f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_zungtr (f08ftc)

## 1  Purpose

nag_zungtr (f08ftc) generates the complex unitary matrix $Q$, which was determined by nag_zhetrd (f08fsc) when reducing a Hermitian matrix to tridiagonal form.

## 2  Specification

 #include #include
 void nag_zungtr (Nag_OrderType order, Nag_UploType uplo, Integer n, Complex a[], Integer pda, const Complex tau[], NagError *fail)

## 3  Description

nag_zungtr (f08ftc) 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 $n-1$ elementary reflectors.
This function may be used to generate $Q$ explicitly as a square matrix.

## 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:     uploNag_UploTypeInput
On entry: this must be the same argument uplo as supplied to nag_zhetrd (f08fsc).
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:     nIntegerInput
On entry: $n$, the order of the matrix $Q$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     a[$\mathit{dim}$]ComplexInput/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_zhetrd (f08fsc).
On exit: the $n$ by $n$ unitary 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 ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$.
5:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6:     tau[$\mathit{dim}$]const ComplexInput
Note: the dimension, dim, of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: further details of the elementary reflectors, as returned by nag_zhetrd (f08fsc).
7:     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_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 unitary matrix by a matrix $E$ such that
 $E2 = Oε ,$
where $\epsilon$ is the machine precision.

The total number of real floating point operations is approximately $\frac{16}{3}{n}^{3}$.
The real analogue of this function is nag_dorgtr (f08ffc).

## 9  Example

This example computes all the eigenvalues and 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 by nag_zhetrd (f08fsc). The program then calls nag_zungtr (f08ftc) to form $Q$, and passes this matrix to nag_zsteqr (f08jsc) which computes the eigenvalues and eigenvectors of $A$.

### 9.1  Program Text

Program Text (f08ftce.c)

### 9.2  Program Data

Program Data (f08ftce.d)

### 9.3  Program Results

Program Results (f08ftce.r)