A library for numerical calculations with multivariate polynomials. This Python script confirms Euler’s four-square identity (in 0.02 seconds):

import time
from mvpoly.dict import *

t0 = time.time()
a1, a2, a3, a4, b1, b2, b3, b4 = MVPolyDict.variables(8)
p = (a1**2 + a2**2 + a3**2 + a4**2) * (b1**2 + b2**2 + b3**2 + b4**2)
q = (a1*b1 - a2*b2 - a3*b3 - a4*b4)**2 + \
    (a1*b2 + a2*b1 + a3*b4 - a4*b3)**2 + \
    (a1*b3 - a2*b4 + a3*b1 + a4*b2)**2 + \
    (a1*b4 + a2*b3 - a3*b2 + a4*b1)**2
assert p == q, "failed Euler-Goldbach"
t1 = time.time()
print("%8.6f" % (t1 - t0))

The Octave package can be installed using the command

sudo octave --eval "pkg install mvpoly-octave-0.98.3.tar.gz"

For the Matlab package, see the included README for installation instructions.

A Python module can be found on GitLab and the documentation here. It can be installed with

python3 -m pip install mvpoly \

NetBSD users can install the Python package py-mvpoly via pkgsrc.

The package source can be found on the GitLab page.

mvpoly plot
A bi-variate polynomial, evaluated by mvpoly, plotted by matplotlib