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 nagmk26.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 its LAPACK name zungtr.

3

Description

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

A

to real symmetric tridiagonal form

T

by a unitary similarity transformation:

A = Q T Q^{H}

. f08fsf (zhetrd) 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 (zhetrd).

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

Constraint: $n \geq 0$ .
3: $a (lda *)$ – Complex (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 f08fsf (zhetrd).

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

Constraint: $lda \geq \max (1, n)$ .
5: $tau (*)$ – 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 f08fsf (zhetrd).
6: $work (\max (1, lwork))$ – Complex (Kind=nag_wp) arrayWorkspace: On exit: if $info = 0$ , the real part of $work (1)$ 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 f08ftf (zungtr) 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

$info < 0$: 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 (zungtr) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08ftf (zungtr) 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 (dorgtr).

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 (zhetrd). The program then calls f08ftf (zungtr) to form

Q

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

A

NAG Library Routine Document

f08ftf (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

1

Purpose

2

Specification