# NAG FL Interfacef08ftf (zungtr)

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## 1Purpose

f08ftf generates the complex unitary matrix $Q$, which was determined by f08fsf when reducing a Hermitian matrix to tridiagonal form.

## 2Specification

Fortran Interface
 Subroutine f08ftf ( uplo, n, a, lda, tau, work, info)
 Integer, Intent (In) :: n, lda, lwork Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (In) :: tau(*) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*) Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: uplo
#include <nag.h>
 void f08ftf_ (const char *uplo, const Integer *n, Complex a[], const Integer *lda, const Complex tau[], Complex work[], const Integer *lwork, Integer *info, const Charlen length_uplo)
The routine may be called by the names f08ftf, nagf_lapackeig_zungtr or its LAPACK name zungtr.

## 3Description

f08ftf is intended to be used after a call to f08fsf, which reduces a complex Hermitian matrix $A$ to real symmetric tridiagonal form $T$ by a unitary similarity transformation: $A=QT{Q}^{\mathrm{H}}$. f08fsf 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 f08fsf.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $Q$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{a}\left({\mathbf{lda}},*\right)$Complex (Kind=nag_wp) array Input/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.
On exit: the $n×n$ unitary matrix $Q$.
4: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08ftf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
5: $\mathbf{tau}\left(*\right)$Complex (Kind=nag_wp) array Input
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.
6: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Complex (Kind=nag_wp) array Workspace
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: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08ftf 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: $\mathbf{info}$Integer Output
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
 $‖E‖2 = O(ε) ,$
where $\epsilon$ is the machine precision.

## 8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08ftf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08ftf 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 f08fff.

## 10Example

This example computes all the eigenvalues and eigenvectors of the matrix $A$, where
 $A = ( -2.28+0.00i 1.78-2.03i 2.26+0.10i -0.12+2.53i 1.78+2.03i -1.12+0.00i 0.01+0.43i -1.07+0.86i 2.26-0.10i 0.01-0.43i -0.37+0.00i 2.31-0.92i -0.12-2.53i -1.07-0.86i 2.31+0.92i -0.73+0.00i ) .$
Here $A$ is Hermitian and must first be reduced to tridiagonal form by f08fsf. The program then calls f08ftf to form $Q$, and passes this matrix to f08jsf which computes the eigenvalues and eigenvectors of $A$.

### 10.1Program Text

Program Text (f08ftfe.f90)

### 10.2Program Data

Program Data (f08ftfe.d)

### 10.3Program Results

Program Results (f08ftfe.r)