A Formula for Iterating

Date: 10/11/2002 at 15:50:23
From: Raed Naser
Subject: Complex equations

Find the value of x if

   3 = sqrt(x+sqrt(x_sqrt(x+sqrt(x))))

When I try to solve this equation, the final equation was of degree 8.
Is there is a law for this kind of equation?

Regards.

Date: 10/12/2002 at 08:44:09
From: Doctor Mitteldorf
Subject: Re: Complex equations

Dear Raed,

Yes, to solve this equation algebraically is a big mess. It may not be 
possible. But you can find a numerical solution without too much 
difficulty.

The general method of Newton-Raphson works.  

But often in this kind of equation, you can find a numerical solution 
more easily by deriving from your equation a formula for iterating.  
The trick is to get x on one side of the equation, and a more slowly 
varying function of x on the other side. In your equation, you can 
square both sides to give 9 = x+sqrt(x+sqrt(x+sqrt(x))), then 
rearrange to give

   x = 9 - sqrt(x+sqrt(x+sqrt(x)))

The right side is "slowly varying" because sqrt(x) changes less 
rapidly than x. Therefore, this arrangement of the equation should 
work for iteration. Choose a guess for x; guess, for example, that
x = 2. Substitute x = 2 on the right side and get

   x = 9 - sqrt(2+sqrt(2+sqrt(2))) = 7.038

Now let x = 7.038, and find the next x:

   x = 9 - sqrt(7.038+sqrt(7.038+sqrt(7.038))) = 5.814

Continue with x = 5.814, and repeat the process. With just a few
iterations, you will home in on the 6.013...

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/