Why is 1 smaller than 0?

*To*: mathgroup at smc.vnet.net*Subject*: [mg73749] Why is 1 smaller than 0?*From*: p at dirac.org (Peter Jay Salzman)*Date*: Tue, 27 Feb 2007 05:49:58 -0500 (EST)

This is an implementation of "steepest descent" to minimize a function. It runs for 36 iterations. On the 36th iteration, it claims: 0.0274 > .00001 True which is true. After the While loop exits, it claims: 0.03 > .00001 False which is true, not false, as Mathematica claims. Why is the While loop exiting prematurely? How do I write this so that it runs for as long as f(x,y) is greater than .00001? Thanks! Pete (* Implements Minimization via method of steepest descent. *) Clear[f, x, s, delf, a , xNew, iteration, delta, tolerance, theA, min]; (* Function to minimize *) f[x_, y_] := 100`50*(y - x^2)^2 + (1 - x)^2; delf[x_, y_] := {-2*(1 - x) - 400.0`50*x(-x^2 + y), 200.0`50*(-x^2 + y)}; iteration = 0.0`50; (* Initial guess. *) x = { {5.0`50,1.0`50} }; (* Points "downhill" from the current position. *) s = {}; While[ f[Last[x][[1]],Last[x][[2]]] > .00001, Print[f[Last[x][[1]],Last[x][[2]]], " ", f[Last[x][[1]],Last[x][[2]]] > .00001]; (* Get direction to travel in (downhill) from grad f. *) s = Append[s, -delf[Last[x][[1]], Last[x][[2]]]]; (* a tells us how far to travel. Need to minimize f to find it. *) xNew = Last[x] + a*delf[Last[x][[1]], Last[x][[2]]]; (* Minimizes f with respect to a. *) {min, theA} = Minimize[f[xNew[[1]], xNew[[2]]], {a}]; (* Update x using the direction *) xNew = xNew /. theA; delta = Norm[Last[x] - xNew]; x = Append[x, xNew /. theA]; ++iteration; ]; Print["Convergence in ", iteration, " iterations.\n", "Delta: ", delta, ", tolerance: ", N[tolerance], "\n", "Minimum point at ", Last[x], "\n", "Value of f at min point: ", f[Last[x][[1]], Last[x][[2]]], "\n"]; Print[f[Last[x][[1]],Last[x][[2]]], " ", f[Last[x][[1]],Last[x][[2]]] > .00001];