Numerical Methods and Optimization

80 Numerical Methods and Optimization: An Introduction

we scale x

(k)

at each iteration:

v

(k)

=

x

(k)

x

(k)



2

,x

(k+1)

= Av

(k)

,k≥ 0. (3.11)

Also,

μ

(k)

=

(v

(k)

)

T

Av

(k)

(v

(k)

)

T

v

(k)

=(v

(k)

)

T

x

(k+1)

. (3.12)

In summary, starting with an initial guess x

(0)

, we proceed by computing

{v

(k)

: k ≥ 1} and {μ

(k)

: k ≥ 1} using (3.11)–(3.12) to find approximations

of v

1

and λ

1

, respectively.

3.5.2 Application: ranking methods

A notable application of the power method for computing the dominant

eigenpair of a matrix can be found in ranking methods. We discuss two exam-

ples, the celebrated PageRank algorithm and the Analytic Hierarchy Process.

PageRank algorithm. The PageRank method w