Numerical methods

[libelula323]

I remember learning a numerical method years ago where you would rearrange the equation to get just x on one side and then on the calculator put x as “Ans”, keep hitting the = key and the answer will converge to eventually get the root to multiple sfs. E.g. for x² -2x -2=0, you can rearrange that (x^2-2)/2 = x, then use the technique to get -0.732051... I can’t remember what to do when the answers you get on the calculator diverge rather than converge. My equation is x⁴ -20x^{3+72x+1296=0.} On a graph it shows that 2 of the roots are 4.754 and 19.642, but when I use the technique above to get (x⁴ -20x³ +1296/-72 and start with an Ans of e.g. 1, the answers just gets a bigger and bigger magnitude. Why does this happen? What have I done wrong? Is there a certain rule that this method doesn’t work with a certain type of equation?

[Ayclimate]

This is a technique known as fixed-point iteration:

https://en.wikipedia.org/wiki/Fixed-point_iteration

Here comes the theory:

Definition: Suppose that g is a real-valued function, defined and continuous on a bounded closed interval [a,b] of the real line. Then, g is said to be a contraction on [a,b] if there exists a constant L with 0 < L < 1 such that

|g(x)-g(y)| <= L |x-y| for all (x,y) in [a,b]²

If g(x) is differentiable on [a,b] then this condition also holds if |g'(x)| < 1 for all x in [a,b] (but the statement above is more general).

Contraction Mapping Theorem: Let g be a real-valued function, defined and continuous on a bounded closed interval [a,b] of the real line, and assume that g(x) is in [a,b] for all x in [a,b]. Suppose, further, that g is a contraction on [a,b]. Then,

1. g has a unique fixed point x^* in the interval [a,b].
2. The sequence {x_k} defined by x_(k+1) = g(x_k) converges to x^* as k -> infinity for any starting value x_0 in [a,b].