In this lab, you will explore what happens when you apply a linear function $ax+b$ again and again to a given initial value $x_0$. Using Sage to help you automate your investigations, you will observe the iteration process for many values of $a$, $b$, and $x_0$. If all goes well, patterns will start to emerge…. I will give you gidance on how to write your findings, and after a couple weeks, you will hand in a fancy article.
A short draft which includes your first section and at least part of your second section is due on Monday, September 9. A final draft is due Monday, September 16.
- To get the last element of an array
a, use the syntax
- To round a quantity
qoff with three decimal points of accuracy, use the syntax
My example code
def iterlin(a,b,x0,n=20): output = [x0] for i in range(n): output.append( a*output[-1]+b ) return [round(x,3) for x in output]
convergent_pairs =  divergent_pairs =  for a in vector(range(-30,31))/10 : for b in vector(range(-30,31))/10 : s = iterlin( a, b, 2 ) if abs(s[-1]-s[-2])<.1 : convergent_pairs.append( (a,b) ) else: divergent_pairs.append( (a,b) ) point(convergent_pairs,color="pink") + point(divergent_pairs,color="lightblue")