Newton’s Method: Learn It 3

Other Iterative Processes

As mentioned earlier, Newton’s method is a type of iterative process. We now look at an example of a different type of iterative process.

Consider a function F and an initial number x0. Define the subsequent numbers xn by the formula xn=F(xn1).

This process is an iterative process that creates a list of numbers x0,x1,x2,,xn,. This list of numbers may approach a finite number x as n gets larger, or it may not.

 In the next example, we see an example of a function F and an initial guess x0 such that the resulting list of numbers approaches a finite value.

Let F(x)=12x+4 and let x0=0. For all n1, let xn=F(xn1). Find the values x1,x2,x3,x4,x5. Make a conjecture about what happens to this list of numbers x1,x2,x3,,xn, as n. If the list of numbers x1,x2,x3, approaches a finite number x, then x satisfies x=F(x), and x is called a fixed point of F.

Consider the function F(x)=13x+6. Let x0=0 and let xn=F(xn1) for n2. Find x1,x2,x3,x4,x5. Make a conjecture about what happens to the list of numbers x1,x2,x3,,xn, as n.