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# NAG Toolbox: nag_lapack_zgehrd (f08ns)

## Purpose

nag_lapack_zgehrd (f08ns) reduces a complex general matrix to Hessenberg form.

## Syntax

[a, tau, info] = f08ns(ilo, ihi, a, 'n', n)
[a, tau, info] = nag_lapack_zgehrd(ilo, ihi, a, 'n', n)

## Description

nag_lapack_zgehrd (f08ns) reduces a complex general matrix $A$ to upper Hessenberg form $H$ by a unitary similarity transformation: $A=QH{Q}^{\mathrm{H}}$. $H$ has real subdiagonal elements.
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_zgebal (f08nv), 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_zgebal (f08nv) can be supplied to the function directly. If nag_lapack_zgebal (f08nv) 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_zgebal (f08nv), 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
a stores the upper Hessenberg matrix $H$ and details of the unitary matrix $Q$. The subdiagonal elements of $H$ are real.
2:     $\mathrm{tau}\left(:\right)$ – complex 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 unitary 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.

## Further Comments

The total number of real floating-point operations is approximately $\frac{8}{3}{q}^{2}\left(2q+3n\right)$, where $q={i}_{\mathrm{hi}}-{i}_{\mathrm{lo}}$; if ${i}_{\mathrm{lo}}=1$ and ${i}_{\mathrm{hi}}=n$, the number is approximately $\frac{40}{3}{n}^{3}$.
To form the unitary matrix $Q$ nag_lapack_zgehrd (f08ns) may be followed by a call to nag_lapack_zunghr (f08nt):
```[a, info] = f08nt(ilo, ihi, a, tau);
```
To apply $Q$ to an $m$ by $n$ complex matrix $C$ nag_lapack_zgehrd (f08ns) may be followed by a call to nag_lapack_zunmhr (f08nu). For example,
```[c, info] = f08nu('Left', 'No Transpose', ilo, ihi, a, tau, c);
```
forms the matrix product $QC$.
The real analogue of this function is nag_lapack_dgehrd (f08ne).

## Example

This example computes the upper Hessenberg form of the matrix $A$, where
 $A = -3.97-5.04i -4.11+3.70i -0.34+1.01i 1.29-0.86i 0.34-1.50i 1.52-0.43i 1.88-5.38i 3.36+0.65i 3.31-3.85i 2.50+3.45i 0.88-1.08i 0.64-1.48i -1.10+0.82i 1.81-1.59i 3.25+1.33i 1.57-3.44i .$
```function f08ns_example

fprintf('f08ns 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);

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

```
```f08ns example results

Upper Hessenberg Form H
-3.9700 - 5.0400i  -1.1318 - 2.5693i  -4.6027 - 0.1426i  -1.4249 + 1.7330i
-5.4797 + 0.0000i   1.8585 - 1.5502i   4.4145 - 0.7638i  -0.4805 - 1.1976i
0.6932 - 0.4829i   6.2673 + 0.0000i  -0.4504 - 0.0290i  -1.3467 + 1.6579i
-0.2113 + 0.0864i   0.1242 - 0.2289i  -3.5000 + 0.0000i   2.5619 - 3.3708i

```

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