Numerically solving an equation

Question

Algorithm to be coded in C#:

fn = f(xn)
f′n = df(xn)/dx
∆xn = -fn / f′n
Update: xn+1 = xn + ∆xn
Repeat the process until ∆xn ≤ e

I must use the Newton-Raphson method to solve but I do not know how to do a loop that puts in the next answer each time. How do I compute this?

This is my broken code

double a = 1, Lspan = 30, Lcable = 33, fn, fdn, dfn, j;
fn = (2 * a * (Math.Sinh(Lspan / 2 * a))) - Lcable;
fdn = (2 * (Math.Sinh(Lspan / 2 * a)) - ((Lspan / 2 * a) * Math.Cosh(Lspan / 2    * a)));
dfn = -fn / fdn;
do
    j = a + dfn;
while (dfn > 0.00000000001);

Console.WriteLine( " {0} ",j) ;
Console.ReadKey();

Answer 1

Your loop performs the same calculation each time, because neither a or dfn change between iterations. I'm sure I've actually implemented a Newton-Raphson method myself years ago, but I don't remember enough about it to check that your arithmetic is correct without looking it up.

I expect that you intended fdn and dfn to be updated on each iteration - although your pseudocode statement of the method is ambiguous since it implies that only the whole solution is updated on each iteration, whereas actually each term needs to be updated or you'll just keep adding the starting value of ∆xn forever. I think the solution is to move the second, third and fourth lines inside the loop.

Does this make sense?

(It looks as though you were expecting C# to work with symbolic mathematics, which isn't the case. C# is basically procedural within the body of a method, so making an assignment statement fn = some terms; happens once, when the program hits that line. There is no knowledge built into that variable of how it was calculated, it's just a box with a number in it.)