How to write Newton-Raphson module and return a value?

Question

I am writing a code which solve nonlinear algebraic systems via Newton-Raphson algorithm. I want to make a Newton Module with another file and use the Module in my test file, but i didn't achieve it. Because, If the method is not converged, I break the code and return "not converged". Hence, iteration is not continuing and my code is not working properly. Here is my code:

F[y1_,y2_] = {
((y1 - 11)^2 / 64) - ((y2-7)^2/100) -1,
(y1-3)^2 + (y2-1)^2 -400
}; (*Define nonlinear algebraic system vector *)

J[y1_,y2_] = Outer[D,F[y1,y2],{y1,y2}]; (* Find symbolic jacobian *)

{y10,y20}= {20,-4}; (* Initial values for newton iterations *)

Tol   = 10^-12;
counter = 0; (*How many newton iterations*)
DeltaY = {0.1,0.1};
MaxIter = 50;
list  = Table[{0,0},{i,1,MaxIter}];
(* Begin loop for newton iterations *)
i=1;
While[Norm[ DeltaY, 2] > Tol,
    Result = Solve[J[y10,y20].{{dy1},{dy2}}==-F[y10,y20],{dy1,dy2}] //N //Flatten ;  (*Solve linear system for Delta y *)
    DeltaY = {dy1,dy2}/.Result;
    {y10,y20} = {y10,y20} + DeltaY; (*Find new Y values with computed delta y solutions *)
    list[[i]] = {y10,y20}; (* Append new computed Y elements into list array *)
    Print[list[[i]]," and error:",DeltaY]       
    Print[Norm[ DeltaY, Infinity]]
    If[i>MaxIter,Print["Not converged"] Return[{0,0}];]
counter++
i++;
] //AbsoluteTiming

How can I make n dimensional newton iteration module without breaking code. Also how to return a value with a module.

Note: I will use my newton module for some implicit numerical schemes.

Best regards.

Answer 1

The simplest code to create your module is

nr[func_, jac_, y1_, y2_, MaxIter_, Tol_] := 
 Module[{y10 = y1, y20 = y2, counter = 0, DeltaY = {0.1, 0.1}, 
   list = Table[{0, 0}, {i, 1, MaxIter}], i = 1},
  While[Norm[DeltaY, 2] > Tol, 
   Result = Solve[jac[y10, y20].{{dy1}, {dy2}} == -func[y10, y20], {dy1, 
        dy2}] // N // Flatten;
   DeltaY = {dy1, dy2} /. Result;
   {y10, y20} = {y10, y20} + DeltaY;
   list[[i]] = {y10, y20};
   Print[list[[i]], " and error:", DeltaY, "  ", i, "  ", counter] ;
   Print[Norm[DeltaY, Infinity]]; 
   If[i > MaxIter, Print["Not converged"] Return[{0, 0}];] ;
   counter++ ;
   i++;]; {y10, y20}]

Call it with

nr[F, J, y10, y20, MaxIter, Tol]

and it will produce the Print statements produced by the code in your question, plus the answer, {22.519, -3.35977}. Of course, your final product should contain options, documentation, additional error checks, and the like. Hope this helps.

Answer 2

Why not make use of Mathematica built-in functional capabilities? The following code can be modified to add whatever stopping condition is desired.

   jac[f_, vars_] := Outer[D, f[Sequence @@ vars], vars]
   jacobian[f_, vars_, pt_] := jac[f, vars] /. Thread[vars -> pt]
   newt[f_, vars_, pt_] := pt - Inverse[jacobian[f, vars, pt]].f[Sequence @@ pt]

   f[x_, y_] := {x y - 3, x^2 + y^2 - 9}
   NestList[newt[f, {x, y}, #] &, {3., 1.}, 5]

(*
    {{3.,1.},{2.8125,1.0625},{2.80256,1.07042},{2.80252,1.07047},{2.80252,1.07047},{2.80252,1.07047}}
*)