f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_dormhr (f08ngc)

## 1  Purpose

nag_dormhr (f08ngc) multiplies an arbitrary real matrix $C$ by the real orthogonal matrix $Q$ which was determined by nag_dgehrd (f08nec) when reducing a real general matrix to Hessenberg form.

## 2  Specification

 #include #include
 void nag_dormhr (Nag_OrderType order, Nag_SideType side, Nag_TransType trans, Integer m, Integer n, Integer ilo, Integer ihi, const double a[], Integer pda, const double tau[], double c[], Integer pdc, NagError *fail)

## 3  Description

nag_dormhr (f08ngc) 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 form one of the matrix products
 $QC , QTC , CQ ​ or ​ CQT ,$
overwriting the result on $C$ (which may be any real rectangular matrix).
A common application of this function is to transform a matrix $V$ of eigenvectors of $H$ to the matrix $\mathit{QV}$ of eigenvectors of $A$.

## 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:     sideNag_SideTypeInput
On entry: indicates how $Q$ or ${Q}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{side}}=\mathrm{Nag_LeftSide}$
$Q$ or ${Q}^{\mathrm{T}}$ is applied to $C$ from the left.
${\mathbf{side}}=\mathrm{Nag_RightSide}$
$Q$ or ${Q}^{\mathrm{T}}$ is applied to $C$ from the right.
Constraint: ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_RightSide}$.
3:     transNag_TransTypeInput
On entry: indicates whether $Q$ or ${Q}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$Q$ is applied to $C$.
${\mathbf{trans}}=\mathrm{Nag_Trans}$
${Q}^{\mathrm{T}}$ is applied to $C$.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ or $\mathrm{Nag_Trans}$.
4:     mIntegerInput
On entry: $m$, the number of rows of the matrix $C$; $m$ is also the order of $Q$ if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$.
Constraint: ${\mathbf{m}}\ge 0$.
5:     nIntegerInput
On entry: $n$, the number of columns of the matrix $C$; $n$ is also the order of $Q$ if ${\mathbf{side}}=\mathrm{Nag_RightSide}$.
Constraint: ${\mathbf{n}}\ge 0$.
6:     iloIntegerInput
7:     ihiIntegerInput
On entry: these must be the same arguments ilo and ihi, respectively, as supplied to nag_dgehrd (f08nec).
Constraints:
• if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ and ${\mathbf{m}}>0$, $1\le {\mathbf{ilo}}\le {\mathbf{ihi}}\le {\mathbf{m}}$;
• if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ and ${\mathbf{m}}=0$, ${\mathbf{ilo}}=1$ and ${\mathbf{ihi}}=0$;
• if ${\mathbf{side}}=\mathrm{Nag_RightSide}$ and ${\mathbf{n}}>0$, $1\le {\mathbf{ilo}}\le {\mathbf{ihi}}\le {\mathbf{n}}$;
• if ${\mathbf{side}}=\mathrm{Nag_RightSide}$ and ${\mathbf{n}}=0$, ${\mathbf{ilo}}=1$ and ${\mathbf{ihi}}=0$.
8:     a[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array a must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{m}}\right)$ when ${\mathbf{side}}=\mathrm{Nag_LeftSide}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$ when ${\mathbf{side}}=\mathrm{Nag_RightSide}$.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_dgehrd (f08nec).
9:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
• if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{side}}=\mathrm{Nag_RightSide}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10:   tau[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array tau must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}-1\right)$ when ${\mathbf{side}}=\mathrm{Nag_LeftSide}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$ when ${\mathbf{side}}=\mathrm{Nag_RightSide}$.
On entry: further details of the elementary reflectors, as returned by nag_dgehrd (f08nec).
11:   c[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array c must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdc}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pdc}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $C$ is stored in
• ${\mathbf{c}}\left[\left(j-1\right)×{\mathbf{pdc}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{c}}\left[\left(i-1\right)×{\mathbf{pdc}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $m$ by $n$ matrix $C$.
On exit: c is overwritten by $QC$ or ${Q}^{\mathrm{T}}C$ or $CQ$ or $C{Q}^{\mathrm{T}}$ as specified by side and trans.
12:   pdcIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
13:   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_ENUM_INT_3
On entry, ${\mathbf{side}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
if ${\mathbf{side}}=\mathrm{Nag_RightSide}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{side}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
if ${\mathbf{side}}=\mathrm{Nag_RightSide}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_ENUM_INT_4
On entry, ${\mathbf{side}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{ilo}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{ihi}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ and ${\mathbf{m}}>0$, $1\le {\mathbf{ilo}}\le {\mathbf{ihi}}\le {\mathbf{m}}$;
if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ and ${\mathbf{m}}=0$, ${\mathbf{ilo}}=1$ and ${\mathbf{ihi}}=0$;
if ${\mathbf{side}}=\mathrm{Nag_RightSide}$ and ${\mathbf{n}}>0$, $1\le {\mathbf{ilo}}\le {\mathbf{ihi}}\le {\mathbf{n}}$;
if ${\mathbf{side}}=\mathrm{Nag_RightSide}$ and ${\mathbf{n}}=0$, ${\mathbf{ilo}}=1$ and ${\mathbf{ihi}}=0$.
NE_INT
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdc}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdc}}>0$.
NE_INT_2
On entry, ${\mathbf{pdc}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pdc}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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 result differs from the exact result by a matrix $E$ such that
 $E2 = Oε C2 ,$
where $\epsilon$ is the machine precision.

## 8  Parallelism and Performance

nag_dormhr (f08ngc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dormhr (f08ngc) 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 $2n{q}^{2}$ if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ and $2m{q}^{2}$ if ${\mathbf{side}}=\mathrm{Nag_RightSide}$, where $q={i}_{\mathrm{hi}}-{i}_{\mathrm{lo}}$.
The complex analogue of this function is nag_zunmhr (f08nuc).

## 10  Example

This example computes all the eigenvalues of the matrix $A$, where
 $A = 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 ,$
and those eigenvectors which correspond to eigenvalues $\lambda$ such that $\mathrm{Re}\left(\lambda \right)<0$. Here $A$ is general and must first be reduced to upper Hessenberg form $H$ by nag_dgehrd (f08nec). The program then calls nag_dhseqr (f08pec) to compute the eigenvalues, and nag_dhsein (f08pkc) to compute the required eigenvectors of $H$ by inverse iteration. Finally nag_dormhr (f08ngc) is called to transform the eigenvectors of $H$ back to eigenvectors of the original matrix $A$.

### 10.1  Program Text

Program Text (f08ngce.c)

### 10.2  Program Data

Program Data (f08ngce.d)

### 10.3  Program Results

Program Results (f08ngce.r)