f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zupgtr (f08gtc)

## 1  Purpose

nag_zupgtr (f08gtc) generates the complex unitary matrix $Q$, which was determined by nag_zhptrd (f08gsc) when reducing a Hermitian matrix to tridiagonal form.

## 2  Specification

 #include #include
 void nag_zupgtr (Nag_OrderType order, Nag_UploType uplo, Integer n, const Complex ap[], const Complex tau[], Complex q[], Integer pdq, NagError *fail)

## 3  Description

nag_zupgtr (f08gtc) is intended to be used after a call to nag_zhptrd (f08gsc), which reduces a complex Hermitian matrix $A$ to real symmetric tridiagonal form $T$ by a unitary similarity transformation: $A=QT{Q}^{\mathrm{H}}$. nag_zhptrd (f08gsc) 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 $\mathrm{Nag_ColMajor}$.
2:     uploNag_UploTypeInput
On entry: this must be the same argument uplo as supplied to nag_zhptrd (f08gsc).
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:     ap[$\mathit{dim}$]const ComplexInput
Note: the dimension, dim, of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_zhptrd (f08gsc).
5:     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_zhptrd (f08gsc).
6:     q[$\mathit{dim}$]ComplexOutput
Note: the dimension, dim, of the array q must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdq}}×{\mathbf{n}}\right)$.
The $\left(i,j\right)$th element of the matrix $Q$ is stored in
• ${\mathbf{q}}\left[\left(j-1\right)×{\mathbf{pdq}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{q}}\left[\left(i-1\right)×{\mathbf{pdq}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $n$ by $n$ unitary matrix $Q$.
7:     pdqIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraint: ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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, ${\mathbf{pdq}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdq}}>0$.
NE_INT_2
On entry, ${\mathbf{pdq}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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.

## 8  Parallelism and Performance

nag_zupgtr (f08gtc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zupgtr (f08gtc) 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 $\frac{16}{3}{n}^{3}$.
The real analogue of this function is nag_dopgtr (f08gfc).

## 10  Example

This example computes all the eigenvalues and eigenvectors of the matrix $A$, where
 $A = -2.28+0.00i 1.78-2.03i 2.26+0.10i -0.12+2.53i 1.78+2.03i -1.12+0.00i 0.01+0.43i -1.07+0.86i 2.26-0.10i 0.01-0.43i -0.37+0.00i 2.31-0.92i -0.12-2.53i -1.07-0.86i 2.31+0.92i -0.73+0.00i ,$
using packed storage. Here $A$ is Hermitian and must first be reduced to tridiagonal form by nag_zhptrd (f08gsc). The program then calls nag_zupgtr (f08gtc) to form $Q$, and passes this matrix to nag_zsteqr (f08jsc) which computes the eigenvalues and eigenvectors of $A$.

### 10.1  Program Text

Program Text (f08gtce.c)

### 10.2  Program Data

Program Data (f08gtce.d)

### 10.3  Program Results

Program Results (f08gtce.r)