NAG Library Routine Document

F06WNF (ZLANHF)

${‖A‖}_{1} = \max_{j} \sum_{i = 1}^{n} \|a_{i j}\|$	(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 http://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 Parameters

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 normal RFP representation of

A

or its conjugate transpose is stored.

$TRANSR ='N'$: The matrix $A$ is stored in normal RFP format.
$TRANSR ='C'$: The conjugate transpose of the RFP representation of the matrix $A$ is stored.

Constraint:

TRANSR ='N'

'C'

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: $N$ – INTEGERInput

On entry:

n

, the order of the matrix

A

When

N = 0

, F06WNF (ZLANHF) returns zero.

Constraint:

N \geq 0

5: $AR (N \times (N + 1) / 2)$ – COMPLEX (KIND=nag_wp) arrayInput

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

n

n

Hermitian 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) arrayWorkspace

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

Not applicable.

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.

NAG Library Routine DocumentF06WNF (ZLANHF)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Library Routine Document

F06WNF (ZLANHF)