# NAG Library Routine Document

## 1Purpose

f08fff (dorgtr) generates the real orthogonal matrix $Q$, which was determined by f08fef (dsytrd) when reducing a symmetric matrix to tridiagonal form.

## 2Specification

Fortran Interface
 Subroutine f08fff ( uplo, n, a, lda, tau, work, info)
 Integer, Intent (In) :: n, lda, lwork Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: tau(*) Real (Kind=nag_wp), Intent (Inout) :: a(lda,*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: uplo
#include <nagmk26.h>
 void f08fff_ (const char *uplo, const Integer *n, double a[], const Integer *lda, const double tau[], double work[], const Integer *lwork, Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name dorgtr.

## 3Description

f08fff (dorgtr) is intended to be used after a call to f08fef (dsytrd), which reduces a real symmetric matrix $A$ to symmetric tridiagonal form $T$ by an orthogonal similarity transformation: $A=QT{Q}^{\mathrm{T}}$. f08fef (dsytrd) 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.
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 f08fef (dsytrd).
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:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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 f08fef (dsytrd).
On exit: the $n$ by $n$ orthogonal matrix $Q$.
4:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08fff (dorgtr) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
5:     $\mathbf{tau}\left(*\right)$ – 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 f08fef (dsytrd).
6:     $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – 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.
7:     $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08fff (dorgtr) 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, ${\mathbf{lwork}}\ge \left({\mathbf{n}}-1\right)×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$ or ${\mathbf{lwork}}=-1$.
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

f08fff (dorgtr) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08fff (dorgtr) 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 f08ftf (zungtr).

## 10Example

This example computes all the eigenvalues and eigenvectors of the matrix $A$, where
 $A = 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 .$
Here $A$ is symmetric and must first be reduced to tridiagonal form by f08fef (dsytrd). The program then calls f08fff (dorgtr) to form $Q$, and passes this matrix to f08jef (dsteqr) which computes the eigenvalues and eigenvectors of $A$.

### 10.1Program Text

Program Text (f08fffe.f90)

### 10.2Program Data

Program Data (f08fffe.d)

### 10.3Program Results

Program Results (f08fffe.r)