# NAG FL Interfacef16ubf (zgb_​norm)

## 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 $m$ by $n$ 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 $m$ by $n$ band matrix, $A$, f16ubf calculates one of the values given by
 ${‖A‖}_{1}=\underset{j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\sum _{i=1}^{m}\left|{a}_{ij}\right|$ (the $1$-norm of $A$), ${‖A‖}_{\infty }=\underset{i}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\sum _{j=1}^{n}\left|{a}_{ij}\right|$ (the $\infty$-norm of $A$), ${‖A‖}_{F}={\left(\sum _{i=1}^{m}\sum _{j=1}^{n}{\left|{a}_{ij}\right|}^{2}\right)}^{1/2}$ (the Frobenius norm of $A$),  or $\underset{i,j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\left|{a}_{ij}\right|$ (the maximum absolute element value of $A$).
If $m$ or $n$ is $1$ then additionally f16ubf can calculate the value ${‖A‖}_{2}=\sqrt{\sum {\left|{a}_{i}\right|}^{2}}$ (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 $\underset{i,j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\left|{a}_{ij}\right|$ (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 ${\mathbf{kl}}\le 0$ on input, f16ubf returns $0$.
5: $\mathbf{ku}$Integer Input
On entry: ${k}_{u}$, the number of superdiagonals within the band of $A$. If ${\mathbf{ku}}\le 0$ 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 $m$ by $n$ 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
 $abku+1+i-jj for ​max1,j-ku≤i≤minm,j+kl.$
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 $6$ by $4$ 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)