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:     orderNag_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:     transaNag_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:     transbNag_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:     mIntegerInput
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:     nIntegerInput
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:     kIntegerInput
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: ${\mathbf{k}}\ge 0$.
On entry: the scalar $\alpha$.
8:     a[$\mathit{dim}$]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:     pdaIntegerInput
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:   b[$\mathit{dim}$]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:   pdbIntegerInput
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 $\mathrm{Nag_ConjTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry: the scalar $\beta$.
13:   c[$\mathit{dim}$]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:   pdcIntegerInput
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:   failNagError *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.
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.

## 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)