Bernoulli's Method For the Solution of Polynomial Equations

A number analytic means of solving polynomial equations, equations of the form
a₀xⁿ + a₁x^n-1 + ... = 0
(Where a_i 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:
s_n = (a₁s_n-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 q_n = s_n+1/s_n.

Simple With An Example

Take the equation x² - x - 1 = 0.

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

Take the first two elements of the sequence S₀, s₁, ... to be 0, 1. Then {s_n} = 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.