#### Bernoulli's Method for the Solution of Polynomial Equations

# This is elegant silly.

The solutions of the equation x^{2} - x - 1 = 0 are (1 + √5)/2 and (1 - √5)/2, the long side and the short side of the Golden Mean respectively. That is (1 + √5) : 1 and 1 : (1 - √5) are both the Golden Mean.

The Corresponding Difference Equation:

s_{n} - s_{n-1} - s_{n-2} = 0 when rearanged means the next element of the sequence equals the sum of the previous two - the famous Fibonacci sequence.

The Fibonacci sequence is: 0, 1, 1, 2, 3, 5, 8, 13, 21 etc....

# Form the corresponding sequence of quotients:

1, 2, 1.5, 5/3, 8/5, 13/8, 21/13 etc....

The sequence of quotients converges to one of the roots of the polynomial equation.

But every polynomial equation has an analogous difference equation. We may extract a root from any polynomial equation therefore, as long as the sequence of quotients converges.

The QD algorithm is a modern version of this which extracts ?all the roots and ?always converges (Help me Mother!)

Isn't life grand.