Integer, Intent (In)	::	n, lda, lwork
Integer, Intent (Out)	::	info
Complex (Kind=nag_wp), Intent (In)	::	tau(*)
Complex (Kind=nag_wp), Intent (Inout)	::	a(lda,*)
Complex (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))
Character (1), Intent (In)	::	uplo

C Header Interface

#include <nag.h>

void	f08ftf_ (const char uplo, const Integer n, Complex a[], const Integer lda, const Complex tau[], Complex work[], const Integer lwork, Integer *info, const Charlen length_uplo)

The routine may be called by the names f08ftf, nagf_lapackeig_zungtr or its LAPACK name zungtr.

3 Description

f08ftf is intended to be used after a call to f08fsf, which reduces a complex Hermitian matrix

A

to real symmetric tridiagonal form

T

by a unitary similarity transformation:

A = Q T Q^{H}

. f08fsf 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 f08fsf.

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

Constraint: $n \geq 0$ .
3: $a (lda, *)$ – Complex (Kind=nag_wp) array Input/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 f08fsf.

On exit: the $n \times n$ unitary matrix $Q$ .
4: $lda$ – Integer Input: On entry: the first dimension of the array a as declared in the (sub)program from which f08ftf is called.

Constraint: $lda \geq \max (1, n)$ .
5: $tau (*)$ – 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 f08fsf.
6: $work (\max (1, lwork))$ – Complex (Kind=nag_wp) array Workspace: On exit: if $info = 0$ , the real part of $work (1)$ contains the minimum value of lwork required for optimal performance.
7: $lwork$ – Integer Input: On entry: the dimension of the array work as declared in the (sub)program from which f08ftf 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: $info$ – 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

f08ftf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08ftf 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 f08fff.

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}) .

Here

A

is Hermitian and must first be reduced to tridiagonal form by f08fsf. The program then calls f08ftf to form

Q

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

A

f08ft: FL CL CPP AD

NAG FL Interfacef08ftf (zungtr)

▸▿ 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
f08ftf (zungtr)