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/
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