NAG Library Function Document

nag_zsyrk (f16zuc)

void	nag_zsyrk (Nag_OrderType order, Nag_UploType uplo, Nag_TransType trans, Integer n, Integer k, Complex alpha, const Complex a[], Integer pda, Complex beta, Complex c[], Integer pdc, NagError *fail)

3 Description

nag_zsyrk (f16zuc) performs one of the symmetric rank-

k

update operations

C \leftarrow α A A^{T} + β C or C \leftarrow α A^{T} A + β C,

where

A

is a complex matrix,

C

is an

n

n

complex symmetric matrix, and

α

and

β

are complex scalars.

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: uplo – Nag_UploTypeInput

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

C

is stored.

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

Constraint:

uplo = Nag_Upper

Nag_Lower

3: trans – Nag_TransTypeInput

On entry: specifies the operation to be performed.

$trans = Nag_NoTrans$: $C \leftarrow α A A^{T} + β C$ .
$trans = Nag_Trans$: $C \leftarrow α A^{T} A + β C$ .

Constraint:

trans = Nag_NoTrans

Nag_Trans

4: n – IntegerInput

On entry:

n

, the order of the matrix

C

; the number of rows of

A

trans = Nag_NoTrans

, or the number of columns of

A

otherwise.

Constraint:

n \geq 0

5: k – IntegerInput

On entry:

k

, the number of columns of

A

trans = Nag_NoTrans

, or the number of rows of

A

otherwise.

Constraint:

k \geq 0

6: alpha – ComplexInput

On entry: the scalar

α

7: a[ $\dim$ ] – const ComplexInput

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

$\max (1, pda \times k)$ when $trans = Nag_NoTrans$ and $order = Nag_ColMajor$ ;
$\max (1, n \times pda)$ when $trans = Nag_NoTrans$ and $order = Nag_RowMajor$ ;
$\max (1, pda \times n)$ when $trans = Nag_Trans$ or $Nag_ConjTrans$ and $order = Nag_ColMajor$ ;
$\max (1, k \times pda)$ when $trans = Nag_Trans$ or $Nag_ConjTrans$ and $order = Nag_RowMajor$ .

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]

On entry: the matrix

A

;

A

n

k

trans = Nag_NoTrans

, or

k

n

otherwise.

8: pda – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraints:

if order=Nag_ColMajor,
- if $trans = Nag_NoTrans$ , $pda \geq \max (1, n)$ ;
- if $trans = Nag_Trans$ or $Nag_ConjTrans$ , $pda \geq \max (1, k)$ ;
if order=Nag_RowMajor,
- if $trans = Nag_NoTrans$ , $pda \geq \max (1, k)$ ;
- if $trans = Nag_Trans$ or $Nag_ConjTrans$ , $pda \geq \max (1, n)$ .

9: beta – ComplexInput

On entry: the scalar

β

10: c[ $\dim$ ] – ComplexInput/Output

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

\max (1, pdc \times n)

On entry: the

n

n

symmetric matrix

C

order = Nag_ColMajor

C_{i j}

is stored in

c [(j - 1) \times pdc + i - 1]

order = Nag_RowMajor

C_{i j}

is stored in

c [(i - 1) \times pdc + j - 1]

uplo = Nag_Upper

, the upper triangular part of

C

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

uplo = Nag_Lower

, the lower triangular part of

C

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

On exit: the updated matrix

C

11: pdc – IntegerInput

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

C

in the array c.

Constraint:

pdc \geq \max (1, n)

12: 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_ENUM_INT_2: On entry, $trans = ⟨value⟩$ , $k = ⟨value⟩$ , $pda = ⟨value⟩$ .
Constraint: if $trans = Nag_NoTrans$ , $pda \geq \max (1, k)$ .
On entry, $trans = ⟨value⟩$ , $k = ⟨value⟩$ , $pda = ⟨value⟩$ .
Constraint: if $trans = Nag_Trans$ or $Nag_ConjTrans$ , $pda \geq \max (1, k)$ .
On entry, $trans = ⟨value⟩$ , $n = ⟨value⟩$ , $pda = ⟨value⟩$ .
Constraint: if $trans = Nag_NoTrans$ , $pda \geq \max (1, n)$ .
On entry, $trans = ⟨value⟩$ , $n = ⟨value⟩$ , $pda = ⟨value⟩$ .
Constraint: if $trans = Nag_Trans$ or $Nag_ConjTrans$ , $pda \geq \max (1, n)$ .
NE_INT: On entry, $k = ⟨value⟩$ .
Constraint: $k \geq 0$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pdc = ⟨value⟩$ , $n = ⟨value⟩$ .
Constraint: $pdc \geq \max (1, n)$ .
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.

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

Perform rank-

k

update of complex symmetric

4

4

matrix

C

using

4

2

matrix

A

(

k = 2

C = C - (1.0 - 1.0 i) A A^{T}

, where

C = (\begin{array}{r} 4.78 + 1.03 i & 2.00 - 0.30 i & 2.89 - 1.34 i & - 1.89 + 1.15 i \\ 2.00 - 0.30 i & - 4.11 - 2.30 i & 2.36 - 4.25 i & 0.04 - 3.69 i \\ 2.89 - 1.34 i & 2.36 - 4.25 i & 4.15 + 0.57 i & - 0.02 + 0.46 i \\ - 1.89 + 1.15 i & 0.04 - 3.69 i & - 0.02 + 0.46 i & 0.33 - 1.91 i \end{array})

and

A = (\begin{array}{r} 1.7 - 2.3 i & - 1.8 + 2.4 i \\ 2.9 - 2.1 i & 1.2 + 1.4 i \\ - 2.9 + 1.0 i & 0.6 + 0.8 i \\ 1.5 + 0.9 i & - 1.4 - 1.7 i \end{array}) .

NAG Library Function Documentnag_zsyrk (f16zuc)

+− 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 Library Function Document

nag_zsyrk (f16zuc)