MATH3076,3976,4076: Mathematical Computing
Semester 1, 2025
Assignment 1
Problem setting
The Lennard-Jones potential provides a phenomenological description for the interaction between molecules. In standardized units, the potential energy V as a function of the distance r is written as
Consider a particle with total energy E moving under the influence ofthe Lennard-Jones potential.
The Kinetic energy K (r) = E − V (r) of the particle as a function of the position r can thus be written as
Mathematically, and throughout this assignment, we consider K (r) defined for allr > 0. Physically, K (r) can be interpreted as the kinetic energy only when K (r) ≥ 0 (because the kinetic energy is proportional to the square of the velocity). We are interest in the minimum distance of the particle, obtained, for instance, when the particle heading towards collision stops. We denote this minimum distance as r* . This corresponds to the smallest root of K (r), i.e., r* is the smallest value of r for which
K (r) = 0.
Questions
1. Characterization of the problem (3 marks).
(a) Write a code in Python that numerically plots the function K (r) as a function of r for E = −0. 1. The region around K (r) = 0 should be clearly visible.
(b) Based on the plot produced in the previous item, how many different solutions of K (r) = 0 exist for E = −0. 1?
(c) Calculate the number of different solutions of K (r) = 0 as a function of E ∈ R. Justify your answer analytically (i.e., not based on numerical computations).
2. Bisection method (7 marks). We are intrested in computing r* using the bisection method initialized in the pair of points a, b with b > a. You can write the answers to the questions below as a function of r* and other roots of K(r).
(a) For E = −0. 1, specify one pair (a, b) for which the bi-section converges to r = r* as the number of iterations n → ∞. Justify your answer.
(b) For all E ∈ R, determine analytically all the possible regions of a, b ∈ (0, ∞) × (a, ∞) for which the bisection method converges to r* or converges to a different root.
(c) Write a code in Python that implements the bi-section method. Use your code in the case E = −0. 1 to estimate r ∗ (precision of at least 6 digits). Choose also one initial pair (a, b) that does not converge to r*.
(d) Using your code, compute the values ofK(rn) after n iterations of your bisection method for n ∈ [0, 100]. Do the values converge to zero? Explain why.
(e) Modify your code to obtain a more precise estimation of r*, i.e., a value of r* such that K(r*) is closer to zero.
3. Newton-Raphson method (7 marks). We are interested in computing r* using the Newton- Raphson method initialized in r0 . You can write the answers to the questions below as a function of r* and other roots of K(r).
(a) Write the expression for the iterative procedure rn+1 = f (rn) obtained applying the Newton-Raphson method to the problem of finding the roots of K(r), i.e., the solutions of K(r) = 0.
(b) For E = -0. 1, consider the following three possible outcomes of the Newton-Raphson methods:
- the method converges to r*.
- the method converges to a root different from r*
- the method does not converge or fail.
Indicate one intial condition r0 which leads to each of the possible outcomes that takes place in this problem. Justify your answer.
(c) Let I(E) be the largest interval containing r* for which the Newton-Raphson method converges, i.e., r* ∈ I and for any r0 ∈ I we have rn → r* as n → ∞. Compute I for E = -0. 1 and discuss what happens at the boundary of this interval.
(d) Write a code which implements the Newton-Raphson method and apply it to the case E = -0. 1. Use your code to
(i) estimate r* with a precision of at least 6 digits;
(ii) test what happens for different values r0 , including r0 ∈ I and r0 ∉ I. Include values of r0 that illustrate all possible behaviours of the method (i.e., not only cases which converge to r*).
(e) For E = -0. 1, are there values r0 ∉ I(E) for which the Newton-Raphson method converges to r*? Use your code to explore possible outcomes. Explain your numerical observations.
(f) For E = +0. 1, are there values r0 ∉ I(E) for which the Newton-Raphson method converges to r*? Use your code to explore possible outcomes. Explain your numerical observations.
4. Evaluate the AI answer (3 marks). Below you find the answer of Microsoft Co-pilot (based on the large language model GPT-4) to the request to compute the roots of K (r) in the case E = 0. 1. The two user inputs (requests) appears in the text within the blue boxes on the right.
(a) Which programming language has Copilot used to find the roots? Is the chosen library suitable for the requested problem?
(b) For each of the two reported roots, indicate whether they are a good approximation (up to the reported precision) of an actual root of the problem.
(c) Consider the six bullet points listed in the 3-point detailed explanation. Identify the bullet points that are not correct (if any) and correct them.