NAG Library Routine Document

F08GFF (DOPGTR)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

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 = 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 Parameters

1: UPLO – CHARACTER(1)Input: On entry: this must be the same parameter 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

Errors or warnings detected by the routine:

$INFO < 0$: 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 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 = (\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 DocumentF08GFF (DOPGTR)