NAG Library Routine Document

F08FFF (DORGTR)

+− 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

F08FFF (DORGTR) generates the real orthogonal matrix

Q

, which was determined by F08FEF (DSYTRD) when reducing a symmetric matrix to tridiagonal form.

2 Specification

SUBROUTINE F08FFF (

UPLO, N, A, LDA, TAU, WORK, LWORK, INFO)

INTEGER	N, LDA, LWORK, INFO
REAL (KIND=nag_wp)	A(LDA,), TAU(), WORK(max(1,LWORK))
CHARACTER(1)	UPLO

The routine may be called by its LAPACK name dorgtr.

3 Description

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 = Q T Q^{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.

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 F08FEF (DSYTRD).
Constraint: $UPLO ='U'$ or $'L'$ .
2: N – INTEGERInput: On entry: $n$ , the order of the matrix $Q$ .
Constraint: $N \geq 0$ .
3: A(LDA, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output: Note: the second dimension of the array A must be at least $\max (1, N)$ .
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: LDA – INTEGERInput: On entry: the first dimension of the array A as declared in the (sub)program from which F08FFF (DORGTR) is called.
Constraint: $LDA \geq \max (1, N)$ .
5: 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 F08FEF (DSYTRD).
6: WORK( $\max (1, LWORK)$ ) – REAL (KIND=nag_wp) arrayWorkspace: On exit: if $INFO = 0$ , $WORK (1)$ contains the minimum value of LWORK required for optimal performance.
7: LWORK – INTEGERInput: On entry: the dimension of the array WORK as declared in the (sub)program from which F08FFF (DORGTR) is called.
If $LWORK = - 1$ , 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, $LWORK \geq (N - 1) \times nb$ , where $nb$ is the optimal block size.
Constraint: $LWORK \geq \max (1, N - 1)$ or $LWORK = - 1$ .
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 F08FTF (ZUNGTR).

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}) .

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

NAG Library Routine DocumentF08FFF (DORGTR)