f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_dgehrd (f08nec)

## 1  Purpose

nag_dgehrd (f08nec) reduces a real general matrix to Hessenberg form.

## 2  Specification

 #include #include
 void nag_dgehrd (Nag_OrderType order, Integer n, Integer ilo, Integer ihi, double a[], Integer pda, double tau[], NagError *fail)

## 3  Description

nag_dgehrd (f08nec) 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 Section 8).
The function can take advantage of a previous call to nag_dgebal (f08nhc), 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_dgebal (f08nhc) can be supplied to the function directly. If nag_dgebal (f08nhc) has not previously been called however, then ${i}_{\mathrm{lo}}$ must be set to $1$ and ${i}_{\mathrm{hi}}$ to $n$.

## 4  References

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

## 5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     iloIntegerInput
4:     ihiIntegerInput
On entry: if $A$ has been output by nag_dgebal (f08nhc), 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 ${\mathbf{ihi}}=0$.
5:     a[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n$ by $n$ general matrix $A$.
On exit: a is overwritten by the upper Hessenberg matrix $H$ and details of the orthogonal matrix $Q$.
6:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7:     tau[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, 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 $Q$.
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INT_3
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, ${\mathbf{ilo}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ihi}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{n}}>0$, $1\le {\mathbf{ilo}}\le {\mathbf{ihi}}\le {\mathbf{n}}$;
if ${\mathbf{n}}=0$, ${\mathbf{ilo}}=1$ and ${\mathbf{ihi}}=0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

## 7  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_dgehrd (f08nec) may be followed by a call to nag_dorghr (f08nfc):
```nag_dorghr(order,n,ilo,ihi,&a,pda,tau,&fail)
```
To apply $Q$ to an $m$ by $n$ real matrix $C$ nag_dgehrd (f08nec) may be followed by a call to nag_dormhr (f08ngc). For example,
```nag_dormhr(order,Nag_LeftSide,Nag_NoTrans,m,n,ilo,ihi,&a,pda,
tau,&c,pdc,&fail)
```
forms the matrix product $QC$.
The complex analogue of this function is nag_zgehrd (f08nsc).

## 9  Example

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

### 9.1  Program Text

Program Text (f08nece.c)

### 9.2  Program Data

Program Data (f08nece.d)

### 9.3  Program Results

Program Results (f08nece.r)