**lvc@cbscd5.UUCP (Larry Cipriani)** (09/28/83)

Can anyone find a clever solution to the following problem? I haven't. Given: Y(n) = Y(n-1) - X(n-1) X(n) = X(n-1) - X(n-1)*X(n-1)/Y(n-1) Choose X(1) so as to maximize X(8) for Y(1) = 4 Larry Cipriani cbosgd!cbscd5!lvc

**g-rh@cca.UUCP (Richard Harter)** (10/02/83)

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.