NAG Library Routine Document

F16RBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F16RBF 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

n

band matrix, stored in banded form.

It can also be used to compute the value of the

2

-norm of a row

n

-vector or a column

m

-vector.

2 Specification

FUNCTION F16RBF (

INORM, M, N, KL, KU, AB, LDAB)

REAL (KIND=nag_wp) F16RBF

INTEGER	INORM, M, N, KL, KU, LDAB
REAL (KIND=nag_wp)	AB(LDAB,*)

3 Description

Given a real

m

n

banded matrix,

A

, F16RBF calculates one of the values given by

${‖A‖}_{1} = \max_{j} \sum_{i = 1}^{m} \|a_{i j}\|$	(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 http://www.netlib.org/blas/blast-forum/blas-report.pdf

5 Parameters

1: INORM – INTEGERInput

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.
$INORM = 175$ (NAG_INF_NORM): The $\infty$ -norm.
$INORM = 174$ (NAG_FROBENIUS_NORM): The Frobenius (or Euclidean) 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 – INTEGERInput

On entry:

m

, the number of rows of the matrix

A

. If

M \leq 0

on input, F16RBF returns

0

3: N – INTEGERInput

On entry:

n

, the number of columns of the matrix

A

. If

N \leq 0

on input, F16RBF returns

0

4: KL – INTEGERInput

On entry:

k_{l}

, the number of subdiagonals within the band of

A

. If

KL \leq 0

on input, F16RBF returns

0

5: KU – INTEGERInput

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) arrayInput

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

\max (1, N)

On entry: 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}) .

7: LDAB – INTEGERInput

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 Further Comments

None.

9 Example

Calculates the various norms of a

6

4

banded matrix with two subdiagonals and one superdiagonal.

NAG Library Routine DocumentF16RBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F16RBF