# NAG Library Function Document

## 1Purpose

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

## 2Specification

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

## 3Description

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 9).
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 $n$.

## 4References

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

## 5Arguments

1:    $\mathbf{order}$Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:    $\mathbf{ilo}$IntegerInput
4:    $\mathbf{ihi}$IntegerInput
On entry: if $A$ has been output by nag_dgebal (f08nhc), 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:    $\mathbf{a}\left[\mathit{dim}\right]$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:    $\mathbf{pda}$IntegerInput
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:    $\mathbf{tau}\left[\mathit{dim}\right]$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:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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.

## 8Parallelism and Performance

nag_dgehrd (f08nec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dgehrd (f08nec) 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 function. 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 $Q$ 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).

## 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 (f08nece.c)

### 10.2Program Data

Program Data (f08nece.d)

### 10.3Program Results

Program Results (f08nece.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017