Simpson's Rule for Numerical Integration: Accuracy and Applications

[unique_pavadrin]

Does the Simpson's rule of numerical integration (\frac{1}{3}h\left( {f_0 + 4f_1 + f_2 } \right)) give exact values for all polynomials to a third degree i.e., linear functions, quadratic functions, and cubic functions?

Is there a better method for numerical integration approximation? One which a better, more accurate result?

Many thanks

unique_pavadrin

 

[HallsofIvy]

Simpson's method approximates the function by a "piecewise" quadratic. If a function is already quadratic,then that is exact. So it will give exact results for second degree, but not third degree polynomials.

You certainly could develop a method that approximated the function by a "piecewise fourth degree polynomial but the extra work necessary would be more than just decreasing the step size in Simpson's rule. Simpson's rule is the most accurate of the regularly used methods.