Bernoulli's Method For the Solution of Polynomial Equations

A number analytic means of solving polynomial equations, equations of the form
a0xn + a1xn-1 + ... = 0
(Where ai is a constant and may be complex).

In its simplest form, given here, it finds the root of maximum modulus.

First form the corresponding difference equation:
sn = (a1sn-1 - ...)/a0.
Then use the implied recurrence relation to find the sequence s0, s1, etc..

The sequence of quotients q0, q1, etc. then converges to the root of maximum modulus. Where qn = sn+1/sn.

# Simple With An Example

Take the equation x2 - x - 1 = 0.

The associated difference equation is: sn = sn-1 + sn-2

Take the first two elements of the sequence S0, s1, ... to be 0, 1. Then {sn} = 0, 1, 1, 2, 3, 5, 8, 13, 21, .... (The famous Fibonacci - the way rabbit populations allegedly grow - sequence).

The sequence of quotients is 1/0, 1/1, 2/1, 3/2, 5/3, .... This converges to (1+sqrt5)/2 - a solution of the original equation and the Greeks' "Golden" ratio.