How to write mathematical sequences in python

Question

I'm completely stuck on a task in one of the exercises we've been given however and was hoping someone could help me with it.

The following is the actual task:

Consider the sequence: x(n+1)= 0.2x(n)−α(x(n)^2−5) with x(0)= 1 for α successively equal to -0.5, +0.5, -0.25, +0.25.

• Check the convergence; if the sequence converges, print the message Sequence converged to x= (the value you got) otherwise print No convergence detected

• Check whether there are negative elements in the sequence

(Hint: If |xn−xn−1| < 10−9 consider a sequence to be convergent)

I'm not sure how to do sequences in python though and haven't found anything that explains how via googling so have been unsuccessful in trying to do it so far. After numerous attempts, I'm completely stuck.

This is the code I've done:

conv = [-0.5, 0.5, -0.25, 0.25]
b=0
a=conv[b]
c=0

for x in conv:
    x1=0.2*x-a*((x**2)-5)
    if abs(x1 - x) < 1.e-9:
        c += 1        
    x = x1
    b += 1

if c > 0:
    print('Sequence converged to x=' + x) 

if c === 0: 
    print('No converge detected')

Answer 1

The procedure is called 'fixed-point iteration'. The question is similar to this SO question, asked yesterday (and likely others).

The sequence definition shows a as a constant. Indeed, letting a vary for a given sequence in the way indicated makes no sense as it would guarantee non-convergence. The instructor's notation is sloppy, but I am sure that the intent is for students to run 4 iterations, one for each value of a1. (I say this also because I know what fixed-point iteration behaviors are illustrated by the particular choices for a.)

The code below mixes my answer from the link above with the update code from this question. N is chosen as large enough for this problem (I started larger). Charon, I leave it to you to use the results to answer the questions posed by the instructor.

for a in [-0.5, 0.5, -0.25, 0.25]:
    x = 1.0
    n = 30
    for i in range(n):
        x1 = 0.2*x - a*(x**2 - 5)
        print(i, x)  # remove before submitting
        if abs(x1 - x) < 1.e-9:
            print('a = {a}, n = {i}, result is {x}.'.format(a=a, i=i, x=x))
            break
        x = x1
    else:
        print('No convergence within {n} iterations.'.format(n=n))