naginterfaces.library.blast.dgb_​norm

naginterfaces.library.blast.dgb_norm(inorm, m, kl, ku, ab)

dgb_norm calculates the value of the $1$-norm, the $\infty$-norm, the Frobenius norm or the maximum absolute value of the elements of a real $m\times n$ band matrix stored in banded form.

For full information please refer to the NAG Library document for f16rb

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f16/f16rbf.html

Parameters

inormint

Specifies the value to be returned.

$inorm=171$

The $1$-norm.

$inorm=173$

The $2$-norm of a row or column vector.

$inorm=174$

The Frobenius (or Euclidean) norm.

$inorm=175$

The $\infty$-norm.

$inorm=177$

The value ${max}_{i,j}\left(\left|a_{ij}\right|\right)$ (not a norm).

mint

$m$, the number of rows of the matrix $A$. If $m\le 0$ on input, dgb_norm returns $0$.

klint

$k_l$, the number of subdiagonals within the band of $A$. If $kl<0$ on input, dgb_norm returns $0$.

kuint

$k_u$, the number of superdiagonals within the band of $A$. If $ku<0$ on input, dgb_norm returns $0$.

abfloat, array-like, shape $(kl+ku+1,n)$

The $m\times n$ band matrix $A$.

Returns

nrmfloat

The value of the required norm of the matrix supplied in $ab$.

Notes

Given a real $m\times n$ banded matrix, $A$, dgb_norm calculates one of the values given by

${\parallel A\parallel }_1={max}_j\left({\sum }_{i=1}^m\left|a_{ij}\right|\right)$	(the $1$-norm of $A$),
${\parallel A\parallel }_{\infty }={max}_i\left({\sum }_{j=1}^n\left|a_{ij}\right|\right)$	(the $\infty$-norm of $A$),
${\parallel A\parallel }_F={\left({\sum }_{i=1}^m{\sum }_{j=1}^n{\left|a_{ij}\right|}^2\right)}^{1/2}$	(the Frobenius norm of $A$), or
${max}_{i,j}\left(\left|a_{ij}\right|\right)$	(the maximum absolute element value of $A$).

If $m$ or $n$ is $1$ then additionally dgb_norm can calculate the value ${\parallel A\parallel }_2=\sqrt{\sum a_i^2}$ (the $2$-norm of $A$).

References

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