C++ for Scientists - Technische Universität Dresden

276 CHAPTER 14. NUMERICAL EXERCISES
14.4.2 Dictionary application
The page ranking algorithm which was described above can also be used to rank different words
in a dictionary. Consider the following small dictionary:
backwoods = bush, jungle
bush = backwoods, jungle, shrub, plant, hedge
flower = plant
hedge = bush
jungle = bush, backwoods
plant = bush, shrub, flower, weed
shrub = bush, plant, tree
tree = shrub
weed = plant
Construct a graph linking every word with the words in its explanation. The first line of the
dictionary, for example, would link bush and jungle to backwoods. The graph can be constructed
on paper. Use equation (14.4) to construct the sparse column stochastic matrix A and use your
power method to rank the words.
o of a function in an interval
In this exercise, we make a programming exercise on a root finding method, called the bisection
method.
14.5.1 Functions in one variable
Suppose we are given a function f in one variable and we want to compute the unique zero
in the interval [a, b]. A method that could be used is the bisection method. It only requires
function evaluations and is thus widely applicable.
The method computes a small interval that contains the minimum. This small interval is
obtained by splitting the interval [a, b] in two parts [a, m] and [m, b], where
The method works as follows :
1
2
3
4
5
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m =
a + b
2
1. Given the interval [a, b] for which f(a)f(b) < 0.
2. Repeat until b − a < τ:
2.1. Compute m from (14.5).
2.2. If f(m)f(a) < 0: b = m.
2.2. Else: a = m.
(14.5)