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# 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$A$ to upper Hessenberg form H$H$ by an orthogonal similarity transformation: A = QHQT$A=QH{Q}^{\mathrm{T}}$.
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_dgebal (f08nh), 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_dgebal (f08nh) can be supplied to the function directly. If nag_lapack_dgebal (f08nh) 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_dgebal (f08nh), 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, : $:$) – double 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, : $:$) – double 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 orthogonal matrix Q$Q$.
2:     tau( : $:$) – double 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 orthogonal 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 floating point operations is approximately (2/3)q2(2q + 3n)$\frac{2}{3}{q}^{2}\left(2q+3n\right)$, 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 (10/3)n3$\frac{10}{3}{n}^{3}$.
To form the orthogonal matrix Q$Q$ 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$Q$ to an m$m$ by n$n$ real matrix C$C$ 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$QC$.
The complex analogue of this function is nag_lapack_zgehrd (f08ns).

## Example

```function nag_lapack_dgehrd_example
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];
[aOut, tau, info] = nag_lapack_dgehrd(ilo, ihi, a)
```
```

aOut =

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

tau =

1.1751
1.9460
0

info =

0

```
```function f08ne_example
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];
[aOut, tau, info] = f08ne(ilo, ihi, a)
```
```

aOut =

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

tau =

1.1751
1.9460
0

info =

0

```

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