Back when I was still student (last year) I studied/performed numerical integrations, everytime the quadrature formula was provided within the exercise. The exercise time came to an end and now I am wondering how is one supposed to choose between several known fomulas facing a situation where one have to integrate a discrete function ? Why a "progressive" method ? Why a method with weights and not a simple trapezium one ? Why Simpson's rules ?
As an exemple, I need to integrate a function for which the value is known for a hundred points or so. The historical method in my problem is to use a "four-point progressive interpolation method" (Mintz&Jordan), I cannot figure out how to confirm whether it is the method I need or not. In the first place why these "M&Jordan" invented a new method and did not use a more commom method (Newton-Cotes, Simpson, Gauss,...) ? Is there a rule to eleminate/chose formulas ?
Thanks a lot