f16 Chapter Contents
f16 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_dgemm (f16yac)

## 1  Purpose

nag_dgemm (f16yac) performs matrix-matrix multiplication for a real general matrix.

## 2  Specification

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

## 3  Description

nag_dgemm (f16yac) performs one of the matrix-matrix operations
 $C←αAB+βC, C←αATB+βC, C←αABT+βC or C←αATBT+βC,$
where $A$, $B$ and $C$ are real matrices, and $\alpha$ and $\beta$ are real scalars; $C$ is always $m$ by $n$.

## 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:    $\mathbf{order}$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{transa}$Nag_TransTypeInput
On entry: specifies whether the operation involves $A$ or ${A}^{\mathrm{T}}$.
${\mathbf{transa}}=\mathrm{Nag_NoTrans}$
It involves $A$.
${\mathbf{transa}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$
It involves ${A}^{\mathrm{T}}$.
Constraint: ${\mathbf{transa}}=\mathrm{Nag_NoTrans}$, $\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
3:    $\mathbf{transb}$Nag_TransTypeInput
On entry: specifies whether the operation involves $B$ or ${B}^{\mathrm{T}}$.
${\mathbf{transb}}=\mathrm{Nag_NoTrans}$
It involves $B$.
${\mathbf{transb}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$
It involves ${B}^{\mathrm{T}}$.
Constraint: ${\mathbf{transb}}=\mathrm{Nag_NoTrans}$, $\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
4:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of rows of the matrix $C$; the number of rows of $A$ if ${\mathbf{transa}}=\mathrm{Nag_NoTrans}$, or the number of columns of $A$ if ${\mathbf{transa}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
Constraint: ${\mathbf{m}}\ge 0$.
5:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of columns of the matrix $C$; the number of columns of $B$ if ${\mathbf{transb}}=\mathrm{Nag_NoTrans}$, or the number of rows of $B$ if ${\mathbf{transb}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
Constraint: ${\mathbf{n}}\ge 0$.
6:    $\mathbf{k}$IntegerInput
On entry: $k$, the number of columns of $A$ if ${\mathbf{transa}}=\mathrm{Nag_NoTrans}$, or the number of rows of $A$ if ${\mathbf{transa}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$; the number of rows of $B$ if ${\mathbf{transb}}=\mathrm{Nag_NoTrans}$, or the number of columns of $B$ if ${\mathbf{transb}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
Constraint: ${\mathbf{k}}\ge 0$.
7:    $\mathbf{alpha}$doubleInput
On entry: the scalar $\alpha$.
8:    $\mathbf{a}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array a must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{k}}\right)$ when ${\mathbf{transa}}=\mathrm{Nag_NoTrans}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pda}}\right)$ when ${\mathbf{transa}}=\mathrm{Nag_NoTrans}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{m}}\right)$ when ${\mathbf{transa}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}×{\mathbf{pda}}\right)$ when ${\mathbf{transa}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${A}_{ij}$ is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${A}_{ij}$ is stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$.
On entry: the matrix $A$; $A$ is $m$ by $k$ if ${\mathbf{transa}}=\mathrm{Nag_NoTrans}$, or $k$ by $m$ if ${\mathbf{transa}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
9:    $\mathbf{pda}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{transa}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{transa}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{transa}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$;
• if ${\mathbf{transa}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
10:  $\mathbf{b}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array b must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{n}}\right)$ when ${\mathbf{transb}}=\mathrm{Nag_NoTrans}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}×{\mathbf{pdb}}\right)$ when ${\mathbf{transb}}=\mathrm{Nag_NoTrans}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{k}}\right)$ when ${\mathbf{transb}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdb}}\right)$ when ${\mathbf{transb}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${B}_{ij}$ is stored in ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${B}_{ij}$ is stored in ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$.
On entry: the matrix $B$; $B$ is $k$ by $n$ if ${\mathbf{transb}}=\mathrm{Nag_NoTrans}$, or $n$ by $k$ if ${\mathbf{transb}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
11:  $\mathbf{pdb}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{transb}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$;
• if ${\mathbf{transb}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{transb}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{transb}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
12:  $\mathbf{beta}$doubleInput
On entry: the scalar $\beta$.
13:  $\mathbf{c}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array c must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdc}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pdc}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${C}_{ij}$ is stored in ${\mathbf{c}}\left[\left(j-1\right)×{\mathbf{pdc}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${C}_{ij}$ is stored in ${\mathbf{c}}\left[\left(i-1\right)×{\mathbf{pdc}}+j-1\right]$.
On entry: the $m$ by $n$ matrix $C$.
If ${\mathbf{beta}}=0$, c need not be set.
On exit: the updated matrix $C$.
14:  $\mathbf{pdc}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
15:  $\mathbf{fail}$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.
See Section 3.2.1.2 in the Essential Introduction for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_ENUM_INT_2
On entry, ${\mathbf{transa}}=〈\mathit{\text{value}}〉$, ${\mathbf{k}}=〈\mathit{\text{value}}〉$, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{transa}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry, ${\mathbf{transa}}=〈\mathit{\text{value}}〉$, ${\mathbf{m}}=〈\mathit{\text{value}}〉$, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{transa}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{transa}}=〈\mathit{\text{value}}〉$, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{transa}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry, ${\mathbf{transa}}=〈\mathit{\text{value}}〉$, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{transa}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{transb}}=〈\mathit{\text{value}}〉$, ${\mathbf{k}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{transb}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry, ${\mathbf{transb}}=〈\mathit{\text{value}}〉$, ${\mathbf{k}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{transb}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry, ${\mathbf{transb}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{transb}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{transb}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{transb}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INT
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\ge 0$.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction 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)).

Not applicable.

None.

## 10  Example

This example computes the matrix-matrix product
 $C=αAB+βC$
where
 $A = 1.0 2.0 3.0 3.0 4.0 5.0 5.0 6.0 -1.0 ,$
 $B = 1.0 2.0 -2.0 1.0 3.0 -1.0 ,$
 $C = -2.0 1.0 1.0 3.0 2.0 -1.0 ,$
 $α=1.5 and β=1.0 .$

### 10.1  Program Text

Program Text (f16yace.c)

### 10.2  Program Data

Program Data (f16yace.d)

### 10.3  Program Results

Program Results (f16yace.r)