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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_dorghr (f08nf)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_lapack_dorghr (f08nf) generates the real orthogonal matrix

Q

which was determined by nag_lapack_dgehrd (f08ne) when reducing a real general matrix

A

to Hessenberg form.

Syntax

[a, info] = f08nf(ilo, ihi, a, tau, 'n', n)

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

Description

nag_lapack_dorghr (f08nf) is intended to be used following a call to nag_lapack_dgehrd (f08ne), which reduces a real general matrix

A

to upper Hessenberg form

H

by an orthogonal similarity transformation:

A = Q H Q^{T}

. nag_lapack_dgehrd (f08ne) represents the matrix

Q

as a product of

i_{hi} - i_{lo}

elementary reflectors. Here

i_{lo}

and

i_{hi}

are values determined by nag_lapack_dgebal (f08nh) when balancing the matrix; if the matrix has not been balanced,

i_{lo} = 1

and

i_{hi} = n

This function may be used to generate

Q

explicitly as a square matrix.

Q

has the structure:

Q = (\begin{matrix} I & 0 & 0 \\ 0 & Q_{22} & 0 \\ 0 & 0 & I \end{matrix})

where

Q_{22}

occupies rows and columns

i_{lo}

i_{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: $ilo$ – int64int32nag_int scalar

2: $ihi$ – int64int32nag_int scalar

These must be the same arguments ilo and ihi, respectively, as supplied to nag_lapack_dgehrd (f08ne).

Constraints:

if $n > 0$ , $1 \leq ilo \leq ihi \leq n$ ;
if $n = 0$ , $ilo = 1$ and $ihi = 0$ .

3: $a (lda :)$ – double 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_dgehrd (f08ne).

4: $tau (:)$ – double 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_dgehrd (f08ne).

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: $a (lda :)$ – double 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 $n$ orthogonal 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: 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 orthogonal matrix by a matrix

E

such that

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision.

Further Comments

The total number of floating-point operations is approximately

\frac{4}{3} q^{3}

, where

q = i_{hi} - i_{lo}

The complex analogue of this function is nag_lapack_zunghr (f08nt).

Example

This example computes the Schur factorization of the matrix

A

, where

A = (\begin{array}{r} 0.35 & 0.45 & - 0.14 & - 0.17 \\ 0.09 & 0.07 & - 0.54 & 0.35 \\ - 0.44 & - 0.33 & - 0.03 & 0.17 \\ 0.25 & - 0.32 & - 0.13 & 0.11 \end{array}) .

Here

A

is general and must first be reduced to Hessenberg form by nag_lapack_dgehrd (f08ne). The program then calls nag_lapack_dorghr (f08nf) to form

Q

, and passes this matrix to nag_lapack_dhseqr (f08pe) which computes the Schur factorization of

A

Open in the MATLAB editor: f08nf_example

function f08nf_example


fprintf('f08nf example results\n\n');

ilo = int64(1);
ihi = int64(4);
a = [ 0.35,  0.45, -0.14, -0.17;
      0.09,  0.07, -0.54,  0.35;
     -0.44, -0.33, -0.03,  0.17;
      0.25, -0.32, -0.13,  0.11];

% Reduce A to upper Hessenberg Form A = QHQ^T
[H, tau, info] = f08ne( ...
                        ilo, ihi, a);

% Form Q
[Q, info] = f08nf( ...
                   ilo, ihi, H, tau);

% Schur factorize H = Y*T*Y' and form Z = QY  A = QY*T*(QQY)'
job   = 'Schur form';
compz = 'Vectors';
[~, wr, wi, Z, info] = f08pe( ...
                              job, compz, ilo, ihi, H, Q);

w = wr + i*wi;
disp('Eigenvalues of A');
disp(w);

f08nf example results

Eigenvalues of A
   0.7995 + 0.0000i
  -0.0994 + 0.4008i
  -0.0994 - 0.4008i
  -0.1007 + 0.0000i

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Chapter Contents

Chapter Introduction

NAG Toolbox