代做MATH6189、代写Python程序语言

日期：2025-03-13 08:19

MATH6189 Assignment 1

Worth 20%

Submission date: 14 March, 2025, 23:59.

Rules

• You must work on your own on this assignment with no help from others or GenAI.

• You must submit a single Jupyter notebook file as a submission.

• Clearly indicate your name and student number on the code. Save the file with your student

number, e.g student number.ipynb

• Ensure that your code is clean, well-structured, and error-free. Non-functioning codes will be

penalized.

Problem 1 (30 points)

Sustainability is a critical global challenge. One important aspect of sustainability is energy consumption.

Efficient energy use can significantly reduce carbon footprints and promote environmental sustainability.

The Energy Efficiency Score (EES) for each household can be calculated using the formula

100 ×

 1 −

Daily Energy Consumption

Household Size  + Renewable Energy Contribution × 10

EES =

2

Given the dataset household energy data.csv which contains ’House ID’, ’Daily energy consumption’,

’Household size’ and ’Renewable energy contribution’ for each household. The dataset is available under

Assignments on Blackboard.

1. Write a Python function called calculate ees that calculates the EES score for each household.

[5 points]

2. Using the calculated EES scores, categorize each household into one of the following sustainability

ratings:

• Excellent: EES ≥ 80

• Good: 60 ≤ EES < 80

• Fair: 40 ≤ EES < 60

• Poor: EES < 40

[5 points]

3. Calculate the mean, median, standard deviation, and range of EES scores. [5 points]

4. Examine and analyze how the household size, daily energy consumption and renewable energy contri bution affect the EES.

[15 points]

Problem 2 (30 points)

The nonlinear equation f(x) = 0 plays an important part in many application models. The solution to such

system can be solved using the Secant approximation method and bisection method described below:

Secant Method

Given a nonlinear equation f(x) = 0, the Secant Method approximates the root iteratively using the formula:

xn − xn−1

xn+1 = xn − f(xn) ×

f(xn) − f(xn−1)

where:

• xn and xn−1 are the current and previous approximations,

• f(x) is the function whose root we seek.

Bisection Method Formula

The Bisection Method works by narrowing down the interval [a, b] where the root lies. At each iteration, the

interval is halved based on the sign of the function at the midpoint.

(i) Compute the midpoint:

a + b

c =

2

(ii) Update the interval:

• If f(a) × f(c) < 0, the root lies in the interval [a, c].

• If f(c) × f(b) < 0, the root lies in the interval [c, b].

(iii) Repeat until the interval width |b − a| is less than the specified tolerance.

1. Write Python functions for both the Secant Method and the Bisection Method:

• secant method(f, x0, x1, tol, max iter)

• bisection method(f, a, b, tol, max iter)

Each function should:

• Take the function f(x), initial guesses, tolerance (10−6

) for convergence, and maximum iterations

(max iter).

• Iteratively apply the respective formula until the approximate root converges within the given

tolerance or reaches the maximum number of iterations.

• Return the approximate root, number of iterations to convergence, and the list of error at each

iteration.

[10 points]

2. Find the root of the following nonlinear equation:

f(x) = x

3 − 6x

2 + 11x − 6.1 = 0

Specify initial guesses, solve the equation using both methods, and record the following:

• The approximate root found.

• The number of iterations required for convergence.

• The error (|xn+1 − xn|) at each iteration.

[10 points]

3. Plot the error for both methods, showing how the error decreases as the number of iterations increases.

[5 points]

4. Compare the performance of both methods and discuss on their computational efficiency. [5 points]

Problem 3 (40 points)

The formula for calculating the probability of rainfall in a city is given by:

P(rain) = 1

1 + exp(−(a · tmax + b · tmin + c · af − d · sun)),

where tmax is the mean monthly maximum temperature, tmin is the mean monthly minimum temperature,

af is the number of air frost days recorded in the month, sun is the total sunshine duration (in hours), and

a,b,c,d are coefficients representing the impact of each variable on the probability of rainfall, with values

a = 0.05, b = 0.04, c = 0.03 and d = 0.02, respectively. Using the provided weather data.csv dataset under

Assignments on Blackboard, perform the following tasks:

1. Calculate the average probability of rainfall in London, Manchester and Southampton for the year

2020. [10 points]

2. Identify the city with the highest probability of rainfall in 2023. [10 points]

3. Plot a bar chart showing the monthly rainfall probability trends in Southampton for the year 2022.

[10 points]

4. Plot the graph of Liverpool’s average annual rainfall probability between 2020 and 2023. [10 points]