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Exercises for IN1900

October 14, 2019

Preface

This document contains a number of programming exercises made for the course

IN1900. The chapter numbers and titles correspond to the chapters of the book

“A primer on Scientific Programming with Python” by Hans Petter Langtangen.

The exercises are meant to be a supplement to the exercise collection in the book,

and most are motivated by applications in science and applied mathematics.

The exercise collection is used for the first time in 2018, and there may be typos

and small errors. If you find any errors, or have other comments or questions

about the exercises, please send them to Joakim Sundnes: sundnes@simula.no.

1

Chapter 1

Computing with Formulas

Problem 1.1. Throw a ball

When throwing a ball in the air, the position of the ball can be calculated using

the acceleration of the ball. When neglecting air resistance, the acceleration will

be the negative of the gravitational constant, −g. The height of the ball relative

to its starting point is

y(t) = v0t −12gt2,

where v0 is the initial velocity of the ball and t is the time after the throw. The

ball reaches its maximum height at time

tmax =v0g.

Write a program computing the maximum height of the ball, that is y(tmax),

when v0 = 8.2m/s and g = 9.81m/s2. Print the result.

Filename: ball.py

Problem 1.2. Population growth

The growth of a population can often be described by a logistic function

N(t) = B1 + Ce−kt ,

where B is the carrying capacity of the species in the environment, i.e., the

maximum size of the population that the environment can sustain indefinitely.

The constant k tells us something about how fast the population grows, while C

is given by the initial conditions. Let us consider a bacterial colony where we

take the carrying capacity to be B = 50000 and k = 0.2h−1. If the population is

5000 at t = 0, find C and write a code that finds the number of bacteria in the

colony after 24 hours.

Filename: population.py

2

Problem 1.3. Solve the quadratic equation

Given a quadratic equation

ax2 + bx + c = 0,

the two roots are.

Make a program evaluating the roots of.

Print out both roots with two decimals.

Filename: find_roots.py

Problem 1.4. Forces in the hydrogen atom

There are two kinds of forces acting between the proton and the electron in the

hydrogen atom; Coulomb force and gravitational force. The Coulomb force can

be expressed as,

where ke is Coulomb’s constant, e is the elementary charge, and r is the distance

between the proton and the electron.

The gravitational force can be expressed as

FG = Gmpmer2,

where G is the gravitational constant, mp is the mass of the proton, me is the

mass of the electron, and r is the distance between the particles.

We can use these expressions for FC and FG to illustrate the difference in

strength of these two forces, i.e., the electromagnetic force and gravitational force.

Use the values ke = 9.0·109 Nm2C

−2

, e = 1.6·10−19 C, G = 6.7·10−11 Nkg−2m2

,

mp = 1.7·10−27 kg and me = 9.1·10−31 kg.You can take the distance between the

proton and electron to be approximately the Bohr radius r = a0 = 5.3 · 10−11 m.

Make a program that computes both the Coulomb force and the gravitational

force between the proton and the electron. Write out the forces in scientific

notation with one decimal in units of Newton (N = kgm/s

2

). Also print the

ratio between the two forces.

Filename: hydrogen.py

3

Chapter 2

Loops and Lists

Problem 2.1. Multiply by five

Write a code printing out 5 · 1, 5 · 2, ..., 5 · 10, using either a for or a while loop.

Filename: multiplication.py

Problem 2.2. Multiplication table

Write a new code based on the one from Problem 2.1. This code should print

the whole miltiplication table from 1 · 1 to 10 · 10.

Hint: You may want to consider using one loop inside another.

Filename: mult_table.py

Problem 2.3. Stirling’s approximation

Stirling’s approximation can be written ln(x!) ≈ x ln x − x. This is a good

approximation for large x. Write out a nicely formatted table of integer x values,

the actual value of ln(x!), and Stirling’s approximation to ln(x!).

Filename: stirling.py

Problem 2.4. Errors in summation

The program has three errors and therefore does not work. Find the three errors

and write a correct program. Put comments in your program to indicate what

the mistakes were.

Hint: There are two basic ways to find errors in a program:

1. read the program carefully and think about the consequences of each

statement,

2. print out inter mediate results and compare with hand calculations.

4

1. First, try method 1 and find as many errors as you can. Thereafter, try method

2 for M D 3 and compare the evolution of s with your own hand calculations.

Lastly, write a similar code evaluating the same sum using a while loop.

Check that the two loops compute the same answer.

Filename: sum_for.py

5

Problem 2.5. Binomial coefficient

The binomial coefficient is indexed by two integers n and k and is written . Compute the same value using Eq. (2.1) and check that the results are

correct.

Hint: The Q

sign is a product sign.

When checking the result you will need math.factorial.

Filename: binomial.py

Problem 2.6. Table showing population growth

Consider again the bacterial colony from Problem 1.2. Let us study the number

of individuals for n + 1 uniformly spaced t values throughout the interval [0, 48].

Set n = 12. First store the t and N values in two lists t and N. Thereafter, write

out a nicely formatted table of t and N values by traversing the two lists with a

(separate) for loop.

Filename: population_table.py

Problem 2.7. Nested list

a) Compute two lists of t and N values as explained in Problem 2.6. Store the

two lists in a new nested list tN1 such that tN1[0] is the list containing t-values

and tN[1] correspond to the list containing N-values. Write out a table with t

and N values in two columns by looping over the data in the tN1 list. Each t

and N value should be written in the table as integers.

b) Make a nested list tN2 where tN2[i] contains the i-th element of both the

t-list and the N-list. Loop over the tN2 list and write out the t and N values in

the table as integers.

Filename: population_table2.py

Problem 2.8. Calculate Cesaro mean

Let (an)∞n=1 be a sequence of numbers, sk =Pkn=0 an = a0 + . . . , +ak,

is called a Catalan number. Compute and print the first 10 Catalan numbers.

Filename: catalan.py

Problem 2.10. Molar Mass of Alkanes

Alkanes are saturated hydrocarbons with the chemical formula CnH2n+2. If

there are n Carbon atoms in the alkane, there will be m = 2n + 2 Hydrogen

atoms. The molar mass of the hydrocarbon is MCnHm = nMC + mMH, where

MC is the molar mass of Carbon and MH is the molar mass of Hydrogen.

Use a for-loop or a while-loop to compute and print out the molar mass

of the alkanes with two through nine Carbon atoms (n ∈ [2, 9]). The output

should specify the chemical formula of the alkane as well as the molar mass. An

example on how the formatted output should look like for n = 2 is given below.

M(C2H6) = 30.069 g/mol

You can set the molar masses of the atoms to be MC = 12.011 g/mol and

MH = 1.0079 g/mol

Filename: alkane.py

Problem 2.11. Matrix elements

This exercise involves no programming.

The answers should be written in a text file called matrix.txt

Consider a two dimensional 3 × 3 matrix

In Python, the matrix A can be represented as a nested list A, either as a

list of rows or a list of columns. Find the indices i, j of the Python list A such

that A[i][j] = a11 and the indices k, l such that A[k][l] = a32 for the two

following cases:

a) When A is represented as a list of rows. This means that A contains three

lists, where each list corresponds to a row in A.

b) When A is represented as a list of columns. This means that each element

in A contains a list with the elements of a column in A.

Filename: matrix.txt

7

Chapter 3

Functions and Branching

Problem 3.1. Implement a function for population growth

Consider again the function

N(t, k, B, C) = B

1 + Ce−kt .

Implement N as a python function population(t, k, B, C) that returns the

number of individuals in a population after a time t.

Write out a nicely formatted table of t and N values for the time interval

t ∈ [0, 48] using the values from Problem 2.6.

Filename: pop_func.py

Problem 3.2. Sum of integers

We consider the sum Pn

i=1 i = 1 + 2 + · · · + n of positive integers up to n. It

can be shown that the sum is equal to n(n+1)

2

.

a) Write a function sumint(n) that returns the sum of all positive integers

up to n.

b) Write a function implementing n(n+1)

2

.

c) Write test functions for both a) and b) testing for specific known values.

Filename: sumint.py

Problem 3.3. Implement the factorial

a) The factorial can be implemented by a so called recursive function call. Use

a recursive function call to implement a function myfactorial(n) that returns

n!.

b) Write a test function where you call the myfactorial function and check

the value of the returned object for one value of n using math.factorial.

Filename: factorial.py

8

Problem 3.4. Half-wave rectifier

In a half-wave rectifier the positive part of a signal passes, while the negative

part is blocked. Thus, for a signal passing through a half-wave rectifier, the

negative values are set to zero. Let us look at a sine signal that has passed

through a half-wave rectifier:

f(x) = (

sin x if sin x > 0

0 if sin x ≤ 0.

Implement f(x) as a Python function f(x) and make a test function for testing

the implementation of f(x) in both cases.

Filename: half_wave.py

Problem 3.5. Compute the area of an arbitrary triangle An arbitrary triangle

can be described by the coordinates of its three vertices: (x1, y1),(x2, y2),(x3, y3),

numbered in a counterclockwise direction. The area of the triangle is given by

the formula

Write a function triangle_area(vertices) that returns the area of a triangle

whose vertices are specified by the argument vertices, which is a nested list of

the vertex coordinates. Make sure your implementation passes the following test

function, which also illustrates how the triangle_area function works:

def test_triangle_area():

"""

Verify the area of a triangle with vertices

(0,0), (1,0), and (0,2).

"""

v1 = (0,0); v2 = (1,0); v3 = (0,2)

vertices = [v1, v2, v3]

expected = 1

computed = triangle_area(vertices)

tol = 1E-14

success = abs(expected - computed) < tol

msg = f"computed area={computed} != {expected}(expected)"

assert success, msg

Filename: triangle_area.py

Problem 3.6. Primality checker

Recall that a prime number is a number greater than 1 that has exactly 2 divisors.

Said differently, a number greater than one is a prime if it is divisible by only

itself and one. A number that is not prime is called composite. Every number n

can be written as a unique product of primes (e.g. 12 = 2 · 2 · 3), this is called

the prime factorization of n.

a) Make a function that takes a number n, and returns true if it’s prime, and

false if it’s not. Use the program to find all prime numbers up to 100.

p

Hint: You will only need to check divisibility for numbers up to and including

(n), because any greater divisor will imply that there is a divisor less than this.

9

b) Make a function that instead finds the prime factorization of the input

number. It should print “prime” and return nothing if the number is prime,

and both print and return the factorization if it’s composite. Find the prime

factorization of 5525612.

c) Make test functions for the two functions above where you check for small

values of n.

d) Compare the runtime of the two functions with the number 33425626272.

Is the difference big? If so, why do you think one is faster than the other? The

following code returns the mean time it takes for your program to run once:

import timeit

timeit.timeit(’your_func(args)’, \

’from __main__ import your_func’,number=1)

Filename: prime.py

Problem 3.7. Eulers totient function

Two numbers n and m are called relatively prime if they have no common divisors

except for 1. That is, no number greater than one should divide both numbers

with no residue.

a) Make a function that takes two numbers and returns true if they’re relatively

prime and false if they’re not.

b) Euler’s totient function is defined as

φ(d) = #{Numbers less than d which are relatively prime to d}.

Implement Eulers totient function and print φ(d) for d = 10, 50, 100, 200.

c) Make a test function for both a) and b).

Filename: euler.py

Problem 3.8. Simple Statistical Functions

In this problem you will implement two statistical functions and test them by

comparing the results with statistical functions from the numpy module. We will

trust that the functions from the numpy module are correct, and will use them

as benchmark values in the test functions. When you import the numpy module

you should follow the convention of renaming it np, as shown below.

import numpy as np

a) The mean of a set of numbers x1, x2, x3, ..., xN is defined as

where N is the size of the set. Implement a function mean(x_list) that returns

the mean value of a list of numbers.

10

b) Make a test function test_mean() that tests the function from a). Compare

the returned value with the result from numpy.mean. (Such that

expected =np.mean(x_test_values)).

c) The standard deviation of a set of numbers x1, x2, x3, ..., xN is defined as.

Implement a function standard_deviation(x_list) which returns the standard

deviation of a list of numbers. The mean value of the list will be necessary

to calculate the relative deviation. Obtain the mean value inside the

standard_deviation function by calling the function you made in a).

d) Make a test function test_standard_deviation() that tests the function

from c). Compare the returned value with the result from numpy.std. (Such

that expected =np.std(x_test_values)).

You may use the list below as an example for your test functions.

x_test_values = [0.699, 0.703, 0.698, 0.688, 0.701]

Filename: stat.py

Problem 3.9. Münchhausen Numbers

A Münchhausen number is a number such that the sum of every digit to the

power of itself equals the original number. E.g. 1

1 = 1 is a Münchhausen number,

and 5

5 + 33 + 22 = 3156 6= 532, so 532 is not.

Make a function that checks if a number is Münchhausen. Find a Münchhausen

number different from one.

Hint: There is only one such number different from 1 and also under one

million

Filename: m_numbers.py

11

Chapter 4

User Input and Error

Handling

Problem 4.1. Quadratic with user input

Consider the usual formula for computing solutions to the quadratic equation

ax2 + bx + c = 0 given by.

Write a program that asks the user for values ( a =, b = and c = ) to get values

for a, b, and c through the users keyboard. Use input (or raw_input if you are

using Python 2). Print the solutions.

Filename: quadratic_roots_input.py

Problem 4.2. Quadratic with command line

Modify the program from 4.1 such that a, b and c are read from the command

line.

Filename: quadratic_roots_cml.py

Problem 4.3. Quadratic with exceptions

Extend the program from 4.2 with exception handling such that missing command

line arguments are detected. In the except IndexError block, use input (or

raw_input if you are using Python 2) to ask the user for missing input data.

Filename: quadratic_roots_error.py

Problem 4.4. Quadratic with raising Error

In this exercise, use the sqrt function imported from math.

Consider the program from Problem 4.1. Not all inputs yield real solutions.

Modify the program such that it raises ValueError if the values for a, b and c

yield complex roots. (That is if b

2 −4ac < 0). Provide a suitable Error-message.

Test that your program prints out real roots and that it raises ValueError when

the roots are complex. (An example of values that provide complex roots could

be a = 1, b = 1, c = 1, while a = 1, b = 0, c = −1 provide real roots).

Filename: quadratic_roots_error2.py

12

Problem 4.5. Estimating harmonic series

Let f(x) be the function

Write a program that approximates f(x) (that is, evaluates fN (x) = PN

with values of x and N given as command line arguments. Run the program for

x = 0.9, x = 1, and N = 10000. Print the results.

Remark. For x = 1 this is known as the harmonic series. Despite the low values

for large N, the series does not converge, but diverges very slowly. Try to run

the program for different values of N to see how big you can get the value of

f(1).

Filename: harmonic.py

Problem 4.6. Estimating harmonic series extended

Using the program from Problem 4.5, consider the following values for x and N

in a text file

x: 0.9 1

N: 500 1000 10 100 50000 10000 5000

a) Write a function to read a file containing information in the above format

that returns two lists containing the values of x and N.

b) Write a test function for a) that generates a file in the given format and

checks that the values returned by the function is correct.

c) Use the program from Problem 4.5 to evaluate fN (x) for the different values

of x and N. Create a function that writes the information to a file in a table

format with the first column containing the values of N in increasing order, and

the second and third the values of fN (x) at 0.9 and 1 respectively.

Filename: harmonic_table.py

Problem 4.7. Read isotope file

Isotopes of a chemical element in its ground state have the same number of

protons but differ in the number of neutrons. The weight of isotopes of the same

chemical element will therefore be different.

The molar mass, M, of a chemical element, can be calculated by summing

over all its isotopes M =Pi miwi, where mi

is the weight of the i-th isotope

and wi the corresponding natural abundance.

The file Oxygen.txt, which is given below, contains the information on

Oxygen’s isotopes (16O,

17O and 18O).

Isotope weight [g/mol] Natural abundance

(16)O 15.99491 0.99759

(17)O 16.99913 0.00037

(18)O 17.99916 0.00204

13

Write a script in Python to read the file Oxygen.txt and extract the weights

and the natural abundance of all the isotopes of Oxygen. Use these to calculate

the molar mass of Oxygen. Print out the result with four decimals and provide

the correct units.

Filename: read_file_isotopes.py

Problem 4.8. A result on prime numbers

A famous result concerning prime numbers states that the number of primes

below a natural number n, denoted π(n), is approximately given by

tends to 1 as n → ∞. The following

table contains the exact values of π(n) for some values of n.

n: 10**20 10**4 10**2 10**1 10**12 10**4 10**6 10**15

pi(n): 2220819602560918840 1229 25 4 37607912018 168

78498 29844570422669

a) Write a function that reads the file given above and returns two tuples

containing sorted values of n and π(n). It is important that the correspondence

in the orderings are correct, that is, the same as in the table above.

b) Write a test function that generates a file with the format above and tests

that the returned values are correct. It should test that the order of the elements

are in correspondence as in the file.

Hint: The == operator on tuples will take the order into account. The same

operator on lists will not.

c) Create a function that writes the values of n and p(n) to a file in a table

format in increasing order with the values of n in the first column and the

corresponding values of p(n) in the second column.

Bonus problem There are better approximations to π(n), for example the

function

Approximate the integral for different values of n and modify the program to

write these into a third column.

Hint: Implement an algorithm for approximating the integral (e.g. the

trapezoidal rule) and compute the difference as before.

Filename: primes.py

14

Problem 4.9. Conversion from other bases

Recall that a binary number is a sequence of zeros and ones which converted to

the decimal system becomes P

i

2

i where i is a term in the sequence containing

a 1 (e.g. 100101 = 25 + 22 + 20 = 37).

a) Write a function that takes a binary number and converts it to a decimal

number. If the argument is not a binary number, a message should be printed

and nothing returned.

Hint: Let the number in the argument be of type string to avoid problems

with numbers starting with a zero.

b) Let the binary number from a) be taken as a command line argument.

Use exceptions (IndexError) to handle missing input. Print the conversion of

100111101.

c) Extend the program with a function to also handle numbers written in base

3.

Hint: An example of a ternary number(a number in base 3) converted to a

decimal number: 1201 = 1 · 3

Filename: base_conversion.py

Problem 4.10. Read temperatures from two files

We consider data sets from the Norwegian Meteorological Institute, containing

daily mean temperatures of any month of any year at Blindern (Oslo).[Ins19]

Each file looks typically like this:

Year: 1997. Month: April. Location: Blindern(Oslo)

9.0 12.3 15.8 13.4 11.0 16.2 13.3

12.9 14.0 14.1 12.0 17.3 15.5 15.4

...

The observations are given chronologically, and the temperatures are given in

degrees Celsius. There are no empty lines in the bottom of the file.

a) Write a function extract_data(filename) that reads any such file and

returns a list of the temperatures from the given month.

b) In the two files temp_oct_1945.dat and temp_oct_2014.dat you will

find observations of daily mean temperatures in October 1945 and October

2014, respectively. Store the temperatures in two lists oct_1945 and oct_2014.

Calculate the average, maximum and minimum value of the temperatures of both

months, and print the results. You may use the numpy.mean(), numpy.max()

and numpy.min() methods.

c) Write a function write_formatting() that takes at least filename, list1

and list2 as parameters, and creates a new file with two nicely formatted

columns containing the temperatures of the given months (you can assume that

the months have equal lengths). Finally, call the function such that the file

temp_formatted.txt is created, using the lists oct_1945 and oct_2014.

Filenames: temp_read_write.py, temp_formatted.txt

15

Chapter 5

Array Computing and

Curve Plotting

Problem 5.1. Fill arrays; loop version

We study the function

f(x) = ln(x).

We want to fill two arrays x and y with x and f(x) values, respectively. Use 101

uniformly spaced x values in the interval [1, 10]. Create empty x and y arrays

and compute each element in x and y with a for loop.

Filename: fill_log_arrays_loop.py

Problem 5.2. Fill arrays; vectorized version

Vectorize the code in Problem 5.1 by creating the x values using the linspace

function from the numpy package and evaluating f(x) with an array argument.

Since the calculation should be vectorized, you may not use any form of loop.

Filename: fill_log_arrays_vectorized.py

Problem 5.3. Plot the population growth

Again, we’re considering a population undergoing logistic growth. The number

of individuals in the population is given by

N(t, k, B, C) = B

1 + Ce−kt .

Plot this function for t ∈ [0, 48] with a carrying capacity B = 50000, C = 9 from

the initial condition that we have 5000 individuals at t = 0 and a steepeness of

k = 0.2.

Filename: population_plot.py

Problem 5.4. Oscillating spring

A rock of mass m is hung from a spring, and pulled down a length A. When

released, the rock will oscillate up and down with a vertical position given by

y(t) = Ae−γt cos r

Here, y is the vertical position of the rock, k is the spring constant, and γ is

a friction coefficient representing air resistance. Set k = 4 kg s−2 and γ = 0.15

s

−1

, m = 9 kg, and A = 0.3 m.

16

a) Create arrays t_array and y_array of size 101, both initially filled with

zeros. Use a for loop to fill them with time values in the range from 0 to 25

seconds, and the corresponding y(t) values.

b) Vectorize your program by using the NumPy’s linspace function to generate

the t_array, and send it into a function y(t) to generate the y_array.

Your program should now be free of for loops. paragraphc) Plot the position

of the rock against time in the given time interval. Use the arrays from both

exercise a) and b), and confirm that they give the same result. Put the correct

units on both axes.

Filename: oscillating_spring.py

Problem 5.5. Plot Stirling’s approximation

Stirling’s approximation is

ln(x!) ≈ x ln x − x.

a) Make two functions stirling(x) and exact(x), returning Stirling’s approximation

and the exact value of ln(x!), respectively. Plot both the approximation

and the exact curve in the same figure.

Hint: To implement a vectorized version of the exact function, you can use

scipy.special.gamma(x). This function is a “generalized factorial” which can find

the “factorial” of float numbers. It works such that n! = gamma(n + 1). You

can also just consider integer values and plot the value of ln(x!) for each integer

x in the interval you’re considering. Keep in mind that math.factorial is not

vectorized.

b) Use a while loop and find the minimal value of x for the relative error to

be less than 0.1%.

Hint: Relative error is given as (a − a˜)/a, where a is the exact value and a˜

is the approximation. Also, do not start with x smaller than or equal to 1, why?

Filename: stirling_plot.py

Problem 5.6. Plotting roots of a complex number

The n’th roots of a complex number z = reiθ can be found by,

for k = 0, 1, ..., n − 1. The roots can be rewritten to separate the real component

xk and the imaginary component yk, such that ωk = xk + iyk. Through the

relation between the exponential function and the sine functions we get,

for k = 0, 1, ..., n − 1.

17

a) Write a function that takes the angle θ, the radius r and the degree n of

the roots as parameters. The function should calculate and return all of the n’th

roots of a complex number reiθ, as two lists or arrays corresponding to the real

part x = x0, x1, ..., xn−1 and the complex part y = y0, y1, ..., yn−1 of the roots.

An example of a function call on the function you will write is given below.

x, y = roots(r, theta, n)b) Consider the complex number z = 10−4e

i2π

. Use the function from a) to

get all the roots of order n = 6, n = 12 and n = 24. Plot the roots as points,

and plot all the three orders of roots in the same plot. Label the different orders

of roots. And example of code for plotting the roots of order n = 6 is given

below.

plt.plot(x_n_6, y_n_6, "o", label="n = 6")

Filename: roots.py

Problem 5.7. Fermi-Dirac distribution

The Fermi-Dirac distribution says something about the probability of an energy

state being occupied by a particle, or more precisely a fermion, e.g. an electron.

It is a function of energy and temperature given by

f(E, T) = 1

1 + e

(E−µ)/kT , (5.1)

where E is energy, T is temperature, k is Boltzmann’s constant and µ is the

so-called chemical potential. Use k = 8.6 · 10−5

eVK−1 and µ = 4.74eV and

make a program that visualizes the Fermi-Dirac distribution on the interval

E ∈ [0, 10]eV when T = 0.1K. (eV is a unit of energy, 1eV = 1.6 · 10−19J.)

Filename: Fermi_Dirac.py

Problem 5.8. Animate the temperature dependence of the FermiDirac

distribution

Make an animation of the Fermi-Dirac distribution f(E, T) from Problem 5.7

We’re interested in studying how the distribution changes when we raise the

temperature. Plot f as a function of E on [0, 10] for a set of temperatures

T ∈ [0.1, 3 · 104

]. Also make an animated GIF file. Remember to label your axes

and include a legend to show the value of the temperature.

Hint: A suitable resolution can be 1000 intervals (1001 points) along the E

axis, 60 intervals (61 points) in temperature, and 6 frames per second in the

animated GIF file. Use the recipe in Section 5.3.4 and remember to remove the

family of old plot files in the beginning of the program.

Filename: Fermi_Dirac_movie.py

Problem 5.9. Bump functions

Consider the function

f(x) = (

ke− 1

1−x2 −1 < x < 1

0 otherwise.

a) Plot the function with k = 1 on the interval −2 ≤ x ≤ 2 by implementing a

vectorized version in your program.

b) Animate the function on the same interval as above when k decreases from

1 to 0.

Filename: bump.py

Problem 5.10. Band structure of solids

Electrons in solids are waves. These waves have different wave lengths λ. Often,

waves are characterised by their wave number k = 2π/λ, and the wave number

is associated with the energy of the electron. The energies of electrons in solids

have a band structure, i.e., there are different bands of energies separated by a

band gap.

The file bands.txt contains k-values and corresponding energies for the

three first bands of a solid. Have your program read the values for k and the

energies and plot the energy bands as functions of k in the same figure. You will

see that some energies can never be obtained by electrons in the solids. These

areas of non-allowed energies are called the band gaps.

Filename: band_structure.py

19

Problem 5.11. Half-wave rectifier vectorized

In Problem 3.4, we implemented a function illustrating a sine signal after it had

passed through a half-wave rectifier. Vectorize this function and plot f(x) for

x ∈ [0, 10π].

Hint: The numpy.where(condition, x1, x2) function returns an array

of the same length as condition, whose element number i equals x1[i] if

condition is True, and x2[i] otherwise.

Filename: half_wave_vec.py

Problem 5.12. Singularity plot

In this problem we consider the function. Create arrays of r and θ values on the unit

circle centered at the origin with n uniformly spaced values. Fix axes between

-0.5 and 0.5 for x and y and visualize the function for n = 10, 50, 100, 500. You

can use the following to generate the correct values for r and θ:

theta = np.linspace(0,2*np.pi,100)

r = np.linspace(0.01,1,100)

r, theta = np.meshgrid(r,theta)

Remark. If we had an ideal computer that could calculate every value in an

interval and plot it, then the image we have plotted would touch every single

value in the plane, except for at most one! In our program we have 0.01 < r < 1.

The remarkable thing is that the same is true if we replace the inequality with

0 < r < for any > 0. Not only that, but all those points are hit an infinite

number of times!

Filename: ess_sing.py

Problem 5.13. Approximate |x|

The absolute value f(x) = |x| can be written as a sum.

Write a program that calculates the first N terms for N = 1, 2, 3, 4 and plots it

against the exact function. Let the x-axis be [−π, π] with a suitable y-axis.

Filename: approx_abs.py

Problem 5.14. Plotting graphs

A graph is a collection of lines and points in the plane such that each line

connects two points. In this exercise we will create functions for plotting graphs

on a set of points.

a) Make a function plot_line(p1,p2) that takes two points as input arguments

and plots the line between them. The two input arguments should be

lists or tuples specifying x- and y-coordinates, i.e. p1 =(x1,y1). Demonstrate

that the function works by plotting a vertical and a horizontal line.

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