Re: Why is 1 smaller than 0?

*To*: mathgroup at smc.vnet.net*Subject*: [mg73762] Re: Why is 1 smaller than 0?*From*: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>*Date*: Wed, 28 Feb 2007 04:24:16 -0500 (EST)*Organization*: Uni Leipzig*References*: <es12vk$nsk$1@smc.vnet.net>*Reply-to*: kuska at informatik.uni-leipzig.de

Hi, you call Minimize[] and every call loose some digits of precision, at least one per iteration. When the precision is sufficently decreased it turn out that 0.0274 plus minus 0.1 > 0.00001 is True. You see the loss of precision in your Print[] statements because the printed, trustable digist become less and less in every iteration. To restore the precision you have to add a SetPrecision[] and While[f @@ Last[x] > .00001, Print[f @@ Last[x], " ", f @@ Last[x] > 0.00001`50]; (*Get direction to travel in (downhill) from grad f.*) s = Append[s, -delf @@ Last[x]]; (*a tells us how far to travel. Need to minimize f to find it.*) xNew = Last[x] + a*delf @@ Last[x]; (*Minimizes f with respect to a.*) {min, theA} = Minimize[f @@ xNew, {a}]; (*Update x using the direction*) (* HERE RESTORE THE PRECISION *) xNew = SetPrecision[xNew /. theA, 50]; delta = Norm[Last[x] - xNew]; x = Append[x, xNew /. theA]; ++iteration; ]; work fine. Regards Jens Peter Jay Salzman wrote: > 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]; >