Maxima: a Computer Algebra System (CAS) for symbolic computation<\/h1>\n
Last updated: Feb 19, 2023 (fixed links). Nov 11, 2022 (added omega-math’s excellent web interface, and generating function calculation of the partition of integers problem).<\/em><\/p>\n Maxima<\/strong> is a computer algebra system (CAS)<\/a> for symbolic computation that is free, open source, runs on multiple operating systems (Win,Mac,Linux), and covers a wide range of mathematical capabilities and graphical capabilities. These include algebraic simplification, polynomials, methods from calculus, matrix equations, differential equations, number theory, combinatorics, hypergeometric functions, tensors, gravitational physics, PDEs, nonlinear systems, plus including 2-D\/3-D plotting and animation.\u00a0 With a large and responsive user community, there is plenty of help to get up the learning curve, and with its active developer base, Maxima and its ecosystem continue to gain capability, including a fantastic web interface<\/a> by Omega-Math\/Vroom-Labs (see the screenshot below, r0*0). The result is a free, versatile, powerful mathematical computing package for engineers, scientists, mathematicians, programmers, and students. This article will help you get started with Maxima and set you up with resources to flatten the learning curve.<\/p>\n Omega-Math’s web interface to Maxima. Used here to calculate the first 10 elements of p(n), the number of ways to partition integer n, using a generating function comprising a truncated series of polynomials up to degree n=10<\/p><\/div> Maxima<\/a> is a free\/open-source computer algebra system (CAS)<\/a> with functionality covering a large part of mathematics and analytical engineering, originating in 1982 as the open-source alternative to MIT’s Macsyma system. It is readily available with a fantastic web front end<\/a> by Omega-Math\/Vroom-Labs, and easily installed desktop versions for all three major operating systems. A free\/open-source alternative is SageMath<\/a> however running this on Windows<\/a> requires installing Windows Subsystem for Linux (WSL)<\/a> (easy, one command, make sure you have 16GB RAM machine, running Windows 10 or 11). SageMath has launched CoCalc which is effectively Sage in the Cloud<\/a>, under CEO William Stein, based in Seattle<\/a> (founded 2013). 2023 marks a big increase in the power of CoCalc: it is now integrated with ChatGPT 3.5 (free) and ChatGPT 4 (cents per query), Machine Learning, Neural Networks, with capabilities in Statistics, Big Data Analytics, production of Dashboards, LaTeX, a number of dynamic programming languages built in as different evaluation kernels, and more! Commercial alternatives are Mathematica<\/a>, Maple<\/a> and Magma<\/a>, but the licensing fees are not cheap (you might find trial or student versions). Mobile apps have also been released, e.g. Maple’s Maple Calculator<\/a>. If you’ve used alternatives and find them much better than Maxima, feel free to leave your views on the comparison in the comments at the end of this article.<\/p>\n <\/a><\/p>\n The fastest way to get up and running with Maxima is to use the fantastic web front end<\/a> by Omega-Math\/Vroom-Labs. This presents a simple user interface, with a calculator like layout, and allows you to just get going without installation, learning specialized syntax, etc. It has all the bells and whistles for those who already know Maxima, so that advanced users will have that power to hand.<\/p>\n To run Maxima, from your desktop computer, you can download the Installation on Windows is straight-forward: <\/p>\n The following resources will help you up the learning ramp.<\/p>\n For more detailed guides, see “Building Proficiency”<\/a> below.<\/p>\n Over the past 10 years (updated in 2023), I’ve used Maxima in a variety of discrete maths research problem. Below are some useful examples and code illustrating a few areas that may be of interest to others.<\/p>\n From the above, it is easy to see that the most expensive fruit basket is the full fruit basket of 9 fruits at $24. Less obvious: 1) there are 22 choices of fruit baskets with either 5 or 4 fruits, of which the most expensive is $18 [4 cantaloupes & 1 banana] and 2) the least expensive is $6 [2 apples & 2 bananas]. Less obvious problem with many more combinations possible can be handled with the same approach.<\/p>\n Three numerical solutions: Rounding to 3 places: Maxima makes it easy to work through hairy symbolic computations accurately leaving more time for applications and research. Below are examples. <\/p>\n <\/a><\/p>\n The following resources will help you up the learning ramp.<\/p>\n <\/a><\/p>\n <\/a><\/p>\n <\/a><\/p>\n![\"\"](\"https:\/\/mathscitech.org\/articles\/wp-content\/uploads\/2022\/11\/screenshot.2349-800x477.png\")
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\n<\/p>\n
\nMaxima and the Alternatives<\/h3>\n
Installing & Using Maxima<\/h3>\n
\nlatest installer for Windows and other OS’s<\/a> from the Maxima project page<\/a>. This will look something like maxima-clisp-sbcl-5.41.0-win64.exe which (as of Oct 2017) included in it both the computational engine (Maxima) and the graphical user interface (wxMaxima). (Installation pre-requisites<\/a>)<\/p>\n\n
\n
\n f(x):= 3*x+3$
\n plot2d(f(x), [x,-50,50])$
\n<\/code>
\nIf you get a strange error<\/a> such as ‘temp directory not found’, create a usable temp directory using the command:
\n
\n maxima_tempdir : \"C:\/totalprogs\/Maxima\/temp\"$
\n<\/code><\/p>\n
\n
\nExample 1: polynomial expansion: telescoping series
\n expand((1+x+x^2+x^3+x^4)*(1-x));
\nResult: 1-x^5
\nRemarks: ending statement with ; shows output, ending with $ suppresses output
\n<\/code><\/p>\n
\n
\n wxbuild_info()$;
\n<\/code>\n<\/ol>\nCheat Sheets \/ Ready Reference Sheets<\/h4>\n
\n
Application Examples<\/h4>\n
\n
\nConsider a monic polynomial of degree 3 with rational coefficients p(x).
\nBy the Fundamental Theorem of Algebra (Gauss), p(x) will have exactly 3 (possibly complex) roots a1,a2,a3.
\nSo we can write the polynomial in terms of the roots as follows:
\n
\n(x-a1)*(x-a2)*(x-a3)
\nexpand(%)
\nratsimp(%)
\n<\/code>
\nResult: you see the polynomial written as a function of the three roots a1, a2, a3
\nYou see the form of the coefficients in terms of sum of product combinations of the roots by size of the term: trace = a1 + a2 + a3, determinant = a1*a2*a3, and the other combinations in between.<\/p>\n
\nc(n,k):=binomial(n,k);
\np(n,k):=c(n,k)*k!;<\/code><\/p>\n
\nConsider an illustrative problem of cataloguing fruit baskets that can be made from 2 Apples, 3 Bananas, or 4 Cantaloupes. Apples are $1, Bananas are $2, Cantaloupes $4. The generating function below lists the 60 (=3x4x5) possible fruit baskets that can be made, including the empty fruit basket (no fruit).
\n
\nAll combinations of a fruit basket:
\nexpand((1+A+A^2)*(1+B+B^2+B^3)*(1+C+C^2+C^3+C^4));
\n<\/code>
\nWhat are the sizes, sizes\/costs, and costs\/compositions (i.e. which fruits to put in) of the fruit baskets? The power of the generating function approach is that you can read off details of the choices from the coefficients of the relevant expanded series.
\n
\nShowing size (x) of baskets:
\nexpand((1+x+x^2)*(1+x+x^2+x^3)*(1+x+x^2+x^3+x^4));
\nShowing size (x) and cost (y) of baskets:
\nexpand((1+y*x+y^2*x^2)*(1+y^2*x+y^4*x^2+y^6*x^3)*(1+y^4*x+y^8*x^2+y^12*x^3+y^16*x^4));
\nShowing cost (y) and composition (A,B,C) of baskets:
\nexpand((1+y*A+y^2*A^2)*(1+y^2*B+y^4*B^2+y^6*B^3)*(1+y^4*C+y^8*C^2+y^12*C^3+y^16*C^4));
\n<\/code><\/p>\n
\nHere’s an example of a truncated generating function that correctly gives the coefficients p(n) up to n=10 for the partition problem up to integer n.
\n
\nexpr:
\n(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10)*
\n(1+x^2+x^4+x^6+x^8+x^10)*
\n(1+x^3+x^6+x^9)*
\n(1+x^4+x^8)*
\n(1+x^5+x^10)*
\n(1+x^6)*
\n(1+x^7)*
\n(1+x^8)*
\n(1+x^9)*
\n(1+x^10);
\npowerdisp:true;
\nexpand(expr);
\n<\/code>
\nHere’s the two-variable generating function that gives p(n,k) as the coefficients, i.e. number of ways to partition n into k parts.
\n
\nexpr:
\n(1+k*x+k^2*x^2+k^3*x^3+k^4*x^4+k^5*x^5+k^6*x^6+k^7*x^7+k^8*x^8+k^9*x^9+k^10*x^10)*
\n(1+k*x^2+k^2*x^4+k^3*x^6+k^4*x^8+k^5*x^10)*
\n(1+k^1*x^3+k^2*x^6+k^3*x^9)*
\n(1+k*x^4+k^2*x^8)*
\n(1+k^1*x^5+k^2*x^10)*
\n(1+k*x^6)*
\n(1+k*x^7)*
\n(1+k*x^8)*
\n(1+k*x^9)*
\n(1+k*x^10);
\npowerdisp:true;
\nexpand(expr);
\n<\/code>
\n<\/a><\/p>\n
\nA American colleague of mine recently asked to check a solution to a curious system of 4 nonlinear algebraic equations.
\nWithout much more information, I threw it into Maxima (wxMaxima v17.0) and got three numerical solutions.
\nTotal elapsed time? 2 minutes. Another example of Maxima’s value as an symbolic calculator.
\n
\nalgsys(
\n [
\n (x+y+z)*t=1,
\n (2*x+y+z)*(t-2)=1,
\n (x+3*y+z)*(t-3)=1,
\n (x+y+4*z)*(t-4)=1
\n ], [x,y,z,t]);
\n<\/code><\/p>\n
\n
\n[x=1.859008365639665,y=-1.098952690501986,z=-0.3503503503503503,t=2.44077834179357],
\n[x=0.3619915308610291,y=0.7611952791495773,z=-0.8418458781362007,t=3.554406288787753],
\n[x=0.04370603203113307,y=0.03955473900356641,z=0.04432037496553626,t=7.838148667601683]
\n<\/code><\/p>\n
\n
\n[x=1.859, y=-1.099,z=-0.350,t=2.441],
\n[x=0.362, y=0.761, z=-0.842,t=3.554],
\n[x=0.0437,y=0.0396,z=0.0443,t=7.838]
\n<\/code><\/p>\n<\/ol>\nLearning through Applications<\/h3>\n
Topics by Example<\/h4>\n
\n
Building Proficiency ‘from the Ground Up’<\/h3>\n
Cheat Sheets \/ Ready Reference Sheets<\/h4>\n
\n
Basic Guides<\/h4>\n
\n
Advanced Guides<\/h4>\n
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“Book-Style” Tutorials (PDF or HTML)<\/h4>\n
\n
System Documentation<\/h4>\n
\n
Getting Help<\/h4>\n