f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_dormbr (f08kgc)

## 1  Purpose

nag_dormbr (f08kgc) multiplies an arbitrary real $m$ by $n$ matrix $C$ by one of the real orthogonal matrices $Q$ or $P$ which were determined by nag_dgebrd (f08kec) when reducing a real matrix to bidiagonal form.

## 2  Specification

 #include #include
 void nag_dormbr (Nag_OrderType order, Nag_VectType vect, Nag_SideType side, Nag_TransType trans, Integer m, Integer n, Integer k, const double a[], Integer pda, const double tau[], double c[], Integer pdc, NagError *fail)

## 3  Description

nag_dormbr (f08kgc) is intended to be used after a call to nag_dgebrd (f08kec), which reduces a real rectangular matrix $A$ to bidiagonal form $B$ by an orthogonal transformation: $A=QB{P}^{\mathrm{T}}$. nag_dgebrd (f08kec) represents the matrices $Q$ and ${P}^{\mathrm{T}}$ as products of elementary reflectors.
This function may be used to form one of the matrix products
 $QC , QTC , CQ , CQT , PC , PTC , CP ​ or ​ CPT ,$
overwriting the result on $C$ (which may be any real rectangular matrix).

## 4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Arguments

Note: in the descriptions below, $\mathit{r}$ denotes the order of $Q$ or ${P}^{\mathrm{T}}$: if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, $\mathit{r}={\mathbf{m}}$ and if ${\mathbf{side}}=\mathrm{Nag_RightSide}$, $\mathit{r}={\mathbf{n}}$.
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 2.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{vect}$Nag_VectTypeInput
On entry: indicates whether $Q$ or ${Q}^{\mathrm{T}}$ or $P$ or ${P}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{vect}}=\mathrm{Nag_ApplyQ}$
$Q$ or ${Q}^{\mathrm{T}}$ is applied to $C$.
${\mathbf{vect}}=\mathrm{Nag_ApplyP}$
$P$ or ${P}^{\mathrm{T}}$ is applied to $C$.
Constraint: ${\mathbf{vect}}=\mathrm{Nag_ApplyQ}$ or $\mathrm{Nag_ApplyP}$.
3:    $\mathbf{side}$Nag_SideTypeInput
On entry: indicates how $Q$ or ${Q}^{\mathrm{T}}$ or $P$ or ${P}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{side}}=\mathrm{Nag_LeftSide}$
$Q$ or ${Q}^{\mathrm{T}}$ or $P$ or ${P}^{\mathrm{T}}$ is applied to $C$ from the left.
${\mathbf{side}}=\mathrm{Nag_RightSide}$
$Q$ or ${Q}^{\mathrm{T}}$ or $P$ or ${P}^{\mathrm{T}}$ is applied to $C$ from the right.
Constraint: ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_RightSide}$.
4:    $\mathbf{trans}$Nag_TransTypeInput
On entry: indicates whether $Q$ or $P$ or ${Q}^{\mathrm{T}}$ or ${P}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$Q$ or $P$ is applied to $C$.
${\mathbf{trans}}=\mathrm{Nag_Trans}$
${Q}^{\mathrm{T}}$ or ${P}^{\mathrm{T}}$ is applied to $C$.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ or $\mathrm{Nag_Trans}$.
5:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of rows of the matrix $C$.
Constraint: ${\mathbf{m}}\ge 0$.
6:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of columns of the matrix $C$.
Constraint: ${\mathbf{n}}\ge 0$.
7:    $\mathbf{k}$IntegerInput
On entry: if ${\mathbf{vect}}=\mathrm{Nag_ApplyQ}$, the number of columns in the original matrix $A$.
If ${\mathbf{vect}}=\mathrm{Nag_ApplyP}$, the number of rows in the original matrix $A$.
Constraint: ${\mathbf{k}}\ge 0$.
8:    $\mathbf{a}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array a must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathit{r},{\mathbf{k}}\right)\right)$ when ${\mathbf{vect}}=\mathrm{Nag_ApplyQ}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathit{r}×{\mathbf{pda}}\right)$ when ${\mathbf{vect}}=\mathrm{Nag_ApplyQ}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×\mathit{r}\right)$ when ${\mathbf{vect}}=\mathrm{Nag_ApplyP}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathit{r},{\mathbf{k}}\right)×{\mathbf{pda}}\right)$ when ${\mathbf{vect}}=\mathrm{Nag_ApplyP}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_dgebrd (f08kec).
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 ${\mathbf{vect}}=\mathrm{Nag_ApplyQ}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathit{r}\right)$;
• if ${\mathbf{vect}}=\mathrm{Nag_ApplyP}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathit{r},{\mathbf{k}}\right)\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{vect}}=\mathrm{Nag_ApplyQ}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathit{r},{\mathbf{k}}\right)\right)$;
• if ${\mathbf{vect}}=\mathrm{Nag_ApplyP}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathit{r}\right)$.
10:  $\mathbf{tau}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathit{r},{\mathbf{k}}\right)\right)$.
On entry: further details of the elementary reflectors, as returned by nag_dgebrd (f08kec) in its argument tauq if ${\mathbf{vect}}=\mathrm{Nag_ApplyQ}$, or in its argument taup if ${\mathbf{vect}}=\mathrm{Nag_ApplyP}$.
11:  $\mathbf{c}\left[\mathit{dim}\right]$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $C$ is stored in
• ${\mathbf{c}}\left[\left(j-1\right)×{\mathbf{pdc}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{c}}\left[\left(i-1\right)×{\mathbf{pdc}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the matrix $C$.
On exit: c is overwritten by $QC$ or ${Q}^{\mathrm{T}}C$ or $CQ$ or ${C}^{\mathrm{T}}Q$ or $PC$ or ${P}^{\mathrm{T}}C$ or $CP$ or ${C}^{\mathrm{T}}P$ as specified by vect, side and trans.
12:  $\mathbf{pdc}$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
13:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_ENUM_INT_2
On entry, ${\mathbf{vect}}=〈\mathit{\text{value}}〉$, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{vect}}=\mathrm{Nag_ApplyQ}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathit{r},{\mathbf{k}}\right)\right)$;
if ${\mathbf{vect}}=\mathrm{Nag_ApplyP}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathit{r}\right)$.
On entry, ${\mathbf{vect}}=〈\mathit{\text{value}}〉$, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{vect}}=\mathrm{Nag_ApplyQ}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathit{r}\right)$;
if ${\mathbf{vect}}=\mathrm{Nag_ApplyP}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathit{r},{\mathbf{k}}\right)\right)$.
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: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}>0$.
NE_INT_2
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
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 computed result differs from the exact result by a matrix $E$ such that
 $E2 = Oε C2 ,$
where $\epsilon$ is the machine precision.

## 8  Parallelism and Performance

nag_dormbr (f08kgc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dormbr (f08kgc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
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.

The total number of floating-point operations is approximately
• if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ and $m\ge k$, $2nk\left(2m-k\right)$;
• if ${\mathbf{side}}=\mathrm{Nag_RightSide}$ and $n\ge k$, $2mk\left(2n-k\right)$;
• if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ and $m, $2{m}^{2}n$;
• if ${\mathbf{side}}=\mathrm{Nag_RightSide}$ and $n, $2m{n}^{2}$,
where $k$ is the value of the argument k.
The complex analogue of this function is nag_zunmbr (f08kuc).

## 10  Example

For this function two examples are presented. Both illustrate how the reduction to bidiagonal form of a matrix $A$ may be preceded by a $QR$ or $LQ$ factorization of $A$.
In the first example, $m>n$, and
 $A = -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 .$
The function first performs a $QR$ factorization of $A$ as $A={Q}_{a}R$ and then reduces the factor $R$ to bidiagonal form $B$: $R={Q}_{b}B{P}^{\mathrm{T}}$. Finally it forms ${Q}_{a}$ and calls nag_dormbr (f08kgc) to form $Q={Q}_{a}{Q}_{b}$.
In the second example, $m, and
 $A = -5.42 3.28 -3.68 0.27 2.06 0.46 -1.65 -3.40 -3.20 -1.03 -4.06 -0.01 -0.37 2.35 1.90 4.31 -1.76 1.13 -3.15 -0.11 1.99 -2.70 0.26 4.50 .$
The function first performs an $LQ$ factorization of $A$ as $A=L{P}_{a}^{\mathrm{T}}$ and then reduces the factor $L$ to bidiagonal form $B$: $L=QB{P}_{b}^{\mathrm{T}}$. Finally it forms ${P}_{b}^{\mathrm{T}}$ and calls nag_dormbr (f08kgc) to form ${P}^{\mathrm{T}}={P}_{b}^{\mathrm{T}}{P}_{a}^{\mathrm{T}}$.

### 10.1  Program Text

Program Text (f08kgce.c)

### 10.2  Program Data

Program Data (f08kgce.d)

### 10.3  Program Results

Program Results (f08kgce.r)