f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zhetrd (f08fsc)

## 1  Purpose

nag_zhetrd (f08fsc) reduces a complex Hermitian matrix to tridiagonal form.

## 2  Specification

 #include #include
 void nag_zhetrd (Nag_OrderType order, Nag_UploType uplo, Integer n, Complex a[], Integer pda, double d[], double e[], Complex tau[], NagError *fail)

## 3  Description

nag_zhetrd (f08fsc) 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). Functions are provided to work with $Q$ in this representation (see Section 9).

## 4  References

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

## 5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:     uploNag_UploTypeInput
On entry: indicates whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
The upper triangular part of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     a[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$.
On entry: the $n$ by $n$ Hermitian matrix $A$.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${A}_{ij}$ is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${A}_{ij}$ is stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$.
If ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, 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.
5:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6:     d[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, 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 $T$.
7:     e[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, 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 $T$.
8:     tau[$\mathit{dim}$]ComplexOutput
Note: the dimension, dim, 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 $Q$.
9:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

## 7  Accuracy

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.

## 8  Parallelism and Performance

nag_zhetrd (f08fsc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zhetrd (f08fsc) 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.

The total number of real floating-point operations is approximately $\frac{16}{3}{n}^{3}$.
To form the unitary matrix $Q$ nag_zhetrd (f08fsc) may be followed by a call to nag_zungtr (f08ftc):
```nag_zungtr(order,uplo,n,&a,pda,tau,&fail)
```
To apply $Q$ to an $n$ by $p$ complex matrix $C$ nag_zhetrd (f08fsc) may be followed by a call to nag_zunmtr (f08fuc). For example,
```nag_zunmtr(order,Nag_LeftSide,uplo,Nag_NoTrans,n,p,&a,pda,
tau,&c,pdc,&fail)
```
forms the matrix product $QC$.
The real analogue of this function is nag_dsytrd (f08fec).

## 10  Example

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

### 10.1  Program Text

Program Text (f08fsce.c)

### 10.2  Program Data

Program Data (f08fsce.d)

### 10.3  Program Results

Program Results (f08fsce.r)