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Chapter Introduction
NAG Toolbox

# 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$A$ to upper Hessenberg form H$H$ by a unitary similarity transformation: A = QHQH$A=QH{Q}^{\mathrm{H}}$. H$H$ has real subdiagonal elements.
The matrix Q$Q$ 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$Q$ in this representation (see Section [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
$A11 A12 A13 A22 A23 A33$
where A11${A}_{11}$ and A33${A}_{33}$ are upper triangular. If so, only the central diagonal block A22${A}_{22}$, in rows and columns ilo${i}_{\mathrm{lo}}$ to ihi${i}_{\mathrm{hi}}$, needs to be reduced to Hessenberg form (the blocks A12${A}_{12}$ and A23${A}_{23}$ will also be affected by the reduction). Therefore the values of ilo${i}_{\mathrm{lo}}$ and ihi${i}_{\mathrm{hi}}$ 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 ilo${i}_{\mathrm{lo}}$ must be set to 1$1$ and ihi${i}_{\mathrm{hi}}$ to n$n$.

## 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
If A$A$ 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$1$ and ihi to n.
Constraints:
• if n > 0${\mathbf{n}}>0$, 1 ilo ihi n $1\le {\mathbf{ilo}}\le {\mathbf{ihi}}\le {\mathbf{n}}$;
• if n = 0${\mathbf{n}}=0$, ilo = 1${\mathbf{ilo}}=1$ and ihi = 0${\mathbf{ihi}}=0$.
3:     a(lda, : $:$) – complex array
The first dimension of the array a must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The n$n$ by n$n$ general matrix A$A$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a The second dimension of the array a.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

lda work lwork

### Output Parameters

1:     a(lda, : $:$) – complex array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
a stores the upper Hessenberg matrix H$H$ and details of the unitary matrix Q$Q$. The subdiagonal elements of H$H$ are real.
2:     tau( : $:$) – complex array
Note: the dimension of the array tau must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
Further details of the unitary matrix Q$Q$.
3:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ 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$H$ is exactly similar to a nearby matrix (A + E)$\left(A+E\right)$, where
 ‖E‖2 ≤ c (n) ε ‖A‖2 , $‖E‖2 ≤ c (n) ε ‖A‖2 ,$
c(n)$c\left(n\right)$ is a modestly increasing function of n$n$, and ε$\epsilon$ is the machine precision.
The elements of H$H$ themselves may be sensitive to small perturbations in A$A$ 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 real floating point operations is approximately (8/3)q2(2q + 3n)$\frac{8}{3}{q}^{2}\left(2q+3n\right)$, where q = ihiilo$q={i}_{\mathrm{hi}}-{i}_{\mathrm{lo}}$; if ilo = 1${i}_{\mathrm{lo}}=1$ and ihi = n${i}_{\mathrm{hi}}=n$, the number is approximately (40/3)n3$\frac{40}{3}{n}^{3}$.
To form the unitary matrix Q$Q$ 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$Q$ to an m$m$ by n$n$ complex matrix C$C$ 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$QC$.
The real analogue of this function is nag_lapack_dgehrd (f08ne).

## Example

```function nag_lapack_zgehrd_example
ilo = int64(1);
ihi = int64(4);
a = [ -3.97 - 5.04i,  -4.11 + 3.7i,  -0.34 + 1.01i,  1.29 - 0.86i;
0.34 - 1.5i,  1.52 - 0.43i,  1.88 - 5.38i,  3.36 + 0.65i;
3.31 - 3.85i,  2.5 + 3.45i,  0.88 - 1.08i,  0.64 - 1.48i;
-1.1 + 0.82i,  1.81 - 1.59i,  3.25 + 1.33i,  1.57 - 3.44i];
[aOut, tau, info] = nag_lapack_zgehrd(ilo, ihi, a)
```
```

aOut =

-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

tau =

1.0620 - 0.2737i
1.8059 + 0.3479i
1.1818 + 0.9833i

info =

0

```
```function f08ns_example
ilo = int64(1);
ihi = int64(4);
a = [ -3.97 - 5.04i,  -4.11 + 3.7i,  -0.34 + 1.01i,  1.29 - 0.86i;
0.34 - 1.5i,  1.52 - 0.43i,  1.88 - 5.38i,  3.36 + 0.65i;
3.31 - 3.85i,  2.5 + 3.45i,  0.88 - 1.08i,  0.64 - 1.48i;
-1.1 + 0.82i,  1.81 - 1.59i,  3.25 + 1.33i,  1.57 - 3.44i];
[aOut, tau, info] = f08ns(ilo, ihi, a)
```
```

aOut =

-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

tau =

1.0620 - 0.2737i
1.8059 + 0.3479i
1.1818 + 0.9833i

info =

0

```