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

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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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)).

8

Parallelism and Performance

nag_zgemm (f16zac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

None.

10

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 Document

nag_zgemm (f16zac)

▸▿ 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

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