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From: g-rh@cca.UUCP (Richard Harter)
Newsgroups: net.math
Subject: Iteration Problem
Message-ID: <5794@cca.UUCP>
Date: Sat, 1-Oct-83 17:02:10 EDT
Article-I.D.: cca.5794
Posted: Sat Oct  1 17:02:10 1983
Date-Received: Sun, 2-Oct-83 21:22:44 EDT
Lines: 60


	The posted solution to the iteration equation by Silberberg is
incorrect.  The expressions for X(n) and Y(n) are correct.  However,
the derivative is calculated incorrectly.  A complete (and, I hope,
correct) solution follows.

Problem:

Given the iteration equations

	X(n)=X(n-1)-X(n-1)^2/Y(n-1)
	Y(n)=X(n-1)-Y(n-1)

Find the value of X(1) that maximizes X(8) when Y(1)=4.

Solution:

Substituting Y(N) in the equation for X(N) yields

	Y(N)/X(N)=Y(N-1)/X(N-1)

By induction Y(N)/X(N)=y/x where x=X(1) and y=Y(1).  It follows
that there there is a function w(N) such that

	X(N)=x*w(N)
	Y(N)=y*w(N)

Subsituting in the equation for Y(N) yields

	w(N)=(1-x/y)*w(N-1)

By induction we get

	w(N)=  (1-x/y)^(N-1)
	X(N)=x*(1-x/y)^(N-1)
	Y(N)=y*(1-x/y)^(N-1)

The inflection points of X(N) as a function of x for fixed y are
given by

	dX(N)/dx = (1-x/y)^N-1 -(x/y)(N-1)(1-x/y)^N-2
		 = (1-N*x/y)*(1-x/y)^(N-2)
		 = 0

which has roots

	x = y/N
	x = y

The location of relative maxima, minima, and stationary points 
is complicated; it depends on whether N is odd or even, whether
x>y, and whether x>0.  In the original problem, x=1/2 is both a
local and absolute maximum and x=4 is a stationary point.  There
is no minimum.

Extra Credit:

I.	Characterize X(N) as a function of x for arbitrary N and y.
II.	Show that the maximum of X(N) is assymptotically e*y/N
	for large y.