NAG Library Function Document

nag_zgemm (f16zac)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_zgemm (f16zac) performs matrix-matrix multiplication for a complex general matrix.

2 Specification

#include <nag.h>

#include <nagf16.h>

void	nag_zgemm (Nag_OrderType order, Nag_TransType transa, Nag_TransType transb, Integer m, Integer n, Integer k, Complex alpha, const Complex a[], Integer pda, const Complex b[], Integer pdb, Complex beta, Complex c[], Integer pdc, NagError *fail)

3 Description

nag_zgemm (f16zac) performs one of the matrix-matrix operations

\begin{array}{l} C \leftarrow α A B + β C, & C \leftarrow α A^{T} B + β C, & C \leftarrow α A^{H} B + β C, \\ C \leftarrow α A B^{T} + β C, & C \leftarrow α A^{T} B^{T} + β C, & C \leftarrow α A^{H} B^{T} + β C, \\ C \leftarrow α A B^{H} + β C, & C \leftarrow α A^{T} B^{H} + β C or & C \leftarrow α A^{H} B^{H} + β C, \end{array}

where

A

B

and

C

are complex matrices, and

α

and

β

are complex scalars;

C

is always

m

n

4 References

The BLAS Technical Forum Standard (2001) http://www.netlib.org/blas/blast-forum

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

or Nag_ColMajor.

2: transa – Nag_TransTypeInput

On entry: specifies whether the operation involves

A

A^{T}

A^{H}

$transa = Nag_NoTrans$: It involves $A$ .
$transa = Nag_Trans$: It involves $A^{T}$ .
$transa = Nag_ConjTrans$: It involves $A^{H}$ .

Constraint:

transa = Nag_NoTrans

Nag_Trans

Nag_ConjTrans

3: transb – Nag_TransTypeInput

On entry: specifies whether the operation involves

B

B^{T}

B^{H}

$transb = Nag_NoTrans$: It involves $B$ .
$transb = Nag_Trans$: It involves $B^{T}$ .
$transb = Nag_ConjTrans$: It involves $B^{H}$ .

Constraint:

transb = Nag_NoTrans

Nag_Trans

Nag_ConjTrans

4: m – IntegerInput

On entry:

m

, the number of rows of the matrix

C

; the number of rows of

A

transa = Nag_NoTrans

, or the number of columns of

A

transa = Nag_Trans

Nag_ConjTrans

Constraint:

m \geq 0

5: n – IntegerInput

On entry:

n

, the number of columns of the matrix

C

; the number of columns of

B

transb = Nag_NoTrans

, or the number of rows of

B

transb = Nag_Trans

Nag_ConjTrans

Constraint:

n \geq 0

6: k – IntegerInput

On entry:

k

, the number of columns of

A

transa = Nag_NoTrans

, or the number of rows of

A

transa = Nag_Trans

Nag_ConjTrans

; the number of rows of

B

transb = Nag_NoTrans

, or the number of columns of

B

transb = Nag_Trans

Nag_ConjTrans

Constraint:

k \geq 0

7: alpha – ComplexInput

On entry: the scalar

α

8: a[ $\dim$ ] – const ComplexInput

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

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

m

k

transa = Nag_NoTrans

, or

k

m

transa = Nag_Trans

Nag_ConjTrans

9: 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 $transa = Nag_NoTrans$ , $pda \geq \max (1, m)$ ;
- if $transa = Nag_Trans$ or $Nag_ConjTrans$ , $pda \geq \max (1, k)$ ;
if order=Nag_RowMajor,
- if $transa = Nag_NoTrans$ , $pda \geq \max (1, k)$ ;
- if $transa = Nag_Trans$ or $Nag_ConjTrans$ , $pda \geq \max (1, m)$ .

10: b[ $\dim$ ] – const ComplexInput

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

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

k

n

transb = Nag_NoTrans

, or

n

k

transb = Nag_Trans

Nag_ConjTrans

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

12: beta – ComplexInput

On entry: the scalar

β

13: c[ $\dim$ ] – ComplexInput/Output

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

$\max (1, pdc \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pdc)$ when $order = Nag_RowMajor$ .

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]

On entry: the

m

n

matrix

C

beta = 0

, c need not be set.

On exit: the updated matrix

C

14: pdc – IntegerInput

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

Constraints:

if $order = Nag_ColMajor$ , $pdc \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pdc \geq \max (1, n)$ .

15: 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, $transa = 〈value〉$ , $k = 〈value〉$ , $pda = 〈value〉$ .
Constraint: if $transa = Nag_NoTrans$ , $pda \geq \max (1, k)$ .; On entry, $transa = 〈value〉$ , $m = 〈value〉$ , $pda = 〈value〉$ .
Constraint: if $transa = Nag_Trans$ or $Nag_ConjTrans$ , $pda \geq \max (1, m)$ .; On entry, $transa = 〈value〉$ , $pda = 〈value〉$ , $k = 〈value〉$ .
Constraint: if $transa = Nag_Trans$ or $Nag_ConjTrans$ , $pda \geq \max (1, k)$ .; On entry, $transa = 〈value〉$ , $pda = 〈value〉$ , $m = 〈value〉$ .
Constraint: if $transa = Nag_NoTrans$ , $pda \geq \max (1, m)$ .; On entry, $transb = 〈value〉$ , $k = 〈value〉$ , $pdb = 〈value〉$ .
Constraint: if $transb = Nag_NoTrans$ , $pdb \geq \max (1, k)$ .; On entry, $transb = 〈value〉$ , $k = 〈value〉$ , $pdb = 〈value〉$ .
Constraint: if $transb = Nag_Trans$ or $Nag_ConjTrans$ , $pdb \geq \max (1, k)$ .; On entry, $transb = 〈value〉$ , $n = 〈value〉$ , $pdb = 〈value〉$ .
Constraint: if $transb = Nag_NoTrans$ , $pdb \geq \max (1, n)$ .; On entry, $transb = 〈value〉$ , $n = 〈value〉$ , $pdb = 〈value〉$ .
Constraint: if $transb = Nag_Trans$ or $Nag_ConjTrans$ , $pdb \geq \max (1, n)$ .
NE_INT: On entry, $k = 〈value〉$ .
Constraint: $k \geq 0$ .; On entry, $m = 〈value〉$ .
Constraint: $m \geq 0$ .; On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pdc = 〈value〉$ , $m = 〈value〉$ .
Constraint: $pdc \geq \max (1, m)$ .; On entry, $pdc = 〈value〉$ and $n = 〈value〉$ .
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 The BLAS Technical Forum Standard (2001)).

8 Further Comments

None.

9 Example

This example computes the matrix-matrix product

C = α A B + β C

where

A = (\begin{array}{r} 1.0 + 1.0 i & 1.0 + 2.0 i & - 2.0 + 3.0 i \\ 2.0 + 1.0 i & 2.0 + 2.0 i & 1.0 + 2.0 i \\ 3.0 + 1.0 i & 3.0 + 2.0 i & - 3.0 + 2.0 i \end{array}),

B = (\begin{array}{r} 1.0 - 1.0 i & 1.0 + 2.0 i \\ - 2.0 + 1.0 i & 2.0 - 2.0 i \\ 3.0 - 1.0 i & - 3.0 + 1.0 i \end{array}),

C = (\begin{array}{r} - 3.5 - 0.5 i & 1.5 + 2.0 i \\ - 4.5 + 1.5 i & - 2.0 + 3.5 i \\ - 5.5 + 3.5 i & 3.0 - 1.5 i \end{array}),

α = 1.0 + 0.0 i and β = 2.0 + 0.0 i .

NAG Library Function Documentnag_zgemm (f16zac)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_zgemm (f16zac)