Newton-Raphson method implementation in Python using SymPy. This Python library for symbolic mathematics allows us to derived whatever may be the function passed into the method.
from sympy import symbols, diff, lambdifyx = symbols('x')def NewRaph(f, x0, ϵ=1e-3, Nmax=100):
'''Approximate solution of f(x)=0 by Newton's method.
Parameters
----------
f : function
Function for which we are searching for a solution f(x)=0.
x0 : number
Initial guess for a solution f(x)=0.
ϵ : number
Stopping criteria is abs(f(x)) < epsilon.
Nmax : integer
Maximum number of iterations of Newton's method.
Returns
-------
xn : number
Implement Newton's method: compute the linear approximation of f(x) at xn and find x intercept by the formula
x = xn - f(xn)/df(xn)
Continue until abs(f(xn)) < epsilon and return xn.
If Df(xn) == 0, return None.
If the number of iterations exceeds Nmax, then return None.
Examples
--------
f = x**3 - x**2 - 1
NewRaph(f, x0=1)
Found solution after 6 iterations.
1.4655713749070918
'''
xn = x0
df = diff(f,x)
for n in range(Nmax):
fxn = lambdify(x, f, 'math')(xn)
if abs(fxn) < ϵ:
print('Found solution after',n,'iterations.')
return xn
dfxn = lambdify(x, df, 'math')(xn)
if dfxn == 0:
print('Zero derivative. No solution found.')
return None
xn = xn - fxn/dfxn
print('Exceeded maximum iterations. No solution found.')
return Nonef = x**3 - x**2 - 1
NewRaph(f, x0=1)Found solution after 5 iterations.
1.4655713749070918