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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zunghr (f08nt)

Purpose

nag_lapack_zunghr (f08nt) generates the complex unitary matrix $Q$ which was determined by nag_lapack_zgehrd (f08ns) when reducing a complex general matrix $A$ to Hessenberg form.

Syntax

[a, info] = f08nt(ilo, ihi, a, tau, 'n', n)
[a, info] = nag_lapack_zunghr(ilo, ihi, a, tau, 'n', n)

Description

nag_lapack_zunghr (f08nt) is intended to be used following a call to nag_lapack_zgehrd (f08ns), which reduces a complex general matrix $A$ to upper Hessenberg form $H$ by a unitary similarity transformation: $A=QH{Q}^{\mathrm{H}}$. nag_lapack_zgehrd (f08ns) represents the matrix $Q$ as a product of ${i}_{\mathrm{hi}}-{i}_{\mathrm{lo}}$ elementary reflectors. Here ${i}_{\mathrm{lo}}$ and ${i}_{\mathrm{hi}}$ are values determined by nag_lapack_zgebal (f08nv) when balancing the matrix; if the matrix has not been balanced, ${i}_{\mathrm{lo}}=1$ and ${i}_{\mathrm{hi}}=n$.
This function may be used to generate $Q$ explicitly as a square matrix. $Q$ has the structure:
 $Q = I 0 0 0 Q22 0 0 0 I$
where ${Q}_{22}$ occupies rows and columns ${i}_{\mathrm{lo}}$ to ${i}_{\mathrm{hi}}$.

References

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

Parameters

Compulsory Input Parameters

1:     $\mathrm{ilo}$int64int32nag_int scalar
2:     $\mathrm{ihi}$int64int32nag_int scalar
These must be the same arguments ilo and ihi, respectively, as supplied to nag_lapack_zgehrd (f08ns).
Constraints:
• if ${\mathbf{n}}>0$, $1\le {\mathbf{ilo}}\le {\mathbf{ihi}}\le {\mathbf{n}}$;
• if ${\mathbf{n}}=0$, ${\mathbf{ilo}}=1$ and ${\mathbf{ihi}}=0$.
3:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_zgehrd (f08ns).
4:     $\mathrm{tau}\left(:\right)$ – complex array
The dimension of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
Further details of the elementary reflectors, as returned by nag_lapack_zgehrd (f08ns).

Optional Input Parameters

1:     $\mathrm{n}$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: ${\mathbf{n}}\ge 0$.

Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $n$ unitary matrix $Q$.
2:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

${\mathbf{info}}=-i$
If ${\mathbf{info}}=-i$, parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: n, 2: ilo, 3: ihi, 4: a, 5: lda, 6: tau, 7: work, 8: lwork, 9: 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 $\epsilon$ is the machine precision.

The total number of real floating-point operations is approximately $\frac{16}{3}{q}^{3}$, where $q={i}_{\mathrm{hi}}-{i}_{\mathrm{lo}}$.
The real analogue of this function is nag_lapack_dorghr (f08nf).

Example

This example computes the Schur factorization of the matrix $A$, where
 $A = -3.97-5.04i -4.11+3.70i -0.34+1.01i 1.29-0.86i 0.34-1.50i 1.52-0.43i 1.88-5.38i 3.36+0.65i 3.31-3.85i 2.50+3.45i 0.88-1.08i 0.64-1.48i -1.10+0.82i 1.81-1.59i 3.25+1.33i 1.57-3.44i .$
Here $A$ is general and must first be reduced to Hessenberg form by nag_lapack_zgehrd (f08ns). The program then calls nag_lapack_zunghr (f08nt) to form $Q$, and passes this matrix to nag_lapack_zhseqr (f08ps) which computes the Schur factorization of $A$.
```function f08nt_example

fprintf('f08nt example results\n\n');

ilo = int64(1);
ihi = int64(4);
a = [ -3.97 - 5.04i, -4.11 + 3.70i, -0.34 + 1.01i,  1.29 - 0.86i;
0.34 - 1.50i,  1.52 - 0.43i,  1.88 - 5.38i,  3.36 + 0.65i;
3.31 - 3.85i,  2.50 + 3.45i,  0.88 - 1.08i,  0.64 - 1.48i;
-1.10 + 0.82i,  1.81 - 1.59i,  3.25 + 1.33i,  1.57 - 3.44i];

% Reduce A to upper Hessenberg Form
[H, tau, info] = f08ns(ilo, ihi, a);

% Form Q
[Q, info] = f08nt(ilo, ihi, H, tau);

% Schur factorize H = Y*T*Y' and form Z = QY  A = QY*T*(QQY)'
job   = 'Schur form';
compz = 'Vectors';
[~, w, Z, info] = f08ps( ...
job, compz, ilo, ihi, H, Q);

disp('Eigenvalues of A');
disp(w);

```
```f08nt example results

Eigenvalues of A
-6.0004 - 6.9998i
-5.0000 + 2.0060i
7.9982 - 0.9964i
3.0023 - 3.9998i

```