# NAG Library Routine Document

## 1Purpose

f08gtf (zupgtr) generates the complex unitary matrix $Q$, which was determined by f08gsf (zhptrd) when reducing a Hermitian matrix to tridiagonal form.

## 2Specification

Fortran Interface
 Subroutine f08gtf ( uplo, n, ap, tau, q, ldq, work, info)
 Integer, Intent (In) :: n, ldq Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (In) :: ap(*), tau(*) Complex (Kind=nag_wp), Intent (Inout) :: q(ldq,*) Complex (Kind=nag_wp), Intent (Out) :: work(n-1) Character (1), Intent (In) :: uplo
#include nagmk26.h
 void f08gtf_ (const char *uplo, const Integer *n, const Complex ap[], const Complex tau[], Complex q[], const Integer *ldq, Complex work[], Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name zupgtr.

## 3Description

f08gtf (zupgtr) is intended to be used after a call to f08gsf (zhptrd), which reduces a complex Hermitian matrix $A$ to real symmetric tridiagonal form $T$ by a unitary similarity transformation: $A=QT{Q}^{\mathrm{H}}$. f08gsf (zhptrd) represents the unitary matrix $Q$ as a product of $n-1$ elementary reflectors.
This routine 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:     $\mathbf{uplo}$ – Character(1)Input
On entry: this must be the same argument uplo as supplied to f08gsf (zhptrd).
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $Q$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{ap}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the dimension 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 f08gsf (zhptrd).
4:     $\mathbf{tau}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the dimension 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 f08gsf (zhptrd).
5:     $\mathbf{q}\left({\mathbf{ldq}},*\right)$ – Complex (Kind=nag_wp) arrayOutput
Note: the second dimension of the array q must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: the $n$ by $n$ unitary matrix $Q$.
6:     $\mathbf{ldq}$ – IntegerInput
On entry: the first dimension of the array q as declared in the (sub)program from which f08gtf (zupgtr) is called.
Constraint: ${\mathbf{ldq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7:     $\mathbf{work}\left({\mathbf{n}}-1\right)$ – Complex (Kind=nag_wp) arrayWorkspace
8:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 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

f08gtf (zupgtr) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08gtf (zupgtr) 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of real floating-point operations is approximately $\frac{16}{3}{n}^{3}$.
The real analogue of this routine is f08gff (dopgtr).

## 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 f08gsf (zhptrd). The program then calls f08gtf (zupgtr) to form $Q$, and passes this matrix to f08jsf (zsteqr) which computes the eigenvalues and eigenvectors of $A$.

### 10.1Program Text

Program Text (f08gtfe.f90)

### 10.2Program Data

Program Data (f08gtfe.d)

### 10.3Program Results

Program Results (f08gtfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017