# NAG Library Routine Document

## 1Purpose

f08fsf (zhetrd) reduces a complex Hermitian matrix to tridiagonal form.

## 2Specification

Fortran Interface
 Subroutine f08fsf ( uplo, n, a, lda, d, e, tau, work, info)
 Integer, Intent (In) :: n, lda, lwork Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: d(*), e(*) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), tau(*) Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: uplo
#include <nagmk26.h>
 void f08fsf_ (const char *uplo, const Integer *n, Complex a[], const Integer *lda, double d[], double e[], Complex tau[], Complex work[], const Integer *lwork, Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name zhetrd.

## 3Description

f08fsf (zhetrd) reduces a complex Hermitian matrix $A$ to real symmetric tridiagonal form $T$ by a unitary similarity transformation: $A=QT{Q}^{\mathrm{H}}$.
The matrix $Q$ is not formed explicitly but is represented as a product of $n-1$ elementary reflectors (see the F08 Chapter Introduction for details). Routines are provided to work with $Q$ in this representation (see Section 9).

## 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: indicates whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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: the $n$ by $n$ Hermitian matrix $A$.
• If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: a is overwritten by the tridiagonal matrix $T$ and details of the unitary matrix $Q$ as specified by uplo.
4:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08fsf (zhetrd) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
5:     $\mathbf{d}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: the diagonal elements of the tridiagonal matrix $T$.
6:     $\mathbf{e}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On exit: the off-diagonal elements of the tridiagonal matrix $T$.
7:     $\mathbf{tau}\left(*\right)$ – Complex (Kind=nag_wp) arrayOutput
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 exit: further details of the unitary matrix $Q$.
8:     $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – 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.
9:     $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08fsf (zhetrd) 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, ${\mathbf{lwork}}\ge {\mathbf{n}}×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{lwork}}\ge 1$ or ${\mathbf{lwork}}=-1$.
10:   $\mathbf{info}$ – 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 tridiagonal matrix $T$ is exactly similar to a nearby matrix $\left(A+E\right)$, where
 $E2≤ c n ε A2 ,$
$c\left(n\right)$ is a modestly increasing function of $n$, and $\epsilon$ is the machine precision.
The elements of $T$ themselves may be sensitive to small perturbations in $A$ or to rounding errors in the computation, but this does not affect the stability of the eigenvalues and eigenvectors.

## 8Parallelism and Performance

f08fsf (zhetrd) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08fsf (zhetrd) 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 $\frac{16}{3}{n}^{3}$.
To form the unitary matrix $Q$ f08fsf (zhetrd) may be followed by a call to f08ftf (zungtr):
```Call zungtr(uplo,n,a,lda,tau,work,lwork,info)
```
To apply $Q$ to an $n$ by $p$ complex matrix $C$ f08fsf (zhetrd) may be followed by a call to f08fuf (zunmtr). For example,
```Call zunmtr('Left',uplo,'No Transpose',n,p,a,lda,tau,c,ldc, &
work,lwork,info)```
forms the matrix product $QC$.
The real analogue of this routine is f08fef (dsytrd).

## 10Example

This example reduces the matrix $A$ to tridiagonal form, where
 $A = -2.28+0.00i 1.78-2.03i 2.26+0.10i -0.12+2.53i 1.78+2.03i -1.12+0.00i 0.01+0.43i -1.07+0.86i 2.26-0.10i 0.01-0.43i -0.37+0.00i 2.31-0.92i -0.12-2.53i -1.07-0.86i 2.31+0.92i -0.73+0.00i .$

### 10.1Program Text

Program Text (f08fsfe.f90)

### 10.2Program Data

Program Data (f08fsfe.d)

### 10.3Program Results

Program Results (f08fsfe.r)