NAG Library Routine Document

F08GTF (ZUPGTR)

+− 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

F08GTF (ZUPGTR) generates the complex unitary matrix

Q

, which was determined by F08GSF (ZHPTRD) when reducing a Hermitian matrix to tridiagonal form.

2 Specification

SUBROUTINE F08GTF (

UPLO, N, AP, TAU, Q, LDQ, WORK, INFO)

INTEGER	N, LDQ, INFO
COMPLEX (KIND=nag_wp)	AP(), TAU(), Q(LDQ,*), WORK(N-1)
CHARACTER(1)	UPLO

The routine may be called by its LAPACK name zupgtr.

3 Description

F08GTF (ZUPGTR) is intended to be used after a call to F08GSF (ZHPTRD), which reduces a complex Hermitian matrix

A

to real symmetric tridiagonal form

T

by a unitary similarity transformation:

A = Q T Q^{H}

. F08GSF (ZHPTRD) represents the unitary 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 F08GSF (ZHPTRD).
Constraint: $UPLO ='U'$ or $'L'$ .
2: N – INTEGERInput: On entry: $n$ , the order of the matrix $Q$ .
Constraint: $N \geq 0$ .
3: AP( $*$ ) – COMPLEX (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 F08GSF (ZHPTRD).
4: TAU( $*$ ) – COMPLEX (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 F08GSF (ZHPTRD).
5: Q(LDQ, $*$ ) – COMPLEX (KIND=nag_wp) arrayOutput: Note: the second dimension of the array Q must be at least $\max (1, N)$ .
On exit: the $n$ by $n$ unitary matrix $Q$ .
6: LDQ – INTEGERInput: On entry: the first dimension of the array Q as declared in the (sub)program from which F08GTF (ZUPGTR) is called.
Constraint: $LDQ \geq \max (1, N)$ .
7: WORK( $N - 1$ ) – COMPLEX (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 unitary matrix by a matrix

E

such that

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision.

8 Further Comments

The total number of real floating point operations is approximately

\frac{16}{3} n^{3}

The real analogue of this routine is F08GFF (DOPGTR).

9 Example

This example computes all the eigenvalues and eigenvectors of the matrix

A

, where

A = (\begin{array}{r} - 2.28 + 0.00 i & 1.78 - 2.03 i & 2.26 + 0.10 i & - 0.12 + 2.53 i \\ 1.78 + 2.03 i & - 1.12 + 0.00 i & 0.01 + 0.43 i & - 1.07 + 0.86 i \\ 2.26 - 0.10 i & 0.01 - 0.43 i & - 0.37 + 0.00 i & 2.31 - 0.92 i \\ - 0.12 - 2.53 i & - 1.07 - 0.86 i & 2.31 + 0.92 i & - 0.73 + 0.00 i \end{array}),

using packed storage. Here

A

is Hermitian and must first be reduced to tridiagonal form by F08GSF (ZHPTRD). The program then calls F08GTF (ZUPGTR) to form

Q

, and passes this matrix to F08JSF (ZSTEQR) which computes the eigenvalues and eigenvectors of

A

NAG Library Routine DocumentF08GTF (ZUPGTR)