1: $uplo$ – string (length ≥ 1): This must be the same argument uplo as supplied to nag_lapack_zhetrd (f08fs).

Constraint: $uplo ='U'$ or $'L'$ .
2: $a (lda :)$ – complex array: The first dimension of the array a must be at least $\max (1, n)$ .
The second dimension of the array a must be at least $\max (1, n)$ .

Details of the vectors which define the elementary reflectors, as returned by nag_lapack_zhetrd (f08fs).
3: $tau (:)$ – complex array: The dimension of the array tau must be at least $\max (1, n - 1)$

Further details of the elementary reflectors, as returned by nag_lapack_zhetrd (f08fs).

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array a and the second dimension of the array a.
$n$ , the order of the matrix $Q$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $a (lda :)$ – complex array: The first dimension of the array a will be $\max (1, n)$ .
The second dimension of the array a will be $\max (1, n)$ .

The $n$ by $n$ unitary matrix $Q$ .
2: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

$info = - i$: If $info = - i$ , parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: a, 4: lda, 5: tau, 6: work, 7: lwork, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

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.

Further Comments

The total number of real floating-point operations is approximately

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

The real analogue of this function is nag_lapack_dorgtr (f08ff).

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 nag_lapack_zhetrd (f08fs). The program then calls nag_lapack_zungtr (f08ft) to form

Q

, and passes this matrix to nag_lapack_zsteqr (f08js) which computes the eigenvalues and eigenvectors of

A

Open in the MATLAB editor: f08ft_example

function f08ft_example


fprintf('f08ft example results\n\n');

% Eigenvalues / vectors of Hermitian matrix A
uplo = 'L';
n = 4;
a = [-2.28 + 0.00i,  0.00 + 0i,     0    + 0i,     0    + 0i;
      1.78 + 2.03i, -1.12 + 0i,     0    + 0i,     0    + 0i;
      2.26 - 0.10i,  0.01 - 0.43i, -0.37 + 0i,     0    + 0i;
     -0.12 - 2.53i, -1.07 - 0.86i,  2.31 + 0.92i, -0.73 + 0i];

% A --> QTQ^H, for tridiagonal T
[QT, d, e, tau, info] = f08fs( ...
                               uplo, a);

% Form Q
[Q, info] = f08ft(uplo, QT, tau);

% Calculate eigenvalues/vectors of A from Q, d and e.
compz = 'V';
[w, ~, z, info] = f08js( ...
                         compz, d, e, Q);

% Normalize vectors, largest element is real and positive.
for i = 1:n
  [~,k] = max(abs(real(z(:,i)))+abs(imag(z(:,i))));
  z(:,i) = z(:,i)*conj(z(k,i))/abs(z(k,i));
end

disp(' Eigenvalues of A:');
disp(w);
disp(' Corresponding eigenvectors:');
disp(z);

f08ft example results

 Eigenvalues of A:
   -6.0002
   -3.0030
    0.5036
    3.9996

 Corresponding eigenvectors:
   0.7299 + 0.0000i  -0.2120 + 0.1497i   0.1000 - 0.3570i   0.1991 + 0.4720i
  -0.1663 - 0.2061i   0.7307 + 0.0000i   0.2863 - 0.3353i  -0.2467 + 0.3751i
  -0.4165 - 0.1417i  -0.3291 + 0.0479i   0.6890 + 0.0000i   0.4468 + 0.1466i
   0.1743 + 0.4162i   0.5200 + 0.1329i   0.0662 + 0.4347i   0.5612 + 0.0000i

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox