🚀 go-pugleaf

Date: 30 Apr 1997 14:19:37 GMT
Message-ID: <5k7kdp$6c8$1@www.umsmed.edu>
From Warren <wlmay@umsmed.edu>:

> A good book on solving these types of equations iteratively is 
> Burden and Faires, Numerical Analysis.  I have the 2nd and 3rd editions, 
> but I think there may be a later edition.
> Guess at the solution then iterate until some convergence criteria is 
> met.  The iterative relationship is
> 
> p_n = p_n-1 - f(p_n-1)/f'(p_n-1).

In very simple cases, the equation can be rearranged to yield x = f(x) 
where |f'(x*)| < 1 at the solution x*. Then the solution can be found by 
simply iterating x(n) = f(x(n-1)). For example, you can solve x - 
invcos(x) = 0 by rearranging to get x = cos(x) which obviously has a 
solution between 0 and PI/2 ~ 1.57. Arbitrarily choosing x(0) = 0.8, just 
keep pushing the cos button on your calculator until the display converges.
Vary x(0) and see what happens. Also try x(n) = invcos(x(n-1)). Similarly, 
solve the equation x + ln(x) = 0.

In general, the technique is inferior. However, it is easy and, for the 
inexperienced, can be both fun and instructive.

P.S. I don't think you'll have much fun with x - exp(x) = 0.

Greg

Hope this helps.

Gregory E. Heath     heath@ll.mit.edu      The views expressed here are
M.I.T. Lincoln Lab   (617) 981-2815        not necessarily shared by 
Lexington, MA        (617) 981-0908(FAX)   M.I.T./LL or its sponsors
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