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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zungtr (f08ft)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zungtr (f08ft) generates the complex unitary matrix Q, which was determined by nag_lapack_zhetrd (f08fs) when reducing a Hermitian matrix to tridiagonal form.

Syntax

[a, info] = f08ft(uplo, a, tau, 'n', n)
[a, info] = nag_lapack_zungtr(uplo, a, tau, 'n', n)

Description

nag_lapack_zungtr (f08ft) is intended to be used after a call to nag_lapack_zhetrd (f08fs), which reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: A=QTQH. nag_lapack_zhetrd (f08fs) represents the unitary matrix Q as a product of n-1 elementary reflectors.
This function may be used to generate Q explicitly as a square matrix.

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

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:     alda: – complex array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,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 max1,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: n0.

Output Parameters

1:     alda: – complex array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,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
E2 = Oε ,  
where ε is the machine precision.

Further Comments

The total number of real floating-point operations is approximately 163n3.
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 = -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 .  
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.
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


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