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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgehrd (f08ne)

## Purpose

nag_lapack_dgehrd (f08ne) reduces a real general matrix to Hessenberg form.

## Syntax

[a, tau, info] = f08ne(ilo, ihi, a, 'n', n)
[a, tau, info] = nag_lapack_dgehrd(ilo, ihi, a, 'n', n)

## Description

nag_lapack_dgehrd (f08ne) reduces a real general matrix $A$ to upper Hessenberg form $H$ by an orthogonal similarity transformation: $A=QH{Q}^{\mathrm{T}}$.
The matrix $Q$ 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:
 $A11 A12 A13 A22 A23 A33$
where ${A}_{11}$ and ${A}_{33}$ are upper triangular. If so, only the central diagonal block ${A}_{22}$, in rows and columns ${i}_{\mathrm{lo}}$ to ${i}_{\mathrm{hi}}$, 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 ${i}_{\mathrm{lo}}$ and ${i}_{\mathrm{hi}}$ 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 ${i}_{\mathrm{lo}}$ must be set to $1$ and ${i}_{\mathrm{hi}}$ to $n$.

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{ilo}$int64int32nag_int scalar
2:     $\mathrm{ihi}$int64int32nag_int scalar
If $A$ 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 ${\mathbf{n}}>0$, $1\le {\mathbf{ilo}}\le {\mathbf{ihi}}\le {\mathbf{n}}$;
• if ${\mathbf{n}}=0$, ${\mathbf{ilo}}=1$ and ${\mathbf{ihi}}=0$.
3:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $n$ general matrix $A$.

### 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 $A$.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
a stores the upper Hessenberg matrix $H$ and details of the orthogonal matrix $Q$.
2:     $\mathrm{tau}\left(:\right)$ – double array
The dimension of the array tau will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
Further details of the orthogonal matrix $Q$.
3:     $\mathrm{info}$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 Hessenberg matrix $H$ is exactly similar to a nearby matrix $\left(A+E\right)$, where
 $E2 ≤ c n ε A2 ,$
$c\left(n\right)$ is a modestly increasing function of $n$, and $\epsilon$ 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.

The total number of floating-point operations is approximately $\frac{2}{3}{q}^{2}\left(2q+3n\right)$, where $q={i}_{\mathrm{hi}}-{i}_{\mathrm{lo}}$; if ${i}_{\mathrm{lo}}=1$ and ${i}_{\mathrm{hi}}=n$, 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 $m$ by $n$ 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 $QC$.
The complex analogue of this function is nag_lapack_zgehrd (f08ns).

## Example

This example computes the upper Hessenberg form of the matrix $A$, where
 $A = 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 .$
```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

```