(* finding local minima for a function of one variable *) 

(*  Value of option Method 
     is either "Automatic", "Gradient", "ConjugateGradient", "QuasiNewton",
     "Newton", or "LevenbergMarquardt".
 *) 


	F[x_, y_] := x^2 + y^2; (* simplest possible *)



(*

 F[x_, y_] := x^20 + y^20;    (* ill-conditioned. Too shallow *) 

*)



(* F[x_, y_] := 100(y - x^2)^2 + (1-x)^2; *) (* Steepest descent (Gradient) fails *)

					(* Need ConjugateGradient or Newton *)

ContourPlot[F[x, y], {x, -1, 1}, {y, -1, 1}, ContourShading->False]

Plot3D[F[x, y], {x, -1, 1}, {y, -1, 1}]


answer = FindMinimum [F[x, y], {x, .5}, {y, .5}, PrecisionGoal -> 0.1, MaxIterations -> 50, Method->Newton] 

xmin =  x /. answer[ [2] ] 
ymin = y /. answer[ [2] ]
Fmin = answer[ [1] ]

Print["  "]
Print["Values of x,y that minimizes F, (xmin, ymin) =   ", xmin, "   ", ymin]
Print["  "]
Print["                          F(xmin, ymin)=     ", Fmin]
Print[" "]