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MathGroup Archive 2007

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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];
> 


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