Modified Secant method Javascript algorithm?

Question

I need your help to code this New Secant Method into Javascript.

The secant method uses two starting parameters, while this new method require only one starting parameter.

Thank you in advance.

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Thank you for your answer [ed: comment], here is what i have tried, but it dosn't work :

<script>
    var x
    var pi = 3.141592653589793, e = 2.7182818284590452;

    function meth_sec() {
        with (Math) {
            f = "sin(x)-x/2"; a = 1; s = 1;
            f = prompt("your function f(x) = ", f)
            while (s >= 0) {
                a = eval(prompt("type the approximation. = ", a))                   
                x = a;                     
            }
        }
    } function iter() {
        with (Math) {
            y = eval(f)
            x = a + (abs(y)/2) - abs(y)*((y(a+abs(y/2)))/((y(a+abs(y/2)))-y)
            return x
        }
    }
    function sgn(y) { return (y > 0) - (y < 0) }

</script>

Answer 1

This looks like a variation of the Steffensen method, known for its close relation to Aitken's delta-squared process. It is less a secant method and more a Newton-like method, replacing the derivative f'(x) in x_new = x-f(x)/f'(x) with a divided difference approximation (f(x+h(x))-f(x))/h(x) resulting in the formula

x_new = x - (f(x)*h(x))/(f(x+h(x))-f(x))
      = x + h(x) + (f(x+h(x))*h(x))/(f(x+h(x))-f(x))

where h(x)=O(f(x)) to force convergence towards the derivative as x approaches the root. Steffensen's method uses h(x)=f(x) while the cited method for some reason prefers h(x)=abs(f(x))/2. Neither is invariant under re-scaling of f, so there is no clear preference, there will be cases where either of the methods has a slight advantage.

By its relation to the Newton method one would expect quadratic convergence to simple roots, if there is convergence at all.

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Your problem is that you use y both as value and as function. Apart from the security problem of using eval on user-supplied code snippets, you need to supply the x argument to supplement the function string, and then call the evaluation of the function string, and that for all the different points you want function values for.

A discussion on how to safely parse and evaluate expression strings is Evaluating a string as a mathematical expression in JavaScript.