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?

Edited June 18, 20241 yr by fireshtormik

The theory of this is given by the Banach fixed point theorem otherwise known as the contraction mapping theorem.

https://web.stanford.edu/class/math51h/contraction.pdf

https://en.wikipedia.org/wiki/Banach_fixed-point_theorem

This gives the range on input values for which the process will converge to the desired solution and the conditions for convergence.

The trick with iterative solutions is to arrange or transform the equation into a form that converges over the desired range.

2 hours ago, studiot said:

The theory of this is given by the Banach fixed point theorem otherwise known as the contraction mapping theorem.

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

 

2 hours ago, studiot said:

The theory of this is given by the Banach fixed point theorem otherwise known as the contraction mapping theorem.

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

 

29 minutes ago, fireshtormik said:

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

 

It doesn't 'explain the equation'.

You supply an equation, the theorem applies in one way or another to all equations.

I will look out a picture that makes this clearer.

Can't seem to find the best one on big G.

OK this (payfor) tutor site works through a question from Kreysig Functional Analysis and its applications

https://www.chegg.com/homework-help/questions-and-answers/kreyszig-introductory-functional-analysis-applications-chapter-5-applications-banach-fixed-q34808855#question-transcript

 

At the moment my copy of Kreysig is buried, I should be able to retrieve it tomorrow but this is the picture I was looking for

 

912 × 996

 

 

I don't know what you maths background is or if you are seeking the underlying pure math or the (very widespread) applications ?

 

26 minutes ago, studiot said:

 

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.

 

Edited June 18, 20241 yr by fireshtormik