## Polynomials in python

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Polynomials can be represented as a list of coefficients. For example, the polynomial $$4*x^3 + 3*x^2 -2*x + 10 = 0$$ can be represented as [4, 3, -2, 10]. Here are some ways to create a polynomial object, and evaluate it.

import numpy as np

ppar = [4, 3, -2, 10]
p = np.poly1d(ppar)

print p(3)
print np.polyval(ppar, 3)

x = 3
print 4*x**3 + 3*x**2 -2*x + 10
139
139
139

numpy makes it easy to get the derivative and integral of a polynomial.

Consider: $$y = 2x^2 - 1$$. We know the derivative is $$4x$$. Here we compute the derivative and evaluate it at x=4.

import numpy as np

p = np.poly1d([2, 0, -1])
p2 = np.polyder(p)
print p2
print p2(4)

4 x
16

The integral of the previous polynomial is $$\frac{2}{3} x^3 - x + c$$. We assume $$C=0$$. Let us compute the integral $$\int_2^4 2x^2 - 1 dx$$.

import numpy as np

p = np.poly1d([2, 0, -1])
p2 = np.polyint(p)
print p2
print p2(4) - p2(2)
3
0.6667 x - 1 x
35.3333333333

One reason to use polynomials is the ease of finding all of the roots using numpy.roots.

import numpy as np
print np.roots([2, 0, -1]) # roots are +- sqrt(2)

# note that imaginary roots exist, e.g. x^2 + 1 = 0 has two roots, +-i
p = np.poly1d([1, 0, 1])
print np.roots(p)
[ 0.70710678 -0.70710678]
[ 0.+1.j  0.-1.j]

There are applications of polynomials in thermodynamics. The van der waal equation is a cubic polynomial $$f(V) = V^3 - \frac{p n b + n R T}{p} V^2 + \frac{n^2 a}{p}V - \frac{n^3 a b}{p} = 0$$, where $$a$$ and $$b$$ are constants, $$p$$ is the pressure, $$R$$ is the gas constant, $$T$$ is an absolute temperature and $$n$$ is the number of moles. The roots of this equation tell you the volume of the gas at those conditions.

import numpy as np
# numerical values of the constants
a = 3.49e4
b = 1.45
p = 679.7   # pressure in psi
T = 683     # T in Rankine
n = 1.136   # lb-moles
R = 10.73       # ft^3 * psi /R / lb-mol

ppar = [1.0, -(p*n*b+n*R*T)/p, n**2*a/p,  -n**3*a*b/p];
print np.roots(ppar)
[ 5.09432376+0.j          4.40066810+1.43502848j  4.40066810-1.43502848j]

Note that only one root is real (and even then, we have to interpet 0.j as not being imaginary. Also, in a cubic polynomial, there can only be two imaginary roots). In this case that means there is only one phase present.

## 1 Summary

Polynomials in numpy are even better than in Matlab, because you get a polynomial object that acts just like a function. Otherwise, they are functionally equivalent.