f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_dorghr (f08nfc)

## 1  Purpose

nag_dorghr (f08nfc) generates the real orthogonal matrix $Q$ which was determined by nag_dgehrd (f08nec) when reducing a real general matrix $A$ to Hessenberg form.

## 2  Specification

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

## 3  Description

nag_dorghr (f08nfc) is intended to be used following a call to nag_dgehrd (f08nec), which reduces a real general matrix $A$ to upper Hessenberg form $H$ by an orthogonal similarity transformation: $A=QH{Q}^{\mathrm{T}}$. nag_dgehrd (f08nec) represents the matrix $Q$ as a product of ${i}_{\mathrm{hi}}-{i}_{\mathrm{lo}}$ elementary reflectors. Here ${i}_{\mathrm{lo}}$ and ${i}_{\mathrm{hi}}$ are values determined by nag_dgebal (f08nhc) when balancing the matrix; if the matrix has not been balanced, ${i}_{\mathrm{lo}}=1$ and ${i}_{\mathrm{hi}}=n$.
This function may be used to generate $Q$ explicitly as a square matrix. $Q$ has the structure:
 $Q = I 0 0 0 Q22 0 0 0 I$
where ${Q}_{22}$ occupies rows and columns ${i}_{\mathrm{lo}}$ to ${i}_{\mathrm{hi}}$.

## 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 $\mathrm{Nag_ColMajor}$.
2:     nIntegerInput
On entry: $n$, the order of the matrix $Q$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     iloIntegerInput
4:     ihiIntegerInput
On entry: these must be the same arguments ilo and ihi, respectively, as supplied to nag_dgehrd (f08nec).
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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_dgehrd (f08nec).
On exit: the $n$ by $n$ orthogonal matrix $Q$.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$.
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[$\mathit{dim}$]const doubleInput
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 entry: further details of the elementary reflectors, as returned by nag_dgehrd (f08nec).
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 matrix $Q$ differs from an exactly orthogonal matrix by a matrix $E$ such that
 $E2 = Oε ,$
where $\epsilon$ is the machine precision.

## 8  Parallelism and Performance

nag_dorghr (f08nfc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dorghr (f08nfc) 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.

The total number of floating-point operations is approximately $\frac{4}{3}{q}^{3}$, where $q={i}_{\mathrm{hi}}-{i}_{\mathrm{lo}}$.
The complex analogue of this function is nag_zunghr (f08ntc).

## 10  Example

This example computes the Schur factorization of the matrix $A$, where
 $A = 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 .$
Here $A$ is general and must first be reduced to Hessenberg form by nag_dgehrd (f08nec). The program then calls nag_dorghr (f08nfc) to form $Q$, and passes this matrix to nag_dhseqr (f08pec) which computes the Schur factorization of $A$.

### 10.1  Program Text

Program Text (f08nfce.c)

### 10.2  Program Data

Program Data (f08nfce.d)

### 10.3  Program Results

Program Results (f08nfce.r)