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A.3 Exercises

625

b) Demonstrate the function n on the e equation x6sinπx x = 0, , with

� = 10−13. Trydiﬀerent t starting valuesfor Newton’smethod: x

−2.6,−1.2,1.5,1.7,0.6.Compare theresultswiththe exact solutions

x= ...,−2− 1,0,1,2,....

c) UsetheNewton_plotfunctiontoexploretheimpactofthestarting

point x

when solving the following nonlinear algebraic equations:

sin x = 0,

(A.50)

x= sinx,

(A.51)

=sinx,

(A.52)

sin x = 0,

(A.53)

=16,

(A.54)

=1,

(A.55)

tanh x = 0.tanhx

(A.56)

Hint. Suchanexperimentalinvestigationisconvenientlyrecordedin

an IPython notebook. See Section H.1.9 for a quick introduction to

notebooks.

Filename: Newton2.py.

Exercise A.10: Implement the secant method

Newton’s method (A.34) for solvingf(x) = 0 requires the derivative

of the functionf(x). Sometimes this is diﬃcult or inconvenient. The

derivative can be approximated using the last two approximations to the

root, x

n−2

and x

n−1

�

n−1

)≈

f(x

n−1

)− f(x

n−2

)

n−1

−x

n−2

Using this approximation in (A.34) leads to the Secant method:

n−1

−

f(x

n−1

)(x

n−1

−x

n−2

)

f(x

n−1

)− f(x

n−2

)

, x

given.

(A.57)

Heren = 2,3,.... Make a program that applies the Secant method to

solve x5 = sinx. Filename: Secant.py.

Exercise A.11: Test diﬀerent methods for root ﬁnding

Make a program for solving g f(x) = 0 by Newton’s method (Sec-

tion A.1.10), the Bisection method (Section 4.10.2), and the Secant

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626

A Sequences anddiﬀerence equations

method (Exercise A.10). For each method, the sequence of root ap-

proximations should be written out (nicely formatted) on the screen.

Readf(x),f�(x),a,b,x

,andx

from the command line. Newton’s

method starts withx

,the Bisection method starts with the interval

[a,b], whereas the Secant method starts with x

and x

Run the program for each of the equations listed in Exercise A.9d.

You should ﬁrst plot thef(x) functions so you know how to choosex

,a, and b in each case. Filename: root_finder_examples.py.

Exercise A.12: Make diﬀerence equations for the Midpoint

rule

Use the ideas of Section A.1.7 to make a similar system of diﬀerence

equations and correspondingimplementation for the Midpoint integration

rule:

�

f(x)dx ≈ h

n�−1

i=0

f(a −

h+ ih),

whereh = (b−a)/n andn counts the number of function evaluations

(i.e., rectangles that approximate the area under the curve). Filename:

diffeq_midpoint.py.

Exercise A.13: Compute the arc length of a curve

Sometimes one wants to measure the length of a curvey =f(x) for

x∈[a,b].Thearclengthfrom f(a)tosomepoint f(x)isdenotedby

s(x) and deﬁned through an integral

s(x) =

�

1+ [f�(ξ)]2dξ.

(A.58)

We can compute s(x) via diﬀerence equations as explained in Sec-

tion A.1.7.

a) MakeaPythonfunctionarclength(f, a, b, n)thatreturnsan

arrays withs(x) values forn uniformly spaced coordinatesx in [a,b].

Heref(x) is the Python implementation of the function that deﬁnes the

curve we want to compute the arc length of.

b)Howcanyouverifythatthearclengthfunctionworkscorrectly?Con-

struct test case(s) and write corresponding test functions for automating

the tests.

Hint. Checktheimplementationforcurveswithknownarclength,e.g.,

asemi-circle and a straight line.

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A.3 Exercises

627

c) Apply the function to

f(x) =

�

−2

√

2π

−4t2

dt, x ∈ [−2,2].

Compute s(x) and plot it together with f(x).

Filename: arclength.py.

Exercise A.14: Find diﬀerence equations for computing sin x

The purpose of this exercise is to derive and implement diﬀerence equa-

tions for computing a Taylor polynomial approximation to sin x:

sin x ≈ S(x;n) =

�n

j=0

(−1)

x2j+1

(2j + 1)!

(A.59)

To computeS(x;n) eﬃciently, write the sum asS(x;n) =

�

j=0

,and

derive a relation between two consecutive terms in the series:

=−

(2j +1)2j

j−1

(A.60)

Introduces

=S(x;j− 1) anda

as the two sequences to compute. We

have s

=0 and a

=x.

a) Formulate the two diﬀerence equations for s

and a

Hint. SectionA.1.8explainshowthistaskandtheassociatedprogram-

ming can be solved for the Taylor polynomial approximation of ex.

b) Implement the system m of diﬀerence equations s in a a function

sin_Taylor(x, n), which h returns s

n+1

and |a

n+1

|. The latter r is

the ﬁrst neglected term in the sum (sinces

n+1

�

j=0

)and may act

as a rough measure of the size of the error in the Taylor polynomial

approximation.

c) Verifytheimplementationbycomputingthe diﬀerenceequations

forn = 2 by hand (or in a separate program) and comparing with the

output from thesin_Taylor function. Automate this comparison in a

test function.

d)Makeatableofs

for variousxandn values to verifythat the accuracy

of a Taylor polynomial improves asn increases andx decreases. Be

aware of the fact thatsine_Taylor(x, n) can give extremely inaccurate

approximations tosinx ifx is not suﬃciently small andn suﬃciently

large.

Filename: sin_Taylor_series_diffeq.py.

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628

A Sequences anddiﬀerence equations

Exercise A.15: Find diﬀerence equations for computing cosx

Solve Exercise A.14 for the Taylor polynomial approximation tocosx.

(The relevant expression for the Taylor series is easily found in a

mathematics textbook or by searching on the Internet.) Filename:

cos_Taylor_series_diffeq.py.

Exercise A.16: Make a guitar-like sound

Given start valuesx

,...,x

, the following diﬀerence equation is

known to create guitar-like sound:

n−p

n−p−1

), n = p+ 1,...,N .

(A.61)

With a sampling rater, the frequency of this sound is given byr/p. Make

aprogram with a functionsolve(x, p) which returns the solution array

xof(A.61).Toinitializethearray x[0:p+1]welookattwomethods,

which can be implemented in two alternative functions:

• x

=1, x

=··· = x

• x

,...,x

are uniformly distributed random numbers in [−1,1]

Import max_amplitude, write, and d play from the scitools.sound

module. Choose a sampling rater and setp =r/440 to create a 440 Hz

tone (A). Create an arrayx1 of zeros with length 3r such that the tone

will last for 3seconds. Initializex1 accordingto method 1above and solve

(A.61). Multiply thex1 array bymax_amplitude. Repeat this process

for an arrayx2 of length 2r, but use method 2 for the initial values and

choose p such that the tone is 392 Hz (G). Concatenate x1 and x2, call

writeandthen play toplaythesound.Asyouwillexperience,this

sound is amazingly similar to the sound of a guitar string, ﬁrst playing

Afor 3 seconds and then playing G for 2 seconds.

The method (A.61) is called the Karplus-Strong algorithm and was

discovered in 1979 by a researcher, Kevin Karplus, and his student

Alexander Strong, at Stanford University. Filename: guitar_sound.py.

Exercise A.17: Damp the bass in a sound ﬁle

Given a sequencex

,...,x

N−1

,the following ﬁlter transforms the se-

quence to a new sequence y

,...,y

N−1











n= 0

−

n−1

−2x

n+1

), 1 ≤ n ≤ N −2

n= N −1

(A.62)

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A.3 Exercises

629

Ifx

represents sound,y

is the same sound but with the bass damped.

Load some sound ﬁle, e.g.,

x = scitools.sound.Nothing_Else_Matters()

# or

x = scitools.sound.Ja_vi_elsker()

to get a sound sequence. Apply the ﬁlter (A.62) and play the resulting

sound. Plot the ﬁrst 300 values in thex

andy

signals to see graphically

what the ﬁlter does with the signal. Filename: damp_bass.py.

Exercise A.18: Damp the treble in a sound ﬁle

Solve Exercise A.17 to get some experience with coding a ﬁlter and

trying it out on a sound. The purpose of this exercise is to explore some

other ﬁlters that reduce the treble instead of the bass. Smoothing the

sound signal will in general damp the treble, and smoothing is typically

obtained by letting the values in the new ﬁltered sound sequence be an

average of the neighboring values in the original sequence.

The simplest smoothing ﬁlter can apply a standard average of three

neighboring values:











n= 0

n−1

n+1

), 1 ≤ n ≤ N −2

n= N −1

(A.63)

Two other ﬁlters put less emphasis on the surrounding values:











n= 0

n−1

+2x

n+1

), 1 ≤ n ≤ N − 2

n= N − 1

(A.64)











n= 0,1

n−2

+4x

n−1

+6x

+4x

n+1

n+2

), 2 ≤ n ≤ N − 3

n= N − 2,N −1

(A.65)

Apply all these three ﬁlters to a sound ﬁle and listen to the result.

Plot the ﬁrst 300 values in thex

andy

signals for each of the three

ﬁlters to see graphically what the ﬁlter does with the signal. Filename:

damp_treble.py.

Exercise A.19: Demonstrate oscillatory solutions of (A.13)

a) Write a program to solve the diﬀerence equation (A.13):

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630

A Sequences anddiﬀerence equations

n−1

+qy

n−1

(1 −y

n−1

), n = 0,...,N .

Read the input parametersy

,q, andN from the command line. The

variables and the equation are explained in Section A.1.5.

b) Equation(A.13)hasthesolution y

=1 asn→∞ . Demonstrate, by

running the program, that this is the case wheny

=0.3,q = 1, and

N= 50.

c) For larger q q values, y

does not approach a constant limit, but

oscillates instead around the limiting value. Such oscillations are

sometimes observed in wildlife populations. Demonstrate oscillatory

solutions when q is changed to 2 and 3.

d) Itcouldhappenthat y

stabilizes at a constant level for largerN.

Demonstrate that this is not the case by running the program with

N= 1000.

Filename: growth_logistic2.py.

Exercise A.20: Automate computer experiments

It is tedious to run a program like the one from Exercise A.19 repeatedly

for a wide range of input parameters. A better approach is to let the

computer do the manual work. Modify the program from Exercise A.19

such that the computation ofy

and the plot is made in a function. Let

the title in the plot contain the parametersy

andq (N is easily visible

from thex axis). Also let the name of the plot ﬁle reﬂect the values of

,q, andN. Then make loops overy

andq to perform the following

more comprehensive set of experiments:

• y

=0.01,0.3

• q = 0.1,1,1.5,1.8,2,2.5,3

• N = 50

How does the initial condition (the value y

) seem to inﬂuence the

solution?

Hint. If you do no want to get a lot of plots on the screen, which must

be killed, drop the call toshow() in Matplotlib or useshow=False as

argument to plot in SciTools.

Filename: growth_logistic3.py.

Exercise A.21: Generate an HTML report

Extend the program made in Exercise A.20 with a report containing all

the plots. The report can be written in HTML and displayed by a web

browser. The plots must then be generated in PNG format. The source

of the HTML ﬁle will typically look as follows:

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A.3 Exercises

631

<html>

<body>

...

</html>

</body>

Let the program write out the HTML text to a ﬁle. You may let the func-

tion makingthe plots return the name of the plot ﬁle such that this string

can be inserted in the HTML ﬁle. Filename: growth_logistic4.py.

Exercise A.22: Use a class to archive and report experiments

The purpose of this exercise is to make the program from Exercise A.21

more ﬂexible by creating a Python class that runs and archives all the

experiments (provided you know how to program with Python classes).

Here is a sketch of the class:

class GrowthLogistic:

def __init__(self, show_plot_on_screen=False):

self.experiments = []

self.show_plot_on_screen = show_plot_on_screen

self.remove_plot_files()

def run_one(self, y0, q, N):

"""Run one experiment."""

# Compute y[n] in a loop...

plotfile = ’tmp_y0_%g_q_%g_N_%d.png’ % (y0, q, N)

self.experiments.append({’y0’: y0, ’q’: q, ’N’: N,

’mean’: mean(y[20:]),

’y’: y, ’plotfile’: plotfile})

# Make plot...

def run_many(self, y0_list, q_list, N):

"""Run many experiments."""

for q in q_list:

for y0 in y0_list:

self.run_one(y0, q, N)

def remove_plot_files(self):

"""Remove plot files with names tmp_y0*.png."""

import os, glob

for plotfile in glob.glob(’tmp_y0*.png’):

os.remove(plotfile)

def report(self, filename=’tmp.html’):

"""

Generate an HTML report with plots of all

experiments generated so far.

"""

# Open file and write HTML header...

for e in self.experiments:

html.write(’<img src="%s">\n’ % e[’plotfile’])

# Write HTML footer and close file...

Each time therun_one method is called, data about the current experi-

ment is stored in theexperiments list. Note thatexperiments contains

632

A Sequences anddiﬀerence equations

alist of dictionaries. When desired, we can call thereport method to

collect all the plots made so far in an HTML report. A typical use of the

class goes as follows:

N = 50

g = GrowthLogistic()

g.run_many(y0_list=[0.01, 0.3],

q_list=[0.1, 1, 1.5, 1.8] + [2, 2.5, 3], N=N)

g.run_one(y0=0.01, q=3, N=1000)

g.report()

Make a complete implementation of classGrowthLogistic and test it

with the small program above. The program ﬁle should be constructed

as a module. Filename: growth_logistic5.py.

Exercise A.23: Explore logistic growth interactively

ClassGrowthLogistic from Exercise A.22 is very well suited for inter-

active exploration. Here is a possible sample session for illustration:

>>> from growth_logistic5 import GrowthLogistic

>>> g = GrowthLogistic(show_plot_on_screen=True)

>>> q = 3

>>> g.run_one(0.01, q, 100)

>>> y = g.experiments[-1][’y’]

>>> max(y)

1.3326056469620293

>>> min(y)

0.0029091569028512065

Extend this session with an investigation of the oscillations in the solution

.For this purpose, make a function for computing the local maximum

valuesy

and the corresponding indices where these local maximum

values occur. We can say that y

is a local maximum value if

i−1

i+1

Plot the sequence of local maximum values in a new plot. IfI

,...

constitute the set of increasing indices corresponding to the local maxi-

mum values, we can deﬁnethe periods of the oscillations asI

−I

and so forth. Plot the length of the periods in a separate plot. Repeat this

investigation for q = 2.5. Filename: GrowthLogistic_interactive.py.

Exercise A.24: Simulate the price of wheat

The demand for wheat in year t is given by

=ap

+b,

wherea< 0,b> 0, andp

is the price of wheat. Let the supply of wheat

A.3 Exercises

633

=Ap

t−1

+B +ln(1 +p

t−1

whereA andB are given constants. We assume that the pricep

adjusts

such that all the produced wheat is sold. That is, D

a) ForA=1, a= −3,b=5,B=0,ﬁndfromnumericalcomputations,

astable price such that the production of wheat from year to year is

constant. That is, ﬁnd p such that ap+ b = Ap+B + ln(1 +p).

b) Assumethatinaverydryyeartheproductionofwheatismuchless

than planned. Given that price this year,p

,is 4.5 andD

,compute

in a program how the pricesp

,...,p

develop. This implies solving

the diﬀerence equation

+b = Ap

t−1

+B +ln(1+ p

t−1

From the p

values, compute S

and plot the points (p

) for t =

0,1,2,...,N. How do the prices move when N → ∞?

Filename: wheat.py.

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