# NAG FL Interfacef08nef (dgehrd)

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## 1Purpose

f08nef reduces a real general matrix to Hessenberg form.

## 2Specification

Fortran Interface
 Subroutine f08nef ( n, ilo, ihi, a, lda, tau, work, info)
 Integer, Intent (In) :: n, ilo, ihi, lda, lwork Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), tau(*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
#include <nag.h>
 void f08nef_ (const Integer *n, const Integer *ilo, const Integer *ihi, double a[], const Integer *lda, double tau[], double work[], const Integer *lwork, Integer *info)
The routine may be called by the names f08nef, nagf_lapackeig_dgehrd or its LAPACK name dgehrd.

## 3Description

f08nef 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). Routines are provided to work with $Q$ in this representation (see Section 9).
The routine can take advantage of a previous call to f08nhf, 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 f08nhf can be supplied to the routine directly. If f08nhf has not previously been called however, then ${i}_{\mathrm{lo}}$ must be set to $1$ and ${i}_{\mathrm{hi}}$ to $n$.
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{ilo}$Integer Input
3: $\mathbf{ihi}$Integer Input
On entry: if $A$ has been output by f08nhf, ilo and ihi must contain the values returned by that routine. 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 ${\mathbf{ihi}}=0$.
4: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n×n$ general matrix $A$.
On exit: a is overwritten by the upper Hessenberg matrix $H$ and details of the orthogonal matrix $Q$.
5: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08nef is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6: $\mathbf{tau}\left(*\right)$Real (Kind=nag_wp) array Output
Note: the dimension of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On exit: further details of the orthogonal matrix $Q$.
7: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Real (Kind=nag_wp) array Workspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
8: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08nef is called.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value: for optimal performance, ${\mathbf{lwork}}\ge {\mathbf{n}}×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ or ${\mathbf{lwork}}=-1$.
9: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

$-999<{\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
An explanatory message is output, and execution of the program is terminated.

## 7Accuracy

The computed Hessenberg matrix $H$ is exactly similar to a nearby matrix $\left(A+E\right)$, where
 $‖E‖2 ≤ c (n) ε ‖A‖2 ,$
$c\left(n\right)$ 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.

## 8Parallelism and Performance

f08nef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08nef makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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 $Q$ f08nef may be followed by a call to f08nff :
`Call dorghr(n,ilo,ihi,a,lda,tau,work,lwork,info)`
To apply $Q$ to an $m×n$ real matrix $C$ f08nef may be followed by a call to f08ngf. For example,
```Call dormhr('Left','No Transpose',m,n,ilo,ihi,a,lda,tau,c,ldc, &
work,lwork,info)```
forms the matrix product $QC$.
The complex analogue of this routine is f08nsf.

## 10Example

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

### 10.1Program Text

Program Text (f08nefe.f90)

### 10.2Program Data

Program Data (f08nefe.d)

### 10.3Program Results

Program Results (f08nefe.r)