       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][],Last[x][]] > .00001,

Print[f[Last[x][],Last[x][]], "  ", f[Last[x][],Last[x][]] > .00001];

(* Get direction to travel in (downhill) from grad f. *)
s = Append[s, -delf[Last[x][], Last[x][]]];

(* a tells us how far to travel.  Need to minimize f to find it. *)
xNew = Last[x] + a*delf[Last[x][], Last[x][]];

(* Minimizes f with respect to a. *)
{min, theA} = Minimize[f[xNew[], xNew[]], {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][], Last[x][]], "\n"];
Print[f[Last[x][],Last[x][]], "  ", f[Last[x][],Last[x][]] > .00001];

```

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