**Maxima** is a computer algebra system (CAS) that is free, open source, runs on Windows, Linux, and Mac, and has reasonable coverage of basic and advanced mathematical functions, from garden-variety algebraic simplification, to polynomials, PDEs, and more. Topic areas include calculus, matrix equations, differential equations, number theory, combinatorics, hypergeometric functions, tensors, gravitational physics, PDEs, nonlinear systems. This makes it an appealing mathematical computing package for students, programmers, engineers, scientists, and mathematicians.

### Maxima for Symbolic Computation

#### What is Maxima?

Maxima is a symbolical computation package that is free, open source, and has an active, responsive developer base and community that ensures both the future lifecycle of this software package and plenty of help when dealing with problems. It falls in the category of Computer Algebra Systems (CAS).

A number of computer algebra systems are available as alternatives to choose from: Mathematica, Maple, Macsyma, MuPAD, Sage, etc. Many are commercial packages and have greater accuracy. 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. As such, Maxima is worth considering as a key tool in the toolbelt of programmers, engineers, scientists, and mathematicians.

Maxima is a mathematician’s playground, with basic algebraic manipulation, polynomials, calculus, matrix equations, differential equations, number theory, combinatorics, hypergeometric functions, to state of the art areas in tensor and gravitational physics, PDEs, nonlinear systems.

This page should get you started with downloading and installing, and then provide a few examples and resources to help you on your way.

#### Obtaining Maxima

You can download Maxima from here (Windows, Linux). (Confused? Instructions here.)

#### Installing Maxima

Paul Lutus has written a step-by-step installation walkthrough here. Specifically, take note about the Windows firewall when first running wxMaxima:

“When you first run wxMaxima (an icon is placed on your desktop by default), 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 to ruthless Russian mobsters. Suspend your paranoia and allow the socket to be created.” (from Arachnoid’s Installation Guide)

#### Getting Started

Depending on your level of experience with computers, you may find the following starting points useful:

**For a Quick Start**

- Here’s a succinct Maxima command reference sheet (cheat sheet).
- If you are familiar with Mathematica, this Mathematica / Maxima Syntax Conversion Chart will get you going quickly.

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 some good examples.

- 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). - 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.

More advanced references are here.

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.

**For Basic Users**

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.

**For Advanced Users**

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.

**Getting Help**

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

#### Basic Guides

- 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

#### Advanced Guides

- 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: 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.

#### System Documentation

- 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

Lovely just what I was searching for. Thanks to the author for taking his time on this one.

Took me time to read all the comments, however I really loved the article. It proved to be very useful to me and I’m positive to all of the commenters here! It’s good when you can not only learn, but are also engaged! In my language, there aren’t many good sources like this.

Appreciate this post. Let me try it out.

@Assad: my pleasure! I am looking forward to being an active member of the Maxima community now that it has become part of my daily life.

– Zak

@Zak — your lab manuals for a two-year calculus sequence are an excellent resource for learning Maxima at the same time as Calculus. Thanks for announcing them here.

I ran into this post what seems like ages ago in the early stages of research for my sabbatical project Spring 2015! Thank you, by the way, for consolidating such a helpful bunch of links — it saved me a lot of work. After four months of writing, I released two CC-BY-NC-SA open-texts for wxMaxima that can be used as lab manuals for first-year calculus or “by-example” references for students learning wxMaxima independently. The e-book is free, and the LaTeX source is available for those who wish to create derivative works. http://wxmaximafor.wordpress.com/

Thanks!

Zak Hannan

Instructor of Math and Physics

Solano Community College, Fairfield, CA

Many thanks for sharing this fine write-up. Very inspiring! (as always, btw)

@gerd: Your blog article points out surprising limitations of Maxima in advanced calculations compared to Mathematica. For a scientist or engineer working in these areas, you’re right that the flaws in Maxima’s ability to calculate these may be a good reason to try another CAS. However, IMO Maxima remains a useful choice for those wanting an open source CAS platform, or for those whose calculational needs are more elementary than the examples you give. Appreciate your posting.

Came across your blog when writing up this one from a somewhat different perspective http://thingwy.blogspot.de/

You write “…Maxima is a mathematical computing package that ought to be in the toolbelt of every programmer, engineering, scientist, and mathematician…” For a different point of see http://thingwy.blogspot.de/

A very helpful and suggestive article. Thank you.

Wonderful article, thanks a lot !!