Lab 4: Numerical Methods And Visual Checks

Goal

Use MATLAB numerical methods responsibly by pairing every computed answer with a visual or residual check.

Setup

Work in a folder called week04. Start scripts from a clean workspace:

clear; clc; close all;

Part A: Roots And Residuals

Open week04_root_integration_checks.m.

Run the script and answer:

  • What equation is being solved?
  • How many roots are found?
  • What brackets are used?
  • What is the residual at each root?
  • What would be missing if the script only printed the first root?

Part B: Change The Equation

Modify the root-finding section for:

f = @(x) cos(x) - x;

Use the interval [0, 2].

Record:

  1. the root;
  2. the residual;
  3. a plot showing the crossing;
  4. one sentence explaining why the answer is credible.

Part C: Integration By Function And By Data

The script compares integral with trapz.

Change the number of sampled points used by trapz:

x_data = linspace(0, 6, 7);
x_data = linspace(0, 6, 31);
x_data = linspace(0, 6, 201);

Answer:

  • How does the estimate change?
  • Which estimate is closest to integral?
  • Why does sample spacing matter?

Part D: ODE Solver Check

Open week04_ode_solver_check.m.

Run the script and answer:

  1. What differential equation is being solved?
  2. What are tspan and y0?
  3. What does the initial-condition check verify?
  4. What changes when output times are requested explicitly?

Part E: Surface And Contour

Open week04_surface_contour_visualization.m.

Run the script and answer:

  • What function of two variables is being plotted?
  • What does meshgrid create?
  • What does the surface plot show well?
  • What does the contour plot show more clearly?

Part F: Modify The Surface

Change the plotted function to:

Z = X .* exp(-X.^2 - Y.^2);

Then:

  1. Update the title.
  2. Find the approximate maximum value.
  3. Record the approximate (x, y) location of that maximum.

Deliverable

Submit a short note containing:

  • one root-finding result with residual;
  • one integration comparison;
  • one ODE solve with an initial-condition check;
  • one surface or contour plot;
  • one paragraph explaining how the checks changed your confidence.

AI Use

If you ask an LLM for help, include:

  • the prompt;
  • the generated code or explanation;
  • the check you used before trusting it;
  • one thing the LLM did not verify.