F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08FTF (ZUNGTR)

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

F08FTF (ZUNGTR) generates the complex unitary matrix $Q$, which was determined by F08FSF (ZHETRD) when reducing a Hermitian matrix to tridiagonal form.

## 2  Specification

 SUBROUTINE F08FTF ( UPLO, N, A, LDA, TAU, WORK, LWORK, INFO)
 INTEGER N, LDA, LWORK, INFO COMPLEX (KIND=nag_wp) A(LDA,*), TAU(*), WORK(max(1,LWORK)) CHARACTER(1) UPLO
The routine may be called by its LAPACK name zungtr.

## 3  Description

F08FTF (ZUNGTR) is intended to be used after a call to F08FSF (ZHETRD), which reduces a complex Hermitian matrix $A$ to real symmetric tridiagonal form $T$ by a unitary similarity transformation: $A=QT{Q}^{\mathrm{H}}$. F08FSF (ZHETRD) 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.

## 4  References

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 F08FSF (ZHETRD).
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:     A(LDA,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: details of the vectors which define the elementary reflectors, as returned by F08FSF (ZHETRD).
On exit: the $n$ by $n$ unitary matrix $Q$.
4:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08FTF (ZUNGTR) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
5:     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 F08FSF (ZHETRD).
6:     WORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)$) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, the real part of ${\mathbf{WORK}}\left(1\right)$ contains the minimum value of LWORK required for optimal performance.
7:     LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08FTF (ZUNGTR) is called.
If ${\mathbf{LWORK}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued.
Suggested value: for optimal performance, ${\mathbf{LWORK}}\ge \left({\mathbf{N}}-1\right)×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}-1\right)$ or ${\mathbf{LWORK}}=-1$.
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 F08FFF (DORGTR).

## 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 F08FSF (ZHETRD). The program then calls F08FTF (ZUNGTR) to form $Q$, and passes this matrix to F08JSF (ZSTEQR) which computes the eigenvalues and eigenvectors of $A$.

### 9.1  Program Text

Program Text (f08ftfe.f90)

### 9.2  Program Data

Program Data (f08ftfe.d)

### 9.3  Program Results

Program Results (f08ftfe.r)