F08GFF (DOPGTR) (PDF version)
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NAG Library Manual

# NAG Library Routine DocumentF08GFF (DOPGTR)

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

F08GFF (DOPGTR) generates the real orthogonal matrix $Q$, which was determined by F08GEF (DSPTRD) when reducing a symmetric matrix to tridiagonal form.

## 2  Specification

 SUBROUTINE F08GFF ( UPLO, N, AP, TAU, Q, LDQ, WORK, INFO)
 INTEGER N, LDQ, INFO REAL (KIND=nag_wp) AP(*), TAU(*), Q(LDQ,*), WORK(N-1) CHARACTER(1) UPLO
The routine may be called by its LAPACK name dopgtr.

## 3  Description

F08GFF (DOPGTR) is intended to be used after a call to F08GEF (DSPTRD), which reduces a real symmetric matrix $A$ to symmetric tridiagonal form $T$ by an orthogonal similarity transformation: $A=QT{Q}^{\mathrm{T}}$. F08GEF (DSPTRD) represents the orthogonal matrix $Q$ as a product of $n-1$ elementary reflectors.
This routine may be used to generate $Q$ explicitly as a square matrix.

## 4  References

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

## 5  Parameters

1:     UPLO – CHARACTER(1)Input
On entry: this must be the same parameter UPLO as supplied to F08GEF (DSPTRD).
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
2:     N – INTEGERInput
On entry: $n$, the order of the matrix $Q$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     AP($*$) – REAL (KIND=nag_wp) arrayInput
Note: the dimension of the array AP must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}×\left({\mathbf{N}}+1\right)/2\right)$.
On entry: details of the vectors which define the elementary reflectors, as returned by F08GEF (DSPTRD).
4:     TAU($*$) – REAL (KIND=nag_wp) arrayInput
Note: the dimension of the array TAU must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}-1\right)$.
On entry: further details of the elementary reflectors, as returned by F08GEF (DSPTRD).
5:     Q(LDQ,$*$) – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array Q must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On exit: the $n$ by $n$ orthogonal matrix $Q$.
6:     LDQ – INTEGERInput
On entry: the first dimension of the array Q as declared in the (sub)program from which F08GFF (DOPGTR) is called.
Constraint: ${\mathbf{LDQ}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
7:     WORK(${\mathbf{N}}-1$) – REAL (KIND=nag_wp) arrayWorkspace
8:     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.

## 8  Further Comments

The total number of floating point operations is approximately $\frac{4}{3}{n}^{3}$.
The complex analogue of this routine is F08GTF (ZUPGTR).

## 9  Example

This example computes all the eigenvalues and eigenvectors of the matrix $A$, where
 $A = 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 ,$
using packed storage. Here $A$ is symmetric and must first be reduced to tridiagonal form by F08GEF (DSPTRD). The program then calls F08GFF (DOPGTR) to form $Q$, and passes this matrix to F08JEF (DSTEQR) which computes the eigenvalues and eigenvectors of $A$.

### 9.1  Program Text

Program Text (f08gffe.f90)

### 9.2  Program Data

Program Data (f08gffe.d)

### 9.3  Program Results

Program Results (f08gffe.r)

F08GFF (DOPGTR) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual