Question

Python 3 programming:

Implementing numpy, create a code for the Broyden 1 method of solving roots.

Answer 1

code:

from __future__ import division
import numpy as np
import scipy.linalg as sla

def broyden_good(x, y, f_equations, J_equations, tol=10e-10, maxIters=50):
steps_taken = 0
f = f_equations(x,y)
J = J_equations(x,y)
  
while np.linalg.norm(f,2) > tol and steps_taken < maxIters:
s = sla.solve(J,-1*f)
x = x + s[0]
y = y + s[1]
newf = f_equations(x,y)
z = newf - f
  
J = J + (np.outer ((z - np.dot(J,s)),s)) / (np.dot(s,s))
  
f = newf
steps_taken += 1
  
return steps_taken, x, y

tol = 10.0** -15
maxIters = 50
x0 = 1
y0 = 2

def fs(x,y):
return np.array([x + 2*y - 2, x**2 + 4*y**2 - 4])

def Js(x,y):
return np.array([[1,2],[2, 16]])

n, x, y = broyden_good(x0, y0, fs, Js, tol, maxIters=50 )
print("iterations: ", n)
print("x and y: ", x, y)

Output: