DEPARTMENT OF MATHEMATICAL SCIENCES

AUTUMN SEMESTER 2024-2025

MATH1109 - CORE MATHEMATICS

Python Coursework

Deadline: 15:00, Wednesday 27/11/2024

Rules

It is not permitted to use generative artificial intelligence (AI) software for this coursework. Ensure that you have read and have understood the Policy on academic misconduct. One of the things stated in this policy is that “The submission of work that is generated and/or improved by software that is not permitted for that assessment, for the purpose of gaining marks will be regarded as false authorship and seen as an attempt to gain an unpermitted academic advantage”.

You are allowed to work together and discuss in small groups (two to four MATH1109 students) but you must write all code by yourself with the exception that you may copy from material provided on the MATH1109 Moodle page.

You may ask for and receive help from the tutor Richard Rankin although not all questions will be answered and those that are will be answered to all students in the class.

Each student is to submit their own coursework.

You must submit one .py file for each question. A full submission contains three Python files:

• legendre.py,

• square_root.py,

• time_steppers.py.

You are strongly encouraged to use the Spyder IDE (integrated development environment).

Marks breakdown

This coursework is out of 100 marks:

• Q1 – 40 marks.

• Q2 – 30 marks.

• Q3 – 30 marks.

Guidelines

For full output data marks, your functions need to return the correct outputs for the particular input arguments.

For full plot marks, your plot(s) should have the correct data plotted, axis labels, a legend if appropriate, and a descriptive title. Note that you will not be penalised for not using TeX/LaTeX fonts.

You should comment your code as discussed in Appendix 3 of the Python Notes and all functions should have a docstring, see Chapter 7.6 of the Python Notes. Some marks will be assigned for commenting and docstrings.

Please be aware that we will test your functions with an automated marker to check that they work and produce the desired output(s), both with the data given in the templates and with different undisclosed data.

.py files that include template functions can be found in the same folder as this file on the MATH1109 Moodle page.

The template .py files also contain some simple tests of your functions. You are encouraged to write your own tests of your functions in a separate file.

If your modules have filenames that differ to what has been explicitly asked for, then your modules will not run through the automarker, and you risk scoring low output/plot marks.

The automated tests for all questions will be “timed out” if they take too long to run. All questions should only need at most a few seconds to run.

When setting up a figure inside a function, you should first use the following syntax before plotting anything:

fig = plt.figure()

fig can then be returned from the function as if it were any other variable.

Questions

1. The Legendre polynomials {Lu (x)}u≥0 are a set of polynomials that are orthogonal over the interval (−1, 1). This means that

The Legendre polynomials can be constructed using the recurrence relationship:

L0 (x) = 1,

L1 (x) = x,

nLu (x) = (2n − 1)xLu−1 (x) − (n − 1)Lu−2 (x),   for n ≥ 2.

Your task:

(a)  Complete the function found in legendre.py (available on Moodle) with prototype:

def  compute _ legendre (x,p)

• This function should compute the Legendre polynomials up to and including degree p at the points contained in x.

• This function should take as input:

– A one-dimensional numpy.ndarray x - the values at which to evaluate the Legendre polynomials;

– An int p - the maximum polynomial degree to evaluate;

• The function should output:

– A numpy.ndarray of shape (p+1,len(x)) - the evaluated Legendre polynomials.

Once you have written compute_legendre(), test it by running legendre.py. You should obtain the following:

Legendre  polynomials:

[[ 1. 1. 1. 1. 1. ]

[-1. -0.5 0. 0.5 1. ]

[ 1. -0.125 -0.5 -0.125 1. ]

[-1. 0.4375 -0. -0.4375 1. ]]

(b)  Complete the function found in legendre.py with prototype:

def  plot _ legendre (p)

• This function should plot the Legendre polynomials up to and including degree p over the interval [−1, 1] on a single set of axes.

• This function should take as input:

– An int p - the maximum polynomial degree to plot;

• The function should output:

– A matplotlib.figure .Figure - a handle to the plot created.

Once you have written plot_legendre(), test it by running legendre.py.

[40 marks]

2.  In this question, you will implement an algorithm to find an approximate square root of a > 0. The mathematical square root function has the following Taylor expansion about a point x2:

This motivates the following iteration

where N is some positive integer.

The iteration should begin with an initial guess  is considered sufficiently accurate when

for some give error tolerance ε .

Your task:

Complete the function found in square_root.py (available on Moodle) with the prototype:

def  approx _ square _root (a,N,x0 , eps)

• This function should apply (1) to find the approximate square root of a.

• This function should take as input:

– A float a - the value to be square rooted;

– An int N - to control how many terms in the Taylor expansion are used;

– A float eps - the tolerance at which to stop the iteration;

– A float x0 - the initial guess for the iteration.

• The function should output (in this order):

– A float - the (approximate) positive value of the square root of a;

– An int - the number of iterations required to obtain a sufficiently accurate approximation.

Once you have written approx_square_root(), test it by running square_root .py. You should obtain the following:

sqrt(25) = 5.0 , No_iterations = 4

sqrt(10) = 3.162277660168379 , No_iterations = 2

Note: Very tiny changes to these values are acceptable due to the way Python carries out floating point arithmetic.

[30 marks]

3.  Suppose we wish to solve the scalar initial value problem

A two-point method for approximating solutions to this problem has the time stepping formulae:

yu+2 = yu+1 + δt(3f(tu+1 , yu+1) − f(tu, yu))/2,   n = 0, … , N − 2                      (4)

where, δt = T/N and tu  = nδt, n = 0, … , N.

In order to findy2 , both y0 andy1 are required. y0  is available from the initial condition, but in general, y1 is not known. y1 can be simply provided, as in part (a), or calculated some other way, as in part (b).

Your task:

(a)  Complete the function found in time_steppers.py (available on Moodle) with the prototype:

def  perform _time _ stepping1 (T,N,y0 ,y1 , f)

• This function should use the method shown in (4) to approximately solve the IVP (2)-(3).

• This function should take as input:

– A float T - the final time T;

– An int N - the number of time-steps to perform;

– A float y0 - the initial value y0;

– A float y1 - the value to use for y1;

–  a function f - the function f(t, y).

• The function should output (in this order):

–  a numpy.ndarray - a vector containing all the time points {tn}Nn=0;

–  a numpy.ndarray - a vector containing all the approximations {yn}Nn=0.

Once you have written perform_time_stepping1() test it by running time_steppers .py. You should obtain the following: perform_time_stepping1:

tn  =    [0 .    0 .5  1 .    1 .5  2 .    2 .5  3 .    3 .5  4 .    4 .5  5 .  ]

yn  =     [1 .                  1 .23606798  1 .44647817  1 .6412379     1 .82300795  1 .99402943 2.15596995  2.31011529  2.45747855  2.59887356  2.73496507]

(b)  Complete the function found in time_steppers.py (available on Moodle) with the prototype: def  perform _time _ stepping2 (T,N,M,y0 , f)

• This function should use the method shown in (4) to approximately solve the IVP (2)-(3).

• To find a value for y1  ≈ y(δt), M steps of the forward Euler method should be used.

• This function should take as input:

– A float T - the final time T;

– An int N - the number of time-steps of (4) to perform;

– An int M - the number of time-steps of forward Euler to perform to findy1;

– A float y0 - the initial value y0;

–  a function f - the function f(t, y).

• The function should output (in this order):

–  a numpy.ndarray - a vector containing all the time points {tn}Nn=0;

–  a numpy.ndarray - a vector containing all the approximations {yn}Nn=0 .

Once you have written perform_time_stepping2() test it by running time_steppers .py. You should obtain the following:

perform_time_stepping2: 

tn  =    [0 .    0 .5  1 .    1 .5  2 .    2 .5  3 .    3 .5  4 .    4 .5  5 .  ]

yn  =     [1 .                  1 .23861555  1 .44864405  1 .64325985  1 .82490307  1 .99581873 2.15766904  2.31173637  2.45903135  2.60036596  2.73640354]

[30 marks]