f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_dopgtr (f08gfc)

## 1  Purpose

nag_dopgtr (f08gfc) generates the real orthogonal matrix $Q$, which was determined by nag_dsptrd (f08gec) when reducing a symmetric matrix to tridiagonal form.

## 2  Specification

 #include #include
 void nag_dopgtr (Nag_OrderType order, Nag_UploType uplo, Integer n, const double ap[], const double tau[], double q[], Integer pdq, NagError *fail)

## 3  Description

nag_dopgtr (f08gfc) is intended to be used after a call to nag_dsptrd (f08gec), which reduces a real symmetric matrix $A$ to symmetric tridiagonal form $T$ by an orthogonal similarity transformation: $A=QT{Q}^{\mathrm{T}}$. nag_dsptrd (f08gec) represents the orthogonal matrix $Q$ as a product of $n-1$ elementary reflectors.
This function 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  Arguments

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 2.3.1.3 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:    $\mathbf{uplo}$Nag_UploTypeInput
On entry: this must be the same argument uplo as supplied to nag_dsptrd (f08gec).
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $Q$.
Constraint: ${\mathbf{n}}\ge 0$.
4:    $\mathbf{ap}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, 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 nag_dsptrd (f08gec).
5:    $\mathbf{tau}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: further details of the elementary reflectors, as returned by nag_dsptrd (f08gec).
6:    $\mathbf{q}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array q must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdq}}×{\mathbf{n}}\right)$.
The $\left(i,j\right)$th element of the matrix $Q$ is stored in
• ${\mathbf{q}}\left[\left(j-1\right)×{\mathbf{pdq}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{q}}\left[\left(i-1\right)×{\mathbf{pdq}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $n$ by $n$ orthogonal matrix $Q$.
7:    $\mathbf{pdq}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraint: ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error 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_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pdq}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdq}}>0$.
NE_INT_2
On entry, ${\mathbf{pdq}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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.

## 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  Parallelism and Performance

nag_dopgtr (f08gfc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dopgtr (f08gfc) 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 is approximately $\frac{4}{3}{n}^{3}$.
The complex analogue of this function is nag_zupgtr (f08gtc).

## 10  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 nag_dsptrd (f08gec). The program then calls nag_dopgtr (f08gfc) to form $Q$, and passes this matrix to nag_dsteqr (f08jec) which computes the eigenvalues and eigenvectors of $A$.

### 10.1  Program Text

Program Text (f08gfce.c)

### 10.2  Program Data

Program Data (f08gfce.d)

### 10.3  Program Results

Program Results (f08gfce.r)