It is a well-known fact that, for solving algebraic equations, the bisection method has a linear rate of convergence, the secant method has a rate of convergence equal to 1.62 (approx.) and the Newton-Raphson method has a rate of convergence equal to 2.
Is there any numerical method for solving algebraic equations with a rate of convergence greater than 2?
Best Answer
As mentioned, the Halley method has cubic convergence, the so-called "Householder methods" in generalization of that have higher orders of convergence. However, if you count the convergence rate in terms of complexity and that in terms of function evaluations, i.e., the "Ostrowski index", where derivatives of order $k$ count as $k$ (or $2k$) function evaluations, then you reach the point of diminishing returns at convergence order 2-3. I put numbers to that at https://en.wikipedia.org/wiki/Householder%27s_method