nag_hermitian_eigenvalues (f02awc) (PDF version)
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NAG Library Manual

# NAG Library Function Documentnag_hermitian_eigenvalues (f02awc)

## 1  Purpose

nag_hermitian_eigenvalues (f02awc) calculates all the eigenvalues of a complex Hermitian matrix.

## 2  Specification

 #include #include
 void nag_hermitian_eigenvalues (Integer n, Complex a[], Integer tda, double r[], NagError *fail)

## 3  Description

The complex Hermitian matrix $A$ is first reduced to a real tridiagonal matrix by $n-2$ unitary transformations, and a subsequent diagonal transformation. The eigenvalues are then derived using the $QL$ algorithm, an adaptation of the $QR$ algorithm.
Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

## 5  Arguments

1:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
2:     a[${\mathbf{n}}×{\mathbf{tda}}$]ComplexInput/Output
Note: the $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{tda}}+j-1\right]$.
On entry: the elements of the lower triangle of the $n$ by $n$ complex Hermitian matrix $A$. Elements of the array above the diagonal need not be set.
On exit: a is overwritten.
3:     tdaIntegerInput
On entry: the stride separating matrix column elements in the array a.
Constraint: ${\mathbf{tda}}\ge {\mathbf{n}}$.
4:     r[n]doubleOutput
On exit: the eigenvalues in ascending order.
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_2_INT_ARG_LT
On entry, ${\mathbf{tda}}=⟨\mathit{\text{value}}⟩$ while ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$. These arguments must satisfy ${\mathbf{tda}}\ge {\mathbf{n}}$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_INT_ARG_LT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_TOO_MANY_ITERATIONS
More than $⟨\mathit{\text{value}}⟩$ iterations are required to isolate all the eigenvalues.

## 7  Accuracy

For a detailed error analysis see page 235 of Wilkinson and Reinsch (1971).

Not applicable.

## 9  Further Comments

The time taken by nag_hermitian_eigenvalues (f02awc) is approximately proportional to ${n}^{3}$.

## 10  Example

To calculate all the eigenvalues of the complex Hermitian matrix:
 $0.50 -1.38 i - 0.00 +0.84 i 1.84 + 1.38 i - 2.08 - 1.56 i 0.00 -1.38 i - 0.50 +0.84 i - 1.12 + 0.84 i -0.56 + 0.42 i 1.84 - 1.38 i - 1.12 - 0.84 i 0.50 +0.84 i - 0.00 +0.84 i 2.08 + 1.56 i -0.56 - 0.42 i 0.00 +0.84 i - 0.50 +0.84 i .$

### 10.1  Program Text

Program Text (f02awce.c)

### 10.2  Program Data

Program Data (f02awce.d)

### 10.3  Program Results

Program Results (f02awce.r)

nag_hermitian_eigenvalues (f02awc) (PDF version)
f02 Chapter Contents
f02 Chapter Introduction
NAG Library Manual