Given f, is there an automatic way to calculate fprime for Newton's method?

Question

The following was ported from the pseudo-code from the Wikipedia article on Newton's method:

#! /usr/bin/env python3
# https://en.wikipedia.org/wiki/Newton's_method
import sys

x0 = 1
f = lambda x: x ** 2 - 2
fprime = lambda x: 2 * x
tolerance = 1e-10
epsilon = sys.float_info.epsilon
maxIterations = 20

for i in range(maxIterations):
    denominator = fprime(x0)
    if abs(denominator) < epsilon:
        print('WARNING: Denominator is too small')
        break
    newtonX = x0 - f(x0) / denominator
    if abs(newtonX - x0) < tolerance:
        print('The root is', newtonX)
        break
    x0 = newtonX
else:
    print('WARNING: Not able to find solution within the desired tolerance of', tolerance)
    print('The last computed approximate root was', newtonX)

Question

Is there an automated way to calculate some form of fprime given some form of f in Python 3.x?

Answer 1

Answer

Define the functions formula and derivative as the following directly after your import.

def formula(*array):
    calculate = lambda x: sum(c * x ** p for p, c in enumerate(array))
    calculate.coefficients = array
    return calculate

def derivative(function):
    return (p * c for p, c in enumerate(function.coefficients[1:], 1))

Redefine f using formula by plugging in the function's coefficients in order of increasing power.

f = formula(-2, 0, 1)

Redefine fprime so that it is automatically created using functions derivative and formula.

fprime = formula(*derivative(f))

That should solve your requirement to automatically calculate fprime from f in Python 3.x.

Summary

This is the final solution that produces the original answer while automatically calculating fprime.

#! /usr/bin/env python3
# https://en.wikipedia.org/wiki/Newton's_method
import sys

def formula(*array):
    calculate = lambda x: sum(c * x ** p for p, c in enumerate(array))
    calculate.coefficients = array
    return calculate

def derivative(function):
    return (p * c for p, c in enumerate(function.coefficients[1:], 1))

x0 = 1
f = formula(-2, 0, 1)
fprime = formula(*derivative(f))
tolerance = 1e-10
epsilon = sys.float_info.epsilon
maxIterations = 20

for i in range(maxIterations):
    denominator = fprime(x0)
    if abs(denominator) < epsilon:
        print('WARNING: Denominator is too small')
        break
    newtonX = x0 - f(x0) / denominator
    if abs(newtonX - x0) < tolerance:
        print('The root is', newtonX)
        break
    x0 = newtonX
else:
    print('WARNING: Not able to find solution within the desired tolerance of', tolerance)
    print('The last computed approximate root was', newtonX)