f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zunmtr (f08fuc)

## 1  Purpose

nag_zunmtr (f08fuc) multiplies an arbitrary complex matrix $C$ by the complex unitary matrix $Q$ which was determined by nag_zhetrd (f08fsc) when reducing a complex Hermitian matrix to tridiagonal form.

## 2  Specification

 #include #include
 void nag_zunmtr (Nag_OrderType order, Nag_SideType side, Nag_UploType uplo, Nag_TransType trans, Integer m, Integer n, const Complex a[], Integer pda, const Complex tau[], Complex c[], Integer pdc, NagError *fail)

## 3  Description

nag_zunmtr (f08fuc) is intended to be used after a call to nag_zhetrd (f08fsc), which reduces a complex Hermitian matrix $A$ to real symmetric tridiagonal form $T$ by a unitary similarity transformation: $A=QT{Q}^{\mathrm{H}}$. nag_zhetrd (f08fsc) represents the unitary matrix $Q$ as a product of elementary reflectors.
This function may be used to form one of the matrix products
 $QC , QHC , CQ ​ or ​ CQH ,$
overwriting the result on $C$ (which may be any complex rectangular matrix).
A common application of this function is to transform a matrix $Z$ of eigenvectors of $T$ to the matrix $\mathit{QZ}$ 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 ${Q}^{\mathrm{H}}$ is to be applied to $C$.
${\mathbf{side}}=\mathrm{Nag_LeftSide}$
$Q$ or ${Q}^{\mathrm{H}}$ is applied to $C$ from the left.
${\mathbf{side}}=\mathrm{Nag_RightSide}$
$Q$ or ${Q}^{\mathrm{H}}$ is applied to $C$ from the right.
Constraint: ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_RightSide}$.
3:     uploNag_UploTypeInput
On entry: this must be the same argument uplo as supplied to nag_zhetrd (f08fsc).
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
4:     transNag_TransTypeInput
On entry: indicates whether $Q$ or ${Q}^{\mathrm{H}}$ is to be applied to $C$.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$Q$ is applied to $C$.
${\mathbf{trans}}=\mathrm{Nag_ConjTrans}$
${Q}^{\mathrm{H}}$ is applied to $C$.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ or $\mathrm{Nag_ConjTrans}$.
5:     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: ${\mathbf{m}}\ge 0$.
6:     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: ${\mathbf{n}}\ge 0$.
7:     a[$\mathit{dim}$]const ComplexInput
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_zhetrd (f08fsc).
8:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ 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 ${\mathbf{side}}=\mathrm{Nag_RightSide}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9:     tau[$\mathit{dim}$]const ComplexInput
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_zhetrd (f08fsc).
10:   c[$\mathit{dim}$]ComplexInput/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 ${Q}^{\mathrm{H}}C$ or $CQ$ or $C{Q}^{\mathrm{H}}$ as specified by side and trans.
11:   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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
12:   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 ${\mathbf{side}}=\mathrm{Nag_RightSide}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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_zunmtr (f08fuc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zunmtr (f08fuc) 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 real floating-point operations is approximately $8{m}^{2}n$ if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ and $8m{n}^{2}$ if ${\mathbf{side}}=\mathrm{Nag_RightSide}$.
The real analogue of this function is nag_dormtr (f08fgc).

## 10  Example

This example computes the two smallest eigenvalues, and the associated eigenvectors, of the matrix $A$, where
 $A = -2.28+0.00i 1.78-2.03i 2.26+0.10i -0.12+2.53i 1.78+2.03i -1.12+0.00i 0.01+0.43i -1.07+0.86i 2.26-0.10i 0.01-0.43i -0.37+0.00i 2.31-0.92i -0.12-2.53i -1.07-0.86i 2.31+0.92i -0.73+0.00i .$
Here $A$ is Hermitian and must first be reduced to tridiagonal form $T$ by nag_zhetrd (f08fsc). The program then calls nag_dstebz (f08jjc) to compute the requested eigenvalues and nag_zstein (f08jxc) to compute the associated eigenvectors of $T$. Finally nag_zunmtr (f08fuc) is called to transform the eigenvectors to those of $A$.

### 10.1  Program Text

Program Text (f08fuce.c)

### 10.2  Program Data

Program Data (f08fuce.d)

### 10.3  Program Results

Program Results (f08fuce.r)