f16 Chapter Contents
f16 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_dtrmm (f16yfc)

## 1  Purpose

nag_dtrmm (f16yfc) performs matrix-matrix multiplication for a real triangular matrix.

## 2  Specification

 #include #include
 void nag_dtrmm (Nag_OrderType order, Nag_SideType side, Nag_UploType uplo, Nag_TransType trans, Nag_DiagType diag, Integer m, Integer n, double alpha, const double a[], Integer pda, double b[], Integer pdb, NagError *fail)

## 3  Description

nag_dtrmm (f16yfc) performs one of the matrix-matrix operations
 $B←αAB, B←αATB, B←αBA or B←αBAT,$
where $B$ is an $m$ by $n$ real matrix, $A$ is a real triangular matrix, and $\alpha$ 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:    $\mathbf{order}$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:    $\mathbf{side}$Nag_SideTypeInput
On entry: specifies whether $B$ is operated on from the left or the right.
${\mathbf{side}}=\mathrm{Nag_LeftSide}$
$B$ is pre-multiplied from the left.
${\mathbf{side}}=\mathrm{Nag_RightSide}$
$B$ is post-multiplied from the right.
Constraint: ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_RightSide}$.
3:    $\mathbf{uplo}$Nag_UploTypeInput
On entry: specifies whether $A$ is upper or lower triangular.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
$A$ is upper triangular.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
$A$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
4:    $\mathbf{trans}$Nag_TransTypeInput
On entry: specifies whether the operation involves $A$ or ${A}^{\mathrm{T}}$.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
It involves $A$.
${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$
It involves ${A}^{\mathrm{T}}$.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, $\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
5:    $\mathbf{diag}$Nag_DiagTypeInput
On entry: specifies whether $A$ has nonunit or unit diagonal elements.
${\mathbf{diag}}=\mathrm{Nag_NonUnitDiag}$
The diagonal elements are stored explicitly.
${\mathbf{diag}}=\mathrm{Nag_UnitDiag}$
The diagonal elements are assumed to be $1$ and are not referenced.
Constraint: ${\mathbf{diag}}=\mathrm{Nag_NonUnitDiag}$ or $\mathrm{Nag_UnitDiag}$.
6:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of rows of the matrix $B$; the order of $A$ if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$.
Constraint: ${\mathbf{m}}\ge 0$.
7:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of columns of the matrix $B$; the order of $A$ if ${\mathbf{side}}=\mathrm{Nag_RightSide}$.
Constraint: ${\mathbf{n}}\ge 0$.
8:    $\mathbf{alpha}$doubleInput
On entry: the scalar $\alpha$.
9:    $\mathbf{a}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array a must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{m}}\right)$ when ${\mathbf{side}}=\mathrm{Nag_LeftSide}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$ when ${\mathbf{side}}=\mathrm{Nag_RightSide}$.
On entry: the triangular matrix $A$; $A$ is $m$ by $m$ if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, or $n$ by $n$ if ${\mathbf{side}}=\mathrm{Nag_RightSide}$.
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 ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$.
If ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, $A$ is upper triangular and the elements of the array corresponding to the lower triangular part of $A$ are not referenced.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, $A$ is lower triangular and the elements of the array corresponding to the upper triangular part of $A$ are not referenced.
If ${\mathbf{diag}}=\mathrm{Nag_UnitDiag}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced.
10:  $\mathbf{pda}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array a.
Constraints:
• if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{side}}=\mathrm{Nag_RightSide}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
11:  $\mathbf{b}\left[\mathit{dim}\right]$doubleInput/Output
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 ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pdb}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
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 $m$ by $n$ matrix $B$.
If ${\mathbf{alpha}}=0$, b need not be set.
On exit: the updated matrix $B$.
12:  $\mathbf{pdb}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
13:  $\mathbf{fail}$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, ${\mathbf{side}}=〈\mathit{\text{value}}〉$, ${\mathbf{m}}=〈\mathit{\text{value}}〉$, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{side}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{side}}=\mathrm{Nag_RightSide}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INT
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, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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

Premultiply real $4$ by $2$ matrix $B$ by lower triangular $4$ by $4$ matrix $A$, $B←AB$, where
 $A = 4.30 -3.96 -4.87 0.40 0.31 -8.02 -0.27 0.07 -5.95 0.12$
and
 $B = -3.0 -5.0 -1.0 1.0 2.0 -1.0 1.0 6.0 .$

### 10.1  Program Text

Program Text (f16yfce.c)

### 10.2  Program Data

Program Data (f16yfce.d)

### 10.3  Program Results

Program Results (f16yfce.r)