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

C Header Interface

#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.

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 = Q T Q^{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

Arguments

1: $uplo$ – Character(1)Input: On entry: this must be the same argument uplo as supplied to f08gef (dsptrd).

Constraint: $uplo ='U'$ or $'L'$ .
2: $n$ – IntegerInput: On entry: $n$ , the order of the matrix $Q$ .

Constraint: $n \geq 0$ .
3: $ap (*)$ – Real (Kind=nag_wp) arrayInput: Note: the dimension of the array ap must be at least $\max (1, n \times (n + 1) / 2)$ .

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 $\max (1, n - 1)$ .

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 $\max (1, n)$ .

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: $ldq \geq \max (1, n)$ .
7: $work (n - 1)$ – Real (Kind=nag_wp) arrayWorkspace
8: $info$ – IntegerOutput: On exit: $info = 0$ unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$info < 0$: If $info = - i$ , 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

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision.

8

Parallelism 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.

9

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).

10

Example

This example computes all the eigenvalues and eigenvectors of the matrix

A

, where

A = (\begin{array}{r} 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 \end{array}),

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

NAG Library Routine Document

f08gff (dopgtr)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification