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

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 https://www.netlib.org/blas/blast-forum/blas-report.pdf

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: $uplo$ – 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

4: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

n = 0

, n is set to zero.

Constraint:

n \geq 0

5: $a [\dim]$ – const double Input

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

\max (1, pda \times n)

On entry: the

n \times 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: $pda$ – Integer Input

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 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_INT_2: On entry, $pda = ⟨ value ⟩$ , $n = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, n)$ .
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)).