Bernoulli's Method For the Solution of Polynomial Equations

A number analytic means of solving polynomial equations, equations of the form

a_{0}x^{n} + a_{1}x^{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} = (a_{1}s_{n-1} - ...)/a_{0}.

Then use the implied recurrence relation to find the sequence s_{0}, s_{1}, etc..

The sequence of quotients q_{0}, q_{1}, etc. then converges to the root of maximum modulus. Where q_{n} = s_{n+1}/s_{n}.

# Simple With An Example

Take the equation x^{2} - x - 1 = 0.

The associated difference equation is: s_{n} = s_{n-1} + s_{n-2}

Take the first two elements of the sequence S_{0}, s_{1}, ... 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.