Maxima is a symbolic computation platform that is free, open source, runs on Windows, Linux, and Mac, and covers a wide range of mathematical functions, including 2-D/3-D plotting and animation. Capabiities include algebraic simplification, polynomials, methods from calculus, matrix equations, differential equations, number theory, combinatorics, hypergeometric functions, tensors, gravitational physics, PDEs, nonlinear systems. With an active developer base and responsive community, a user gets a secure future lifecycle of the product and plenty of help when dealing with problems. The result: an essential mathematical computing package for students, programmers, engineers, scientists, and mathematicians. This article will help you get started with Maxima.
Obtaining Maxima, and Alternatives
Alternative choices of Computer Algebra System (CAS) are Mathematica, Maple, Macsyma, MuPAD, Sage, etc. The first 4 are commercial packages, the last is an ambitious comprehensive mathematical platform that includes much more than a CAS, but (as of this writing) requires running on Windows within a virtual machine. Maxima combines low cost (free), with ready availability for all three major operating systems, and basic coverage of a large part of mathematics and analytical engineering.
Installation on Windows is relatively straight-forward. However note that:
“when you first run wxMaxima, your firewall software may complain that a socket is being opened. This is a local socket that wxMaxima (the user-friendly graphical front end) uses to communicate with Maxima (the computation engine), it is not an attempt to take over your computer or communicate your personal secrets…” – Paul Lutus, detailed installation walkthrough
- Here’s a succinct Maxima command reference sheet (cheat sheet) by Ruben Puentedura
- If you are familiar with Mathematica, this Mathematica / Maxima Syntax Conversion Chart will get you going quickly.
- Antonio Cangiano’s 10-minute Tutorial for Solving Math Problems with Maxima is a good start.
- Follow this with Richard Rand’s more advanced whirlwind tour: Introduction to Maxima, including a discussion of writing subroutines/scripts/programs for Maxima. Available here (PDF).
A full list of the mathematical packages and capabilities built into Maxima can be found in the 1,000 page (5MB) Maxima Manual. (You’ll probably want to download a PDF version (5.24) for offline reading.
Paul Lutus has leisurely hands-on tutorial style introduction to Maxima. In addition, there are a number of good “book-style” tutorials that develop familiarity with Maxima thoroughly.
Robert Dodier’s Minimal Maxima (PDF) breaks down the syntactical, evaluation, and data structures underlying Maxima. A good understanding of this is essential when you are trying to go beyond using Maxima as a powerful calculator, or when writing your own functions/subroutines in Maxima.
More advanced references are here.
Learning through Problem Solving applications
Using Maxima makes it relatively easy to work hairy computations with symbolic accuracy, leaving more time for advancing the application or research work. Below are examples.
- To see how Maxima can be used in verification and validation of a result in discrete mathematics, see Finite Summation by Recurrence Relations, Part 1 (Motivation and Low Order Examples) and Part 2 (General Recurrence Solution).
- To see how Maxima can be used to solve linear systems (matrix equations), see Finite Summation by Recurrence Relations, Part 3 (Closed Form Direct Matrix Solution).
- For an example Maxima script (iteratively solving the finite sums p-th order recurrence relation), see sumkp_recurrence.mac
- If you’re looking for using Maxima for solving real Engineering problems, consider Javed Alam’s 22 sessions of Maxima, Paul Lutus’s differential equations and circuit theory and Fourier (Spectral) Analysis.
The Maxima mailing list is a responsive, expert community that can not only help you out of a jam, but also raise the level of your proficiency and your familiarity with “natural” Maxima programming style.
Links and References
Obtaining and Installing
- Download Maxima from here (Windows, Linux)
- The Maxima Page for Windows, Linux
- Installation Walk-Through — take note about allowing the firewall exception for wxMaxima!. Also, how to get Greek fonts to display properly in your Maxima session.
- Installation Prerequisites for Maxima for Mac, Windows, Linux
Cheat Sheets / Ready Reference Sheets
- A Maxima CheatSheet / Ready Reference Sheet
- Mathematica / Maxima Syntax Conversion Chart, and Maxima Cheat Sheet: Harvard University, Department of Mathematics
- Antonio Cangiano’s 10-minute Tutorial for Solving Math Problems with Maxima: Math~Blog
- Richard Rand’s Introduction to Maxima, a more advanced whirlwind tour of maxima, including a discussion of writing subroutines/scripts/programs for Maxima — You’ll like it much better as a formatted PDF.
- Paul Lutus’ Symbolic Mathematics Using MaximaA Leisurely Tutorial in 9 parts.
Topics by Example
- Javed Alam’s22 sessions of Maxima for solving real-world Engineering problems
- Paul Lutus’s differential equations and circuit theory and Fourier (Spectral) Analysis.
- Tensor Algebra in Maxima
- Leon Brin’s Maxima and the Calculus
- Gregory Astley’s Using Maxima for Plotting Direction Fields of First Order ODEs
- Robert Dodier’s Minimal Maxima (PDF) breaks down the syntactical, evaluation, and data structures underlying Maxima. A good understanding of this is essential when you are trying to go beyond using Maxima as a powerful calculator, or when writing your own functions/subroutines in Maxima.
- Maxima mailing list is a responsive, expert community that can not only help you in a jam, but also raise the level of your proficiency and your familiarity with idiomatic Maxima (that intangible called Maxima style).
“Book-Style” Tutorials (PDF or HTML)
- Gilberto Urroz’s Maxima Book (Science & Engineering Applications): Comprehensive, each chapter organized by mathematical area.
- The Maxima Book (2003), P. de Souza, R. Fateman, J. Moses, C. Yapp: Comprehensive, well-written, well-organized. Not the most up-to-date, but the organization, comprehensiveness, and quality of the material makes this a valuable reference.
- Edwin Woollet’s 11 chapterMaxima By Example: a leisurely description of Maxima’s capabilities.
- Maxima Manual: 1000+ page (5MB) comprehensive manual and listing of all mathematical functions and capabilities built into Maxima.
- Maxima Homepage: Maxima is a system for the manipulation of symbolic and numerical expressions, including differentiation, integration, Taylor series, Laplace transforms, ordinary differential equations, systems of linear equations, polynomials, and sets, lists, vectors, matrices, and tensors. Maxima yields high precision numeric results by using exact fractions, arbitrary precision integers, and variable precision floating point numbers. Maxima can plot functions and data in two and three dimensions.
- wxMaxima Homepage: A Windows GUI for Maxima