F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08KFF (DORGBR)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08KFF (DORGBR) generates one of the real orthogonal matrices $Q$ or ${P}^{\mathrm{T}}$ which were determined by F08KEF (DGEBRD) when reducing a real matrix to bidiagonal form.

## 2  Specification

 SUBROUTINE F08KFF ( VECT, M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
 INTEGER M, N, K, LDA, LWORK, INFO REAL (KIND=nag_wp) A(LDA,*), TAU(*), WORK(max(1,LWORK)) CHARACTER(1) VECT
The routine may be called by its LAPACK name dorgbr.

## 3  Description

F08KFF (DORGBR) is intended to be used after a call to F08KEF (DGEBRD), which reduces a real rectangular matrix $A$ to bidiagonal form $B$ by an orthogonal transformation: $A=QB{P}^{\mathrm{T}}$. F08KEF (DGEBRD) represents the matrices $Q$ and ${P}^{\mathrm{T}}$ as products of elementary reflectors.
This routine 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 parameters 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 $Q$:
```CALL DORGBR('Q',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 $Q$:
```CALL DORGBR('Q',m,n,n,...)
```
3. To form the full $n$ by $n$ matrix ${P}^{\mathrm{T}}$:
```CALL DORGBR('P',n,n,m,...)
```
(note that the array A must have at least $n$ rows).
4. If $m, to form the $m$ leading rows of ${P}^{\mathrm{T}}$:
```CALL DORGBR('P',m,n,m,...)
```
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Parameters

1:     VECT – CHARACTER(1)Input
On entry: indicates whether the orthogonal matrix $Q$ or ${P}^{\mathrm{T}}$ is generated.
${\mathbf{VECT}}=\text{'Q'}$
$Q$ is generated.
${\mathbf{VECT}}=\text{'P'}$
${P}^{\mathrm{T}}$ is generated.
Constraint: ${\mathbf{VECT}}=\text{'Q'}$ or $\text{'P'}$.
2:     M – INTEGERInput
On entry: $m$, the number of rows of the orthogonal matrix $Q$ or ${P}^{\mathrm{T}}$ to be returned.
Constraint: ${\mathbf{M}}\ge 0$.
3:     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 ${\mathbf{VECT}}=\text{'Q'}$ and ${\mathbf{M}}>{\mathbf{K}}$, ${\mathbf{M}}\ge {\mathbf{N}}\ge {\mathbf{K}}$;
• if ${\mathbf{VECT}}=\text{'Q'}$ and ${\mathbf{M}}\le {\mathbf{K}}$, ${\mathbf{M}}={\mathbf{N}}$;
• if ${\mathbf{VECT}}=\text{'P'}$ and ${\mathbf{N}}>{\mathbf{K}}$, ${\mathbf{N}}\ge {\mathbf{M}}\ge {\mathbf{K}}$;
• if ${\mathbf{VECT}}=\text{'P'}$ and ${\mathbf{N}}\le {\mathbf{K}}$, ${\mathbf{N}}={\mathbf{M}}$.
4:     K – INTEGERInput
On entry: if ${\mathbf{VECT}}=\text{'Q'}$, the number of columns in the original matrix $A$.
If ${\mathbf{VECT}}=\text{'P'}$, the number of rows in the original matrix $A$.
Constraint: ${\mathbf{K}}\ge 0$.
5:     A(LDA,$*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: details of the vectors which define the elementary reflectors, as returned by F08KEF (DGEBRD).
On exit: the orthogonal matrix $Q$ or ${P}^{\mathrm{T}}$, or the leading rows or columns thereof, as specified by VECT, M and N.
6:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08KFF (DORGBR) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
7:     TAU($*$) – REAL (KIND=nag_wp) arrayInput
Note: the dimension of the array TAU must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{M}},{\mathbf{K}}\right)\right)$ if ${\mathbf{VECT}}=\text{'Q'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{N}},{\mathbf{K}}\right)\right)$ if ${\mathbf{VECT}}=\text{'P'}$.
On entry: further details of the elementary reflectors, as returned by F08KEF (DGEBRD) in its parameter TAUQ if ${\mathbf{VECT}}=\text{'Q'}$, or in its parameter TAUP if ${\mathbf{VECT}}=\text{'P'}$.
8:     WORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)$) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, ${\mathbf{WORK}}\left(1\right)$ contains the minimum value of LWORK required for optimal performance.
9:     LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08KFF (DORGBR) is called.
If ${\mathbf{LWORK}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued.
Suggested value: for optimal performance, ${\mathbf{LWORK}}\ge \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{M}},{\mathbf{N}}\right)×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{M}},{\mathbf{N}}\right)\right)$ or ${\mathbf{LWORK}}=-1$.
10:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7  Accuracy

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 ${P}^{\mathrm{T}}$.

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 routine is F08KTF (ZUNGBR).

## 9  Example

For this routine 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 F08KEF (DGEBRD). The program then calls F08KFF (DORGBR) twice to form $Q$ and ${P}^{\mathrm{T}}$, and passes these matrices to F08MEF (DBDSQR), which computes the singular value decomposition of $A$.

### 9.1  Program Text

Program Text (f08kffe.f90)

### 9.2  Program Data

Program Data (f08kffe.d)

### 9.3  Program Results

Program Results (f08kffe.r)