${‖A‖}_{1} = \max_{j} \sum_{i = 1}^{m} \|a_{i j}\|$	(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 nag_blast_zgb_norm (f16ub) can calculate the value

{‖A‖}_{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 http://www.netlib.org/blas/blast-forum/blas-report.pdf

Parameters

Compulsory Input Parameters

1: $job$ – string (length ≥ 1)

Specifies the value to be returned.

$job ='O'$: The $1$ -norm.
$job ='T'$: The $2$ -norm of a row or column vector.
$job ='I'$: The $\infty$ -norm.
$job ='F'$: The Frobenius (or Euclidean) norm.
$job ='M'$: The value $\max_{i, j} |a_{i j}|$ (not a norm).

Constraints:

$job ='O'$ , $'T'$ , $'I'$ , $'F'$ or $'M'$ ;
if $job ='T'$ , $m = 1$ or $n = 1$ .

2: $m$ – int64int32nag_int scalar

m

, the number of rows of the matrix

A

. If

m \leq 0

on input, nag_blast_zgb_norm (f16ub) returns

0

3: $kl$ – int64int32nag_int scalar

k_{l}

, the number of subdiagonals within the band of

A

. If

kl \leq 0

on input, nag_blast_zgb_norm (f16ub) returns

0

4: $ku$ – int64int32nag_int scalar

k_{u}

, the number of superdiagonals within the band of

A

. If

ku \leq 0

on input, nag_blast_zgb_norm (f16ub) returns

0

5: $ab (ldab :)$ – complex array

The first dimension of the array ab must be at least

kl + ku + 1

The second dimension of the array ab must be at least

\max (1, n)

The

m

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}) .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the second dimension of the array ab.
$n$ , the number of columns of the matrix $A$ . If $n \leq 0$ on input, nag_blast_zgb_norm (f16ub) returns $0$ .

Output Parameters

1: $result$ – double scalar: The result of the function.

Error Indicators and Warnings

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

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)).

Further Comments

None.

Example

Reads in a

6

4

banded complex matrix

A

with two subdiagonals and one superdiagonal, and prints the four norms of

A

Open in the MATLAB editor: f16ub_example

function f16ub_example


fprintf('f16ub example results\n\n');

kl = int64(2);
ku = int64(1);
m  = int64(6);
a = [ 1+i, 1+2i, 0,    0;
      2+i, 2+2i, 2+3i, 0;
      3+i, 3+2i, 3+3i, 3+4i;
      0,   4+2i, 4+3i, 4+4i;
      0,   0,    5+3i, 5+4i;
      0,   0,    0,    6+4i];
ab = complex(zeros(4, 6));
% Convert a to packed storage
[a, ab, ifail] = f01zd( ...
			'p', kl, ku, a, ab);

fprintf('\nNorms of banded matrix ab:\n\n');

r_one = f16ub('o', m, kl, ku, ab);
fprintf('One norm           = %9.4f\n', r_one);

r_inf = f16ub('i', m, kl, ku, ab);
fprintf('Infinity norm      = %9.4f\n', r_inf);

r_fro = f16ub('f', m, kl, ku, ab);
fprintf('Frobenious norm    = %9.4f\n', r_fro);

r_max = f16ub('m', m, kl, ku, ab);
fprintf('Maximum norm       = %9.4f\n', r_max);

f16ub example results


Norms of banded matrix ab:

One norm           =   24.2711
Infinity norm      =   16.0105
Frobenious norm    =   17.4069
Maximum norm       =    7.2111

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox