#### fireshtorm1k

##### New member

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- Jun 18, 2024

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[imath]ln(f(x))=0[/imath], 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:

[math]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[/math]

Example:

* Using the classical Newton's method, the initial approximation [imath]x_0=111.625[/imath] leads to [imath]x_1=148.474[/imath]

* Using the above method, the initial value [imath]x_0=111.625[/imath] yields [imath]x_1=166.560[/imath], which is closer to the exact answer [imath]166.420[/imath]

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?