The purpose of generating polynomials for a multistep method is to find a relationship between the previous and current values of a function, in order to predict future values. This is particularly useful in numerical analysis and solving differential equations.
Polynomials for a multistep method are generated using a combination of the Taylor series expansion and the method of undetermined coefficients. This involves finding a polynomial that satisfies the desired accuracy and stability criteria for the multistep method.
Explicit multistep methods use only known values of a function to calculate the next value, while implicit methods also use future values. This makes implicit methods more accurate, but also more computationally intensive.
Multistep methods are advantageous because they use multiple previous values of a function to predict future values, resulting in a more accurate solution. They also have a larger stability region, meaning they can handle larger step sizes without becoming unstable.
Some common types of multistep methods include the Adams-Bashforth method, the Adams-Moulton method, and the backward differentiation formula (BDF) method. Each of these methods has its own unique algorithm for generating polynomials and predicting future values of a function.