MAD 4401 — Introduction to Numerical Analysis
Numerical analysis is where mathematical theory meets computation.
Ever wondered how a calculator finds square roots, how a computer approximates the solution of a differential equation, or how much confidence we should place in a numerical answer? MAD 4401 studies the algorithms that allow computers to solve mathematical problems approximately, efficiently, and reliably.
The central idea of the course is not simply to obtain a numerical answer, but to understand why an algorithm works, how quickly it converges, how error enters the computation, and when a numerical result can be trusted.
Course Development
I have taught MAD 4401 regularly at the University of Florida and have continually redesigned the course as computational tools have changed. Earlier versions of the course used Mathematica and other proprietary software; I later converted the computational examples and projects to Python so that students could work with a free, open-source language widely used in mathematics, scientific computing, and data science.
I also adopted an open-source numerical analysis textbook, expanded the course’s MyOpenMath question bank, developed new Python-based computational projects, and created assessments that combine mathematical derivation with computation.
During Spring 2025 Professional Development Leave, I worked with the University of Florida’s Center for Instructional Technology and Training (CITT) to redesign MAD 4401 as a fully online course. This work included restructuring the course for online delivery, recording instructional materials, developing online homework and computer projects, and building a reusable online course structure for future offerings.
What Will We Study?
The course focuses on numerical methods that appear throughout mathematics, science, engineering, data analysis, and scientific computing. Core topics include:
- Error analysis and computer arithmetic — floating-point representation, round-off error, truncation error, and accuracy.
- Root-finding methods — bisection, fixed-point iteration, Newton’s method, and related algorithms.
- Interpolation and polynomial approximation — Lagrange interpolation, Newton divided differences, and spline ideas.
- Least-squares approximation and regression — fitting mathematical models to data.
- Numerical differentiation and integration — finite-difference approximations, trapezoidal methods, Simpson’s rules, and composite methods.
- Numerical methods for initial-value differential equations — Euler methods, Runge-Kutta methods, step size, error, and stability.
Mathematics First, Computation Second
Although Python is an important tool in the course, MAD 4401 is not primarily a programming class. The mathematics behind the algorithms remains central. Students are expected to understand the assumptions behind a method, analyze its error, compare competing approaches, and explain why one numerical method may succeed where another fails.
The computer then becomes a laboratory: students can test theoretical results, investigate convergence, visualize error, work with problems too large for hand calculation, and compare mathematical predictions with actual numerical behavior.
How Will We Learn?
MAD 4401 combines mathematical analysis with hands-on computation. Depending on the section and mode of delivery, course activities may include:
- Homework focused on mathematical derivations, error analysis, and numerical techniques.
- Python computer projects and labs in which students implement and compare numerical algorithms.
- MyOpenMath assignments with randomized numerical problems and computational exercises.
- Discussion and explanation of convergence, stability, approximation, and error rather than simply reporting a numerical answer.
- Examinations that assess both conceptual understanding and the ability to carry out numerical analysis by hand.
Why Take Numerical Analysis?
- You want to understand what is happening underneath the numerical tools used by computers.
- You are interested in scientific computing, applied mathematics, data science, engineering, or computational modeling.
- You want a deeper understanding of approximation, numerical error, convergence, and stability.
- You want experience translating mathematical algorithms into working computational methods.
- You enjoy courses that combine proofs and analysis with practical computation.
Open-Source Course Materials
A major goal of the course redesign has been to reduce dependence on expensive proprietary software and textbooks. Current versions of MAD 4401 use Python and open-source numerical analysis resources whenever possible, giving students access to the same computational tools after the course ends.
Syllabi
Full details about grading, course policies, schedules, prerequisites, required materials, and section-specific activities are available through Simple Syllabus.
Different sections cover the same core numerical-analysis material, but meeting times, exam logistics, and some course activities may differ. Students should consult the syllabus for their registered section.