (f16rdc) calculates one of the values given by
 $A1=maxj⁡∑i=1naij,$
 $A∞=maxi⁡∑j= 1naij,$
 $AF=∑i=1n∑j=1n aij21/2$
or
 $maxi,jaij.$
Note that, since $A$ is symmetric, ${‖A‖}_{1}={‖A‖}_{\infty }$.

4References

Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001) Basic Linear Algebra Subprograms Technical (BLAST) Forum Standard University of Tennessee, Knoxville, Tennessee http://www.netlib.org/blas/blast-forum/blas-report.pdf

5Arguments

1:    $\mathbf{order}$Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{norm}$Nag_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: ${\mathbf{norm}}=\mathrm{Nag_OneNorm}$, $\mathrm{Nag_InfNorm}$, $\mathrm{Nag_FrobeniusNorm}$ or $\mathrm{Nag_MaxNorm}$.
3:    $\mathbf{uplo}$Nag_UploTypeInput
On entry: specifies whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
The upper triangular part of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
4:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
If $n=0$, n is set to zero.
Constraint: ${\mathbf{n}}\ge 0$.
5:    $\mathbf{ap}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the $n$ by $n$ symmetric matrix $A$, packed by rows or columns.
The storage of elements ${A}_{ij}$ depends on the order and uplo arguments as follows:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(j-1\right)×j/2+i-1\right]$, for $i\le j$;
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-j\right)×\left(j-1\right)/2+i-1\right]$, for $i\ge j$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-i\right)×\left(i-1\right)/2+j-1\right]$, for $i\le j$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(i-1\right)×i/2+j-1\right]$, for $i\ge j$.
6:    $\mathbf{r}$double *Output
On exit: the value of the norm specified by norm.
7:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7Accuracy

The BLAS standard requires accurate implementations which avoid unnecessary over/underflow (see Section 2.7 of Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001)).

8Parallelism and Performance

nag_dsp_norm (f16rdc) is not threaded in any implementation.