# NAG Library Function Document

## 1Purpose

nag_dorgbr (f08kfc) generates one of the real orthogonal matrices $Q$ or ${P}^{\mathrm{T}}$ which were determined by nag_dgebrd (f08kec) when reducing a real matrix to bidiagonal form.

## 2Specification

 #include #include
 void nag_dorgbr (Nag_OrderType order, Nag_VectType vect, Integer m, Integer n, Integer k, double a[], Integer pda, const double tau[], NagError *fail)

## 3Description

nag_dorgbr (f08kfc) 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 generate $Q$ or ${P}^{\mathrm{T}}$ explicitly as square matrices, or in some cases just the leading columns of $Q$ or the leading rows of ${P}^{\mathrm{T}}$.
The various possibilities are specified by the arguments vect, m, n and k. The appropriate values to cover the most likely cases are as follows (assuming that $A$ was an $m$ by $n$ matrix):
 1 To form the full $m$ by $m$ matrix $Q$: ```nag_dorgbr(order,Nag_FormQ,m,m,n,...) ``` (note that the array a must have at least $m$ columns). 2 If $m>n$, to form the $n$ leading columns of $Q$: ```nag_dorgbr(order,Nag_FormQ,m,n,n,...) ``` 3 To form the full $n$ by $n$ matrix ${P}^{\mathrm{T}}$: ```nag_dorgbr(order,Nag_FormP,n,n,m,...) ``` (note that the array a must have at least $n$ rows). 4 If $m, to form the $m$ leading rows of ${P}^{\mathrm{T}}$: ```nag_dorgbr(order,Nag_FormP,m,n,m,...) ```

## 4References

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

## 5Arguments

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 3.3.1.3 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 the orthogonal matrix $Q$ or ${P}^{\mathrm{T}}$ is generated.
${\mathbf{vect}}=\mathrm{Nag_FormQ}$
$Q$ is generated.
${\mathbf{vect}}=\mathrm{Nag_FormP}$
${P}^{\mathrm{T}}$ is generated.
Constraint: ${\mathbf{vect}}=\mathrm{Nag_FormQ}$ or $\mathrm{Nag_FormP}$.
3:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of rows of the orthogonal matrix $Q$ or ${P}^{\mathrm{T}}$ to be returned.
Constraint: ${\mathbf{m}}\ge 0$.
4:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of columns of the orthogonal matrix $Q$ or ${P}^{\mathrm{T}}$ to be returned.
Constraints:
• ${\mathbf{n}}\ge 0$;
• if ${\mathbf{vect}}=\mathrm{Nag_FormQ}$ and ${\mathbf{m}}>{\mathbf{k}}$, ${\mathbf{m}}\ge {\mathbf{n}}\ge {\mathbf{k}}$;
• if ${\mathbf{vect}}=\mathrm{Nag_FormQ}$ and ${\mathbf{m}}\le {\mathbf{k}}$, ${\mathbf{m}}={\mathbf{n}}$;
• if ${\mathbf{vect}}=\mathrm{Nag_FormP}$ and ${\mathbf{n}}>{\mathbf{k}}$, ${\mathbf{n}}\ge {\mathbf{m}}\ge {\mathbf{k}}$;
• if ${\mathbf{vect}}=\mathrm{Nag_FormP}$ and ${\mathbf{n}}\le {\mathbf{k}}$, ${\mathbf{n}}={\mathbf{m}}$.
5:    $\mathbf{k}$IntegerInput
On entry: if ${\mathbf{vect}}=\mathrm{Nag_FormQ}$, the number of columns in the original matrix $A$.
If ${\mathbf{vect}}=\mathrm{Nag_FormP}$, the number of rows in the original matrix $A$.
Constraint: ${\mathbf{k}}\ge 0$.
6:    $\mathbf{a}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array a must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pda}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_dgebrd (f08kec).
On exit: the orthogonal matrix $Q$ or ${P}^{\mathrm{T}}$, or the leading rows or columns thereof, as specified by vect, m and n.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$.
7:    $\mathbf{pda}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8:    $\mathbf{tau}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array tau must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{k}}\right)\right)$ when ${\mathbf{vect}}=\mathrm{Nag_FormQ}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},{\mathbf{k}}\right)\right)$ when ${\mathbf{vect}}=\mathrm{Nag_FormP}$.
On entry: further details of the elementary reflectors, as returned by nag_dgebrd (f08kec) in its argument tauq if ${\mathbf{vect}}=\mathrm{Nag_FormQ}$, or in its argument taup if ${\mathbf{vect}}=\mathrm{Nag_FormP}$.
9:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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_3
On entry, ${\mathbf{vect}}=〈\mathit{\text{value}}〉$, ${\mathbf{m}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ and ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$ and
if ${\mathbf{vect}}=\mathrm{Nag_FormQ}$ and ${\mathbf{m}}>{\mathbf{k}}$, ${\mathbf{m}}\ge {\mathbf{n}}\ge {\mathbf{k}}$;
if ${\mathbf{vect}}=\mathrm{Nag_FormQ}$ and ${\mathbf{m}}\le {\mathbf{k}}$, ${\mathbf{m}}={\mathbf{n}}$;
if ${\mathbf{vect}}=\mathrm{Nag_FormP}$ and ${\mathbf{n}}>{\mathbf{k}}$, ${\mathbf{n}}\ge {\mathbf{m}}\ge {\mathbf{k}}$;
if ${\mathbf{vect}}=\mathrm{Nag_FormP}$ and ${\mathbf{n}}\le {\mathbf{k}}$, ${\mathbf{n}}={\mathbf{m}}$.
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, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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.

## 7Accuracy

The computed matrix $Q$ differs from an exactly orthogonal matrix by a matrix $E$ such that
 $E2 = Oε ,$
where $\epsilon$ is the machine precision. A similar statement holds for the computed matrix ${P}^{\mathrm{T}}$.

## 8Parallelism and Performance

nag_dorgbr (f08kfc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dorgbr (f08kfc) 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 for the cases listed in Section 3 are approximately as follows:
 1 To form the whole of $Q$: $\frac{4}{3}n\left(3{m}^{2}-3mn+{n}^{2}\right)$ if $m>n$,$\frac{4}{3}{m}^{3}$ if $m\le n$; 2 To form the $n$ leading columns of $Q$ when $m>n$: $\frac{2}{3}{n}^{2}\left(3m-n\right)$; 3 To form the whole of ${P}^{\mathrm{T}}$: $\frac{4}{3}{n}^{3}$ if $m\ge n$,$\frac{4}{3}m\left(3{n}^{2}-3mn+{m}^{2}\right)$ if $m; 4 To form the $m$ leading rows of ${P}^{\mathrm{T}}$ when $m: $\frac{2}{3}{m}^{2}\left(3n-m\right)$.
The complex analogue of this function is nag_zungbr (f08ktc).

## 10Example

For this function two examples are presented, both of which involve computing the singular value decomposition of a matrix $A$, where
 $A = -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50$
in the first example and
 $A = -5.42 3.28 -3.68 0.27 2.06 0.46 -1.65 -3.40 -3.20 -1.03 -4.06 -0.01 -0.37 2.35 1.90 4.31 -1.76 1.13 -3.15 -0.11 1.99 -2.70 0.26 4.50$
in the second. $A$ must first be reduced to tridiagonal form by nag_dgebrd (f08kec). The program then calls nag_dorgbr (f08kfc) twice to form $Q$ and ${P}^{\mathrm{T}}$, and passes these matrices to nag_dbdsqr (f08mec), which computes the singular value decomposition of $A$.

### 10.1Program Text

Program Text (f08kfce.c)

### 10.2Program Data

Program Data (f08kfce.d)

### 10.3Program Results

Program Results (f08kfce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017