Numeric calculations

Maxima specializes in symbolic operations, but it also offers numerical capabilities^[3] such as arbitrary-precision integer, rational number, and floating-point numbers, limited only by space and time constraints.

Programming

Maxima includes a complete programming language with ALGOL-like syntax but Lisp-like semantics. It is written in Common Lisp and can be accessed programmatically and extended, as the underlying Lisp can be called from Maxima. It uses gnuplot for drawing.

For calculations using floating point and arrays heavily, Maxima has translators from the Maxima language to other programming languages (notably Fortran), which may execute more efficiently.

Basic operations

bfloat(sqrt(2)), fpprec=40;

f(x):=x^3$
f(4);

expand((a-b)^3);

factor(x^2-1);

Solving equations

${\displaystyle x^{2}+a\ x+1=0}$

solve(x^2 + a*x + 1, x);

${\displaystyle [x=-{\Biggl (}{\frac {{\sqrt {a^{2}-4}}+a}{2}}{\Biggr )},x={\frac {{\sqrt {a^{2}-4}}-a}{2}}]}$

Solving equations numerically

${\displaystyle \cos x=x}$

find_root(cos(x) = x, x, 0, 1);

${\displaystyle 0.7390851332151607}$

bf_find_root(cos(x) = x, x, 0, 1), fpprec = 50;

${\displaystyle 7.3908513321516064165531208767387340401341175890076\cdot 10^{-1}}$

Indefinite integral

${\displaystyle \int x^{2}+\cos x\ dx}$

integrate(x^2 + cos(x), x);

${\displaystyle \sin x+{\frac {x^{3}}{3}}}$

Definite integral

${\displaystyle \int _{0}^{1}{\frac {1}{x^{3}+1}}\,dx}$

integrate(1/(x^3 + 1), x, 0, 1), ratsimp;

${\displaystyle {\frac {{\sqrt {3}}\log 2+\pi }{3^{\frac {3}{2}}}}}$

Numerical integral

${\displaystyle \int _{0}^{2}\sin(\sin(x))\,dx}$

quad_qags(sin(sin(x)), x, 0, 2)[1];

${\displaystyle 1.247056058244003}$

Derivative

${\displaystyle {d^{3} \over dx^{3}}\cos ^{2}x}$

diff(cos(x)^2, x, 3);

${\displaystyle 8\cos {x}\sin {x}}$

Limit

${\displaystyle \lim _{x\to \infty }{\frac {1+\sinh {x}}{e^{x}}}}$

limit((1+sinh(x))/exp(x), x, inf);

${\displaystyle {\frac {1}{2}}}$

Number theory

primes(10, 20);

${\displaystyle [11,13,17,19]}$

fib(10);

${\displaystyle 55}$

Series

${\displaystyle \sum _{x=1}^{\infty }{\frac {1}{x^{2}}}}$

sum(1/x^2, x, 1, inf), simpsum;

${\displaystyle {\frac {\pi ^{2}}{6}}}$

Series expansion

taylor(sin(x), x, 0, 9);

${\displaystyle x-{\frac {x^{3}}{6}}+{\frac {x^{5}}{120}}-{\frac {x^{7}}{5040}}+{\frac {x^{9}}{362880}}}$

niceindices(powerseries(cos(x), x, 0));

${\displaystyle \sum _{i=0}^{\infty }{\frac {(-1)^{i}x^{2i}}{(2i)!}}}$

Special functions

bessel_j(0, 4.5);

${\displaystyle -0.3205425089851214}$

airy_ai(1.5);

${\displaystyle 0.07174949700810543}$