Real (Kind=nag_wp)	::	f06waf
Integer, Intent (In)	::	n
Real (Kind=nag_wp), Intent (In)	::	a(n*(n+1)/2)
Real (Kind=nag_wp), Intent (Inout)	::	work(*)
Character (1), Intent (In)	::	norm, transr, uplo

C Header Interface

#include <nag.h>

double

f06waf_ (const char *norm, const char *transr, const char *uplo, const Integer *n, const double a[], double work[], const Charlen length_norm, const Charlen length_transr, const Charlen length_uplo)

The routine may be called by the names f06waf, nagf_blas_dlansf or its LAPACK name dlansf.

3 Description

Given a real

n \times n

symmetric matrix,

A

, f06waf calculates one of the values given by

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

A

is stored in compact form using the RFP format. The RFP storage format is described in Section 3.3.3 in the F07 Chapter Introduction.

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

Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

5 Arguments

1: $norm$ – Character(1) Input

On entry: specifies the value to be returned.

$norm ='1'$ or $'O'$: The $1$ -norm.
$norm ='I'$: The $\infty$ -norm.
$norm ='F'$ or $'E'$: The Frobenius (or Euclidean) norm.
$norm ='M'$: The value $\max_{i, j} | a_{i j} |$ (not a norm).

Constraint:

norm ='1'

'O'

'I'

'F'

'E'

'M'

2: $transr$ – Character(1) Input

On entry: specifies whether the RFP representation of

A

is normal or transposed.

$transr ='N'$: The matrix $A$ is stored in normal RFP format.
$transr ='T'$: The matrix $A$ is stored in transposed RFP format.

Constraint:

transr ='N'

'T'

3: $uplo$ – Character(1) Input

On entry: specifies whether the upper or lower triangular part of

A

is stored.

$uplo ='U'$: The upper triangular part of $A$ is stored.
$uplo ='L'$: The lower triangular part of $A$ is stored.

Constraint:

uplo ='U'

'L'

4: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

When

n = 0

, f06waf returns zero.

Constraint:

n \geq 0

5: $a (n \times (n + 1) / 2)$ – Real (Kind=nag_wp) array Input

On entry: the upper or lower triangular part (as specified by uplo) of the

n \times n

symmetric matrix

A

, in either normal or transposed RFP format (as specified by transr). The storage format is described in detail in Section 3.3.3 in the F07 Chapter Introduction.

6: $work (*)$ – Real (Kind=nag_wp) array Workspace

Note: the dimension of the array work must be at least

\max (1, n)

norm ='1'

'O'

'I'

, and at least

1

otherwise.

6 Error Indicators and Warnings

None.

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

Background information to multithreading can be found in the Multithreading documentation.

f06waf is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example reads in the lower triangular part of a symmetric matrix, converts this to RFP format, then calculates the norm of the matrix for each of the available norm types.

f06wa: FL CL CPP AD PY MB

NAG FL Interfacef06waf (dlansf)

▸▿ 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
f06waf (dlansf)