Integer, Intent (In)	::	n, ldq
Integer, Intent (Out)	::	info
Complex (Kind=nag_wp), Intent (In)	::	ap(), tau()
Complex (Kind=nag_wp), Intent (Inout)	::	q(ldq,*)
Complex (Kind=nag_wp), Intent (Out)	::	work(n-1)
Character (1), Intent (In)	::	uplo

C Header Interface

#include <nag.h>

void	f08gtf_ (const char uplo, const Integer n, const Complex ap[], const Complex tau[], Complex q[], const Integer ldq, Complex work[], Integer info, const Charlen length_uplo)

The routine may be called by the names f08gtf, nagf_lapackeig_zupgtr or its LAPACK name zupgtr.

3 Description

f08gtf is intended to be used after a call to f08gsf, which reduces a complex Hermitian matrix

A

to real symmetric tridiagonal form

T

by a unitary similarity transformation:

A = Q T Q^{H}

. f08gsf 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 Arguments

1: $uplo$ – Character(1) Input: On entry: this must be the same argument uplo as supplied to f08gsf.

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

Constraint: $n \geq 0$ .
3: $ap (*)$ – Complex (Kind=nag_wp) array Input: 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.
4: $tau (*)$ – Complex (Kind=nag_wp) array Input: 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.
5: $q (ldq, *)$ – Complex (Kind=nag_wp) array Output: Note: the second dimension of the array q must be at least $\max (1, n)$ .

On exit: the $n \times n$ unitary matrix $Q$ .
6: $ldq$ – Integer Input: On entry: the first dimension of the array q as declared in the (sub)program from which f08gtf is called.

Constraint: $ldq \geq \max (1, n)$ .
7: $work (n - 1)$ – Complex (Kind=nag_wp) array Workspace
8: $info$ – Integer Output: 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 unitary matrix by a matrix

E

such that

{‖ E ‖}_{2} = O (ε),

where

ε

is the machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f08gtf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08gtf 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 real floating-point operations is approximately

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

The real analogue of this routine is f08gff.

10 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. The program then calls f08gtf to form

Q

, and passes this matrix to f08jsf which computes the eigenvalues and eigenvectors of

A

f08gt: FL CL CPP AD PY MB

NAG FL Interfacef08gtf (zupgtr)

▸▿ 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

NAG FL Interface
f08gtf (zupgtr)