fireshtormik

I understand what you're talking about, but as far as I know, the Fixed-point iteration method converges slower than Newton's method. I note that there is a derivative in this formula.

Can you explain in more detail, and suggest how specifically this theorem explains this equation ?

Can you explain in more detail, and suggest how specifically this theorem explains this equation ?

I came across an intriguing iterative algorithm for solving a nonlinear equation of the form ln(f(x))=0 , which differs from the classical Newton's method. This method utilizes a logarithmic difference to calculate the next approximation of the root. A notable feature of this method is its faster convergence compared to the traditional Newton’s method. The formula for the method is as follows: $$x_{n+1} = \frac{\ln(f(x + dx)) - \ln(f(x))}{\ln(f(x + dx)) - \ln(f(x)) \cdot \frac{x_n}{x + dx}} \cdot x_n$$ Example: Using the classical Newton's method, the initial approximation x0=111.625 leads to x1=148.474 Using the above method, the initial value x0=111.625 yields x1=166.560 which is closer to the exact answer 166.420 Questions: 1. How is this formula derived? 2. Can this method be expected to provide a higher rate of convergence for a broad class of nonlinear functions? 3. What are the possible limitations or drawbacks of this method?