NAG Library Function Document

nag_zsy_norm (f16ufc)

+− 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

1 Purpose

nag_zsy_norm (f16ufc) calculates the value of the

1

-norm, the

\infty

-norm, the Frobenius norm or the maximum absolute value of the elements of a complex

n

n

symmetric matrix.

2 Specification

#include <nag.h>

#include <nagf16.h>

void	nag_zsy_norm (Nag_OrderType order, Nag_NormType norm, Nag_UploType uplo, Integer n, const Complex a[], Integer pda, double r, NagError fail)

3 Description

Given a complex

n

n

symmetric matrix,

A

, nag_zsy_norm (f16ufc) calculates one of the values given by

{‖A‖}_{1} = \max_{j} \sum_{i = 1}^{n} |a_{i j}|,

{‖A‖}_{\infty} = \max_{i} \sum_{j = 1}^{n} |a_{i j}|,

{‖A‖}_{F} = {(\sum_{i = 1}^{n} \sum_{j = 1}^{n} {|a_{i j}|}^{2})}^{1 / 2}

\max_{i, j} |a_{i j}| .

Note that, since

A

is symmetric,

{‖A‖}_{1} = {‖A‖}_{\infty}

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 Arguments

1: order – Nag_OrderTypeInput

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: norm – Nag_NormTypeInput

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: uplo – Nag_UploTypeInput

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

4: n – IntegerInput

On entry:

n

, the order of the matrix

A

n = 0

, then n is set to zero.

Constraint:

n \geq 0

5: a[ $\dim$ ] – const ComplexInput

Note: the dimension, dim, of the array a must be at least

\max (1, pda \times n)

On entry: the

n

n

symmetric matrix

A

order = Nag_ColMajor

A_{i j}

is stored in

a [(j - 1) \times pda + i - 1]

order = Nag_RowMajor

A_{i j}

is stored in

a [(i - 1) \times pda + j - 1]

uplo = Nag_Upper

, the upper triangular part of

A

must be stored and the elements of the array below the diagonal are not referenced.

uplo = Nag_Lower

, the lower triangular part of

A

must be stored and the elements of the array above the diagonal are not referenced.

6: pda – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) of the matrix

A

in the array a.

Constraint:

pda \geq \max (1, n)

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pda = ⟨value⟩$ , $n = ⟨value⟩$ .
Constraint: $pda \geq \max (1, n)$ .

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)).