is not formed explicitly, but is represented as a product of elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with

Q

in this representation (see Further Comments).

The function can take advantage of a previous call to nag_lapack_dgebal (f08nh), which may produce a matrix with the structure:

(\begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{22} & A_{23} \\ A_{33} \end{matrix})

where

A_{11}

and

A_{33}

are upper triangular. If so, only the central diagonal block

A_{22}

, in rows and columns

i_{lo}

i_{hi}

, needs to be reduced to Hessenberg form (the blocks

A_{12}

and

A_{23}

will also be affected by the reduction). Therefore the values of

i_{lo}

and

i_{hi}

determined by nag_lapack_dgebal (f08nh) can be supplied to the function directly. If nag_lapack_dgebal (f08nh) has not previously been called however, then

i_{lo}

must be set to

1

and

i_{hi}

n

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

A

has been output by nag_lapack_dgebal (f08nh), then ilo and ihi must contain the values returned by that function. Otherwise, ilo must be set to

1

and ihi to n.

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)

The

n

n

general matrix

A

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 $A$ .

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 $\max (1, n)$ .

a stores the upper Hessenberg matrix $H$ and details of the orthogonal matrix $Q$ .
2: $tau (:)$ – double array: The dimension of the array tau will be $\max (1, n - 1)$

Further details of the orthogonal matrix $Q$ .
3: $info$ – 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 Hessenberg matrix

H

is exactly similar to a nearby matrix

(A + E)

, where

{‖E‖}_{2} \leq c (n) ε {‖A‖}_{2},

c (n)

is a modestly increasing function of

n

, and

ε

is the machine precision.

The elements of

H

themselves may be sensitive to small perturbations in

A

or to rounding errors in the computation, but this does not affect the stability of the eigenvalues, eigenvectors or Schur factorization.

Further Comments

The total number of floating-point operations is approximately

\frac{2}{3} q^{2} (2 q + 3 n)

, where

q = i_{hi} - i_{lo}

; if

i_{lo} = 1

and

i_{hi} = n

, the number is approximately

\frac{10}{3} n^{3}

To form the orthogonal matrix

Q

nag_lapack_dgehrd (f08ne) may be followed by a call to nag_lapack_dorghr (f08nf):

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

To apply

Q

to an

m

n

real matrix

C

nag_lapack_dgehrd (f08ne) may be followed by a call to nag_lapack_dormhr (f08ng). For example,

[c, info] = f08ng('Left', 'No Transpose', ilo, ihi, a, tau, c);

forms the matrix product

Q C

The complex analogue of this function is nag_lapack_zgehrd (f08ns).

Example

This example computes the upper Hessenberg form 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}) .

Open in the MATLAB editor: f08ne_example

function f08ne_example


fprintf('f08ne 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
[H, tau, info] = f08ne( ...
                        ilo, ihi, a);

disp('Upper Hessenberg Form H');
disp(H);

f08ne example results

Upper Hessenberg Form H
    0.3500   -0.1160   -0.3886   -0.2942
   -0.5140    0.1225    0.1004    0.1126
   -0.7285    0.6443   -0.1357   -0.0977
    0.4139   -0.1665    0.4262    0.1632

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

Chapter Contents

Chapter Introduction

NAG Toolbox