NAG Library Function Document

nag_zher2k (f16zrc)

void	nag_zher2k (Nag_OrderType order, Nag_UploType uplo, Nag_TransType trans, Integer n, Integer k, Complex alpha, const Complex a[], Integer pda, const Complex b[], Integer pdb, double beta, Complex c[], Integer pdc, NagError *fail)

3 Description

nag_zher2k (f16zrc) performs one of the Hermitian rank-

2 k

update operations

C \leftarrow α A B^{H} + \bar{α} A^{H} + β C or C \leftarrow \bar{α} A^{H} B + α B^{H} A + β C,

where

A

and

B

are complex matrices,

C

is an

n

n

complex Hermitian matrix,

α

is a complex scalar, and

β

is a real scalar.

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.

$trans = Nag_NoTrans$: $C \leftarrow α B^{H} A + \bar{α} B A^{H} + β C$ .
$trans = Nag_ConjTrans$: $C \leftarrow \bar{α} A^{H} B + α B^{H} A + β C$ .

Constraint:

trans = Nag_NoTrans

Nag_ConjTrans

4: n – IntegerInput

On entry:

n

, the order of the matrix

C

; the number of rows of

A

and

B

trans = Nag_NoTrans

, or the number of columns of

A

and

B

otherwise.

Constraint:

n \geq 0

5: k – IntegerInput

On entry:

k

, the number of columns of

A

and

B

trans = Nag_NoTrans

, or the number of rows of

A

and

B

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: b[ $\dim$ ] – const ComplexInput

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

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

order = Nag_ColMajor

B_{i j}

is stored in

b [(j - 1) \times pdb + i - 1]

order = Nag_RowMajor

B_{i j}

is stored in

b [(i - 1) \times pdb + j - 1]

On entry: the matrix

B

;

B

n

k

trans = Nag_NoTrans

, or

k

n

otherwise.

10: pdb – IntegerInput

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

Constraints:

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

11: beta – doubleInput

On entry: the scalar

β

12: 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

Hermitian 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

. The imaginary parts of the diagonal elements are set to zero.

13: 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)

14: 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, $trans = ⟨value⟩$ , $k = ⟨value⟩$ , $pdb = ⟨value⟩$ .
Constraint: if $trans = Nag_NoTrans$ , $pdb \geq \max (1, k)$ .
On entry, $trans = ⟨value⟩$ , $k = ⟨value⟩$ , $pdb = ⟨value⟩$ .
Constraint: if $trans = Nag_Trans$ or $Nag_ConjTrans$ , $pdb \geq \max (1, k)$ .
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 $Nag_ConjTrans$ , $pda \geq \max (1, n)$ .
On entry, $trans = ⟨value⟩$ , $n = ⟨value⟩$ , $pdb = ⟨value⟩$ .
Constraint: if $trans = Nag_NoTrans$ , $pdb \geq \max (1, n)$ .
On entry, $trans = ⟨value⟩$ , $n = ⟨value⟩$ , $pdb = ⟨value⟩$ .
Constraint: if $trans = Nag_Trans$ or $Nag_ConjTrans$ , $pdb \geq \max (1, n)$ .
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-

2 k

update of complex Hermitian

4

4

matrix

C

using

4

2

matrices

A

and

B

C = - C + (- 0.5 + 0.5 i) A B^{T} + (- 0.5 - 0.5 i) B A^{T}

, where

C = (\begin{array}{r} 4.78 + 0.00 i & 2.00 + 0.30 i & 2.89 + 1.34 i & - 1.89 - 1.15 i \\ 2.00 - 0.30 i & - 4.11 + 0.00 i & 2.36 + 4.25 i & 0.04 + 3.69 i \\ 2.89 - 1.34 i & 2.36 - 4.25 i & 4.15 + 0.00 i & - 0.02 - 0.46 i \\ - 1.89 + 1.15 i & 0.04 - 3.69 i & - 0.02 + 0.46 i & 0.33 + 0.00 i \end{array}),

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 \\ - 0.3 - 1.9 i & 2.1 - 1.1 i \end{array})

and

B = (\begin{array}{r} - 2.4 + 1.4 i & 0.6 - 2.9 i \\ - 0.2 - 2.9 i & - 1.5 + 0.1 i \\ 3.5 + 0.8 i & 2.2 + 3.7 i \end{array}) .

NAG Library Function Documentnag_zher2k (f16zrc)

+− 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_zher2k (f16zrc)