F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08GTF (ZUPGTR)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08GTF (ZUPGTR) generates the complex unitary matrix $Q$, which was determined by F08GSF (ZHPTRD) when reducing a Hermitian matrix to tridiagonal form.

## 2  Specification

 SUBROUTINE F08GTF ( UPLO, N, AP, TAU, Q, LDQ, WORK, INFO)
 INTEGER N, LDQ, INFO COMPLEX (KIND=nag_wp) AP(*), TAU(*), Q(LDQ,*), WORK(N-1) CHARACTER(1) UPLO
The routine may be called by its LAPACK name zupgtr.

## 3  Description

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

## 5  Parameters

1:     UPLO – CHARACTER(1)Input
On entry: this must be the same parameter UPLO as supplied to F08GSF (ZHPTRD).
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
2:     N – INTEGERInput
On entry: $n$, the order of the matrix $Q$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     AP($*$) – 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:     TAU($*$) – 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:     Q(LDQ,$*$) – 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:     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:     WORK(${\mathbf{N}}-1$) – COMPLEX (KIND=nag_wp) arrayWorkspace
8:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 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 routine is F08GFF (DOPGTR).

## 9  Example

This example computes all the eigenvalues and eigenvectors of the matrix $A$, where
 $A = -2.28+0.00i 1.78-2.03i 2.26+0.10i -0.12+2.53i 1.78+2.03i -1.12+0.00i 0.01+0.43i -1.07+0.86i 2.26-0.10i 0.01-0.43i -0.37+0.00i 2.31-0.92i -0.12-2.53i -1.07-0.86i 2.31+0.92i -0.73+0.00i ,$
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 $A$.

### 9.1  Program Text

Program Text (f08gtfe.f90)

### 9.2  Program Data

Program Data (f08gtfe.d)

### 9.3  Program Results

Program Results (f08gtfe.r)
{"ft_lang_label":"__label__en","ft_lang_prob":0.71836734,"math_prob":0.99611473,"size":2833,"snap":"2023-40-2023-50","text_gpt3_token_len":845,"char_repetition_ratio":0.11523507,"word_repetition_ratio":0.08438819,"special_character_ratio":0.2573244,"punctuation_ratio":0.13872832,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9981001,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-11-30T00:37:28Z\",\"WARC-Record-ID\":\"<urn:uuid:47a4349c-1bbb-4303-884d-58e36745fa7d>\",\"Content-Length\":\"16173\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:637693d7-23e4-4a8c-9f7a-2c9ccb606237>\",\"WARC-Concurrent-To\":\"<urn:uuid:70745008-b1da-4aee-8d77-60f09bafe0e9>\",\"WARC-IP-Address\":\"78.129.168.4\",\"WARC-Target-URI\":\"https://support.nag.com/numeric/nl/nagdoc_24/nagdoc_fl24/html/f08/f08gtf.html\",\"WARC-Payload-Digest\":\"sha1:Z4G5QTKD2DQHAP3VMRDPDDDLFDTW3RTR\",\"WARC-Block-Digest\":\"sha1:4DDGD2A5S3LXLIX4VT5NL6JZGNP35OZU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100164.15_warc_CC-MAIN-20231130000127-20231130030127-00359.warc.gz\"}"}