F08GTF (ZUPGTR) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08GTF (ZUPGTR)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

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=QTQH. 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
Constraint: UPLO='U' or 'L'.
2:     N – INTEGERInput
On entry: n, the order of the matrix Q.
Constraint: N0.
3:     AP(*) – COMPLEX (KIND=nag_wp) arrayInput
Note: the dimension of the array AP must be at least max1,N×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 max1,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 max1,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: LDQmax1,N.
7:     WORK(N-1) – COMPLEX (KIND=nag_wp) arrayWorkspace
8:     INFO – INTEGEROutput

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
E2 = Oε ,
where ε is the machine precision.

8  Further Comments

The total number of real floating point operations is approximately 163n3.
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 = -2.28+0.00i 1.78-2.03i 2.26+0.10i -0.12+2.53i 1.78+2.03i -1.12+0.00i 0.01+0.43i -1.07+0.86i 2.26-0.10i 0.01-0.43i -0.37+0.00i 2.31-0.92i -0.12-2.53i -1.07-0.86i 2.31+0.92i -0.73+0.00i ,
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.

9.1  Program Text

Program Text (f08gtfe.f90)

9.2  Program Data

Program Data (f08gtfe.d)

9.3  Program Results

Program Results (f08gtfe.r)


F08GTF (ZUPGTR) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2011