Newton's Method Glitch

Question

I wonder if you can help me with my program - it involves using Newton's Method but my main issue is getting a defined function to return a given value.

Essentially, I want to get my function findRoot to return the value of the root or Null if it doesn't find one. Here is the code for this function:

findRoot[x0_] := Module[{Root},
(*Initialise dummy variable and counter index thing *)
dumx = x0;
l = 0;

While[l < 100,
Root = Iter[dumx];
If[Norm[dumx - Root] < TOL, Break[], dumx = Root]

 (*If true, break the while loop and return root, if false let dumx=Root*)
 ; l++]


If[l == 100, Null, Root]
];

Iter[] is just another function that performs iterations of Newton's method, I made it separate to allow ease of debugging!

Essentially, if I plug

{-0.3,-0.3,-0.3,-0.3}

into this algorithm, (With certain global variables set in ways where I already know what roots should come out of this function) I get

{-0.2Null,-0.2Null,-0.2Null,-0.2Null}

and I have no idea where the null is coming from. The result should read {-0.2,-0.2,-0.2,-0.2}. I suspect the semi-colon on line 7 is responsible for this Null, but removing it causes greater issues.

Answer 1

Here's an attempt at a clean example using a stand-in definition for Iter. Note: it's generally better to avoid creating symbols that start with a capital letter or shadow built-in functions, so I've changed a couple variable names in obvious ways:

findRoot[x0_] := Module[{myRoot, iter, tol, result},
  iter[x_] = RandomReal[];
  dumx = x0;
  tol = 1;
  l = 0;
  While[l < 100,
        myRoot = iter[dumx];
        If[Norm[dumx - myRoot] < tol, Break[], dumx = myRoot]; l++];
        result = If[l == 100, Null, myRoot]]

findRoot[10]
(*0.371588*)

Newton's method is the default approach implemented in the built-in function FindRoot, so the following is analogous to your code above, operating on the function Sin[x - 10] - x + 10:

Module[{f},
  f[x_] = Sin[x - 10] - x + 10;
  FindRoot[f[x], {x, 0}, Method -> "Newton", AccuracyGoal -> 1, PrecisionGoal -> 1]]