Muller's method is the same as Newton's method with a quadratic interpolating polynomial?

Question

I'm new to numerical analysis, and have been learning root finding algorithms. I am a bit confused about the difference between Muller's method, and Newton's method using an n-degree interpolating polynomial.

How is the Muller's method, which approximates f(x) using a quadratic polynomial, different from the Newton's method, where lets say we use a 2 degree interpolating polynomial to find roots of f(x)?

I could not find clear and concise theory on using the Newton method with a 2 degree interpolating polynomial to identify if its the same as or is different than the Muller's method.

Answer 1

One of the most basic techniques in numerical analysis, when solving a complicated problem, is to construct an approximately-similar easy problem, solve that to obtain an approximate solution, and restart the process using that solution as a new starting point. This is a very general technique, and it’s not even restricted to root finding methods.

Newton’s and Halley’s methods both share this construction, but that’s not enough to call them the “same” method: most iterative methods are constructed this way. So picking which specific approximation (a line, a quadratic, etc) to use and how is what defines the method, not the general approach.