# NAG Library Routine Document

## 1Purpose

f08gff (dopgtr) generates the real orthogonal matrix $Q$, which was determined by f08gef (dsptrd) when reducing a symmetric matrix to tridiagonal form.

## 2Specification

Fortran Interface
 Subroutine f08gff ( uplo, n, ap, tau, q, ldq, work, info)
 Integer, Intent (In) :: n, ldq Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: ap(*), tau(*) Real (Kind=nag_wp), Intent (Inout) :: q(ldq,*) Real (Kind=nag_wp), Intent (Out) :: work(n-1) Character (1), Intent (In) :: uplo
#include nagmk26.h
 void f08gff_ ( const char *uplo, const Integer *n, const double ap[], const double tau[], double q[], const Integer *ldq, double work[], Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name dopgtr.

## 3Description

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.

## 4References

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

## 5Arguments

1:     $\mathbf{uplo}$ – Character(1)Input
On entry: this must be the same argument uplo as supplied to f08gef (dsptrd).
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $Q$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{ap}\left(*\right)$ – 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:     $\mathbf{tau}\left(*\right)$ – 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:     $\mathbf{q}\left({\mathbf{ldq}},*\right)$ – 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:     $\mathbf{ldq}$ – 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:     $\mathbf{work}\left({\mathbf{n}}-1\right)$ – Real (Kind=nag_wp) arrayWorkspace
8:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 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.

## 8Parallelism and Performance

f08gff (dopgtr) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08gff (dopgtr) 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is approximately $\frac{4}{3}{n}^{3}$.
The complex analogue of this routine is f08gtf (zupgtr).

## 10Example

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 $A$.

### 10.1Program Text

Program Text (f08gffe.f90)

### 10.2Program Data

Program Data (f08gffe.d)

### 10.3Program Results

Program Results (f08gffe.r)

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