A pseudo Laguerre method

Aaron Melman

Newton's method to find the zero of a function in one variable uses the ratio of the function and derivative values, but it does not use the information provided by these quantities separately. It is a natural question to ask what a method would look like that does take into account these values instead of just their ratio. We answer that question in the case of a polynomial with all real zeros, the result being a method that is somewhat reminiscent of Laguerre's method.