Numerical Methods for Engineers

Why this course

Most real engineering problems have no closed-form answer. This course teaches students to build, test, and trust the numerical tools that solve them, and to know when not to trust them. Every method comes paired with its error, so students understand what a root-finder, a spline, or a Runge–Kutta step is actually doing. Later, the software they rely on is never a black box.

Who it’s for

Engineering BS/BSE students, typically in their second or third year.

Prerequisites

• MAT 274 or MAT 275 with a grade of C or better
• MAT 242 or MAT 343 (or MAT 342 for non-MAE students) with a grade of C or better
• MAT 272 or MAT 267 with a grade of C or better, as a pre- or co-requisite
• Credit is allowed for only one of MAE 384 or CEE 384

What you’ll be able to do

1. Explain the types and sources of numerical error, including the remainder term in Taylor’s theorem.
2. Quantify error and judge the accuracy of a numerical solution.
3. Choose an appropriate method for a problem, weighing data availability, accuracy, and computational cost.
4. Compute derivatives and integrals, solve nonlinear equations and linear systems, and fit data by interpolation and regression.
5. Solve ordinary differential equations, both initial- and boundary-value problems.
6. Implement these methods in MATLAB and critically interpret the results in an engineering context.

Course plan

1. Weeks 1–2

   MATLAB bootcamp

   • Orientation
   • Working with real data (the “Baby Names” exercise)
   • Machine Learning Toolbox basics
2. Weeks 2–3

   Errors and representation

   • Taylor series and truncation error
   • Floating point and round-off error
3. Weeks 3–4

   Numerical differentiation

   • Continuous functions
   • Discrete data points
4. Weeks 4–5

   Nonlinear equations

   • Bisection method
   • Newton–Raphson method
5. Weeks 5–6

   Systems of linear equations

   • Gaussian elimination
   • LU decomposition
6. Weeks 7–8

   Interpolation

   • Direct interpolation
   • Splines and length of a curve
7. Weeks 9–10

   Regression

   • Linear regression and its adequacy
   • Nonlinear regression
8. Weeks 11–12

   Numerical integration

   • Trapezoidal rule
   • Simpson’s rules
   • Gauss quadrature
9. Weeks 13–14

   Ordinary differential equations

   • Euler’s method
   • Runge–Kutta methods
   • Higher-order ODEs
10. Week 15

    Project presentations

    • Team presentations and final reports

The project

Teams apply numerical methods to an engineering problem of their choosing. The project runs alongside the course with an initial meeting, introductions, an implementation check-in, a final presentation, and a written report with MATLAB code. Each student also writes a short individual discussion of the method, results, and limitations.

Readings

• Holistic Numerical Methods: open textbook, videos, and MATLAB examples (nm.mathforcollege.com)
• Kaw, Rigsby, Miller & Handžić, Introduction to Programming Concepts with MATLAB