${‖ A ‖}_{1} = \max_{j} \sum_{i = 1}^{n} \| a_{i j} \|$	(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 3.4.3 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: $order$ – Nag_OrderType Input

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $norm$ – Nag_NormType Input

On entry: specifies the value to be returned.

$norm = Nag_OneNorm$: The $1$ -norm.
$norm = Nag_InfNorm$: The $\infty$ -norm.
$norm = Nag_FrobeniusNorm$: The Frobenius (or Euclidean) norm.
$norm = Nag_MaxNorm$: The value $\max_{i, j} | a_{i j} |$ (not a norm).

Constraint:

norm = Nag_OneNorm

Nag_InfNorm

Nag_FrobeniusNorm

Nag_MaxNorm

3: $transr$ – Nag_RFP_Store Input

On entry: specifies whether the RFP representation of

A

is normal or transposed.

$transr = Nag_RFP_Normal$: The matrix $A$ is stored in normal RFP format.
$transr = Nag_RFP_Trans$: The matrix $A$ is stored in transposed RFP format.

Constraint:

transr = Nag_RFP_Normal

Nag_RFP_Trans

4: $uplo$ – Nag_UploType Input

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

A

is stored.

$uplo = Nag_Upper$: The upper triangular part of $A$ is stored.
$uplo = Nag_Lower$: The lower triangular part of $A$ is stored.

Constraint:

uplo = Nag_Upper

Nag_Lower

5: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

n = 0

, f16rkc returns immediately.

Constraint:

n \geq 0

6: $ar [n \times (n + 1) / 2]$ – const double 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 3.4.3 in the F07 Chapter Introduction.

7: $r$ – double * Output

On exit: the value of the norm specified by norm.

8: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

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.

f16rkc 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.

f16rk: FL CL CPP AD PY MB

NAG CL Interfacef16rkc (dsf_​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 CL Interface
f16rkc (dsf_norm)