numpy.polynomials

Goal

Understanding polynomials handling methods of numpy.

Includes

1. poly1d
2. polyval
3. polyfit

import numpy as np

poly1d

Creates a polynomial

Creating polynomial

# Creating polynomial of order 1
# 2x + 1
pol = np.poly1d([2, 1])
print(pol)

2 x + 1

# Creating a polynomal of order 2
# 2x^2 + 3x + 4
pol = np.poly1d([2, 3, 4])
print(pol)

   2
2 x + 3 x + 4

Evaluation

# Evaluating pol with x = 2
print(pol)
print(pol(2))
print(pol([2, 3, 4]))

   2
2 x + 3 x + 4
18
[18 31 48]

Arithmetic operations

# addition of pols
 
print(pol + pol)
 
# multiplication
print(pol * pol)
 
# division
 
print(pol / pol) # returns quotient and remainder
 
# square pols
print(pol ** 3)

   2
4 x + 6 x + 8
   4      3      2
4 x + 12 x + 25 x + 24 x + 16
(poly1d([1.]), poly1d([0.]))
   6      5       4       3       2
8 x + 36 x + 102 x + 171 x + 204 x + 144 x + 64

polyval

np.polynomial.polynomial.polyval([1,2,3, 4], [2, 3, 4])

array([ 9., 24., 47., 78.])

print(pol(4))

48

Issue?

print(pol)
print(pol(2))
print(np.polynomial.polynomial.polyval(2, pol))
# why both evaluations result different values?

   2
2 x + 3 x + 4
18
24.0

polyfit

polyfit is used to find the cofficient of a polynomial that best fits the given data points.

For example, there are data points such as $(1,1),(2,1),(3,4)$ and we need to find a polynomial that best fit these data points, we can use polyfit numpy method

How it finds the best fit polynomial?

numpy.polyfit uses least squares method.

# data points
x = np.array([1, 2, 3, 4], dtype="int64")
y = np.array([2, 3, 5, 7], dtype="int64")

# finding the coefficients using polyfit
 
coef = np.polynomial.polynomial.polyfit(x, y, 1)
coef

array([-1.38690251e-16,  1.70000000e+00])

# creating the polynomial
pol = np.poly1d(coef)
pol

poly1d([-1.38690251e-16,  1.70000000e+00])

_y_fit = np.polyval(coef, x) # another method
_y_fit

array([1.7, 1.7, 1.7, 1.7])

# fitted y_axis
y_fit = pol(x)

import matplotlib.pyplot as plt

plt.scatter(x, y, label="Data points")
plt.plot(x, y_fit, label="fitted line")
plt.xlabel("x")
plt.ylabel("y")
plt.legend()
plt.show()
print(y_fit)

[1.7 1.7 1.7 1.7]

Outputs

1. poly1d creates the polynomial with takes a list coefficients. The number of coefficients determine the degree of the polynomial. The resultant polynomial can be evaluated by calling it as a function and passing value of x.

2. np.polyval and np.polynomial.polynomials.polyval work different and give different results. See the section of polyval.

3. for polyfit, need a better data points to work with.