# NAG C Library Function Document

## 1Purpose

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

## 2Specification

 #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)

## 3Description

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.

## 4References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1:    $\mathbf{order}$Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{uplo}$Nag_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:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $Q$.
Constraint: ${\mathbf{n}}\ge 0$.
4:    $\mathbf{ap}\left[\mathit{dim}\right]$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:    $\mathbf{tau}\left[\mathit{dim}\right]$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:    $\mathbf{q}\left[\mathit{dim}\right]$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:    $\mathbf{pdq}$IntegerInput
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:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The computed matrix $Q$ differs from an exactly unitary matrix by a matrix $E$ such that
 $E2 = Oε ,$
where $\epsilon$ is the machine precision.

## 8Parallelism 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.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

## 9Further Comments

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).

## 10Example

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 $A$.

### 10.1Program Text

Program Text (f08gtce.c)

### 10.2Program Data

Program Data (f08gtce.d)

### 10.3Program Results

Program Results (f08gtce.r)