${‖ A ‖}_{1} = \max_{j} \sum_{i = 1}^{m} \| a_{i j} \|$	(the $1$ -norm of $A$ ),
${‖ A ‖}_{\infty} = \max_{i} \sum_{j = 1}^{n} \| a_{i j} \|$	(the $\infty$ -norm of $A$ ),
${‖ A ‖}_{F} = {(\sum_{i = 1}^{m} \sum_{j = 1}^{n} {\| a_{i j} \|}^{2})}^{1 / 2}$	(the Frobenius norm of $A$ ), or
$\max_{i, j} \| a_{i j} \|$	(the maximum absolute element value of $A$ ).

m

n

1

then additionally f16rbf can calculate the value

{‖ A ‖}_{2} = \sqrt{\sum a_{i}^{2}}

(the

2

-norm of

A

4 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

5 Arguments

1: $inorm$ – 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.

$inorm = 171$ (NAG_ONE_NORM): The $1$ -norm.
$inorm = 173$ (NAG_TWO_NORM): The $2$ -norm of a row or column vector.
$inorm = 174$ (NAG_FROBENIUS_NORM): The Frobenius (or Euclidean) norm.
$inorm = 175$ (NAG_INF_NORM): The $\infty$ -norm.
$inorm = 177$ (NAG_MAX_NORM): The value $\max_{i, j} | a_{i j} |$ (not a norm).

Constraints:

$inorm = 171$ , $173$ , $174$ , $175$ or $177$ ;
if $inorm = 173$ , $m = 1$ or $n = 1$ .

2: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

. If

m \leq 0

on input, f16rbf returns

0

3: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

A

. If

n \leq 0

on input, f16rbf returns

0

4: $kl$ – Integer Input

On entry:

k_{l}

, the number of subdiagonals within the band of

A

. If

kl \leq 0

on input, f16rbf returns

0

5: $ku$ – Integer Input

On entry:

k_{u}

, the number of superdiagonals within the band of

A

. If

ku \leq 0

on input, f16rbf returns

0

6: $ab (ldab, *)$ – Real (Kind=nag_wp) array Input

Note: the second dimension of the array ab must be at least

\max (1, n)

On entry: the

m \times n

band matrix

A

The matrix is stored in rows

1

k_{l} + k_{u} + 1

, more precisely, the element

A_{i j}

must be stored in

ab (k_{u} + 1 + i - j, j) for ​ \max (1, j - k_{u}) \leq i \leq \min (m, j + k_{l}) .

7: $ldab$ – Integer Input

On entry: the first dimension of the array ab as declared in the (sub)program from which f16rbf is called.

Constraint:

ldab \geq kl + ku + 1

6 Error Indicators and Warnings

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

7 Accuracy

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

8 Parallelism and Performance

f16rbf is not threaded in any implementation.

9 Further Comments

None.

10 Example

Calculates the various norms of a

6 \times 4

banded matrix with two subdiagonals and one superdiagonal.

f16rb: FL CL CPP AD

NAG FL Interfacef16rbf (dgb_​norm)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
f16rbf (dgb_norm)