NAG FL Interfacef16ubf (zgb_​norm)

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1Purpose

f16ubf calculates the value of the $1$-norm, the $\infty$-norm, the Frobenius norm or the maximum absolute value of the elements of a complex $m×n$ band matrix stored in banded packed form.
It can also be used to compute the value of the $2$-norm of a row $n$-vector or a column $m$-vector.

2Specification

Fortran Interface
 Function f16ubf ( m, n, kl, ku, ab, ldab)
 Real (Kind=nag_wp) :: f16ubf Integer, Intent (In) :: inorm, m, n, kl, ku, ldab Complex (Kind=nag_wp), Intent (In) :: ab(ldab,*)
#include <nag.h>
 double f16ubf_ (const Integer *inorm, const Integer *m, const Integer *n, const Integer *kl, const Integer *ku, const Complex ab[], const Integer *ldab)
The routine may be called by the names f16ubf or nagf_blast_zgb_norm.

3Description

Given a complex $m×n$ band matrix, $A$, f16ubf calculates one of the values given by
 ${‖A‖}_{1}=\underset{j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\sum _{i=1}^{m}|{a}_{ij}|$ (the $1$-norm of $A$), ${‖A‖}_{\infty }=\underset{i}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\sum _{j=1}^{n}|{a}_{ij}|$ (the $\infty$-norm of $A$), ${‖A‖}_{F}={\left(\sum _{i=1}^{m}\sum _{j=1}^{n}{|{a}_{ij}|}^{2}\right)}^{1/2}$ (the Frobenius norm of $A$),  or $\underset{i,j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}|{a}_{ij}|$ (the maximum absolute element value of $A$).
If $m$ or $n$ is $1$ then additionally f16ubf can calculate the value ${‖A‖}_{2}=\sqrt{\sum {|{a}_{i}|}^{2}}$ (the $2$-norm of $A$).

4References

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

5Arguments

1: $\mathbf{inorm}$Integer Input
On entry: specifies the value to be returned. The integer codes shown below can be replaced by the equivalent named constants of the form NAG_?_NORM. These named constants are available via the nag_library module and are also used in the example program for clarity.
${\mathbf{inorm}}=171$ (NAG_ONE_NORM)
The $1$-norm.
${\mathbf{inorm}}=173$ (NAG_TWO_NORM)
The $2$-norm of a row or column vector.
${\mathbf{inorm}}=174$ (NAG_FROBENIUS_NORM)
The Frobenius (or Euclidean) norm.
${\mathbf{inorm}}=175$ (NAG_INF_NORM)
The $\infty$-norm.
${\mathbf{inorm}}=177$ (NAG_MAX_NORM)
The value $\underset{i,j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}|{a}_{ij}|$ (not a norm).
Constraints:
• ${\mathbf{inorm}}=171$, $173$, $174$, $175$ or $177$;
• if ${\mathbf{inorm}}=173$, ${\mathbf{m}}=1$ or ${\mathbf{n}}=1$.
2: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $A$. If ${\mathbf{m}}\le 0$ on input, f16ubf returns $0$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $A$. If ${\mathbf{n}}\le 0$ on input, f16ubf returns $0$.
4: $\mathbf{kl}$Integer Input
On entry: ${k}_{l}$, the number of subdiagonals within the band of $A$. If ${\mathbf{kl}}<0$ on input, f16ubf returns $0$.
5: $\mathbf{ku}$Integer Input
On entry: ${k}_{u}$, the number of superdiagonals within the band of $A$. If ${\mathbf{ku}}<0$ on input, f16ubf returns $0$.
6: $\mathbf{ab}\left({\mathbf{ldab}},*\right)$Complex (Kind=nag_wp) array Input
Note: the second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $m×n$ band matrix $A$.
The matrix is stored in rows $1$ to ${k}_{l}+{k}_{u}+1$, more precisely, the element ${A}_{ij}$ must be stored in
 $ab(ku+1+i-j,j) for ​max(1,j-ku)≤i≤min(m,j+kl).$
7: $\mathbf{ldab}$Integer Input
On entry: the first dimension of the array ab as declared in the (sub)program from which f16ubf is called.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{kl}}+{\mathbf{ku}}+1$.

6Error Indicators and Warnings

If any constraint on an input parameter is violated, an error message is printed and program execution is terminated.

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

f16ubf is not threaded in any implementation.

None.

10Example

Reads in a $6×4$ banded complex matrix $A$ with two subdiagonals and one superdiagonal, and prints the four norms of $A$.

10.1Program Text

Program Text (f16ubfe.f90)

10.2Program Data

Program Data (f16ubfe.d)

10.3Program Results

Program Results (f16ubfe.r)