How do i add Aitken Sequence in my root finding algorithm

Question

from numpy import log as ln def g(x): return ln(4+x-x**2) def FixedPoint(p0,tolerance): p = g(p0) abs_error = abs(p-p0) p0 = p while abs_error>=tolerance: p = g(p0) abs_error = abs(p-p0) p0 = p return p starting_point = 2 tolerance = 10**-10 fixed_point = FixedPoint(starting_point,tolerance) print('Fixed-point of g(x) = {0} is x = {1:.7f}'.format(formula,fixed_point))

So, i have this fixed point root finding algorithm to find the root of ln(4+x-x^2), how do i add the aitken sequence program to increase the rate of convergence of finding the root of ln(4+x-x^2). im having trouble in keeping the value of each iterations.

I hope you are solving exp(x)=4+x-x^2. What did you already try for Aitken's delta squared process? It could look similar to math.stackexchange.com/a/3109837/115115 — Lutz Lehmann
– Lutz Lehmann, Commented Nov 5, 2019 at 11:00

Lutz Lehmann · Accepted Answer · 2019-11-05 12:25:47Z

I would reduce the number of lines in the fixed-point iteration via

def FixedPoint(p0,tolerance): p1 = g(p0) while abs(p1-p0)>=tolerance: p0,p1 = p1,g(p1) print(p0) return p1

This requires 29 evaluations of g for the given test case

The Aitken delta-squared process can be implemented as

def FixedPointAitken(p0,tolerance): while True: p1=g(p0); p2=g(p1); print(p0,p1,p2) if abs(p1-p0)<tolerance: break p0 = p0 - (p1-p0)**2/(p0+p2-2*p1) return p0

This requires 5 steps with 15 evaluations of g to reach the target accuracy.

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