Numerical Analysis Questions
Simpson's rule is a numerical integration method used to approximate the definite integral of a function. It is based on approximating the function by a series of quadratic polynomials and then integrating each polynomial over a small interval. The rule states that the integral of a function f(x) over an interval [a, b] can be approximated by the formula:
∫[a,b] f(x) dx ≈ (h/3) [f(a) + 4f(a+h) + 2f(a+2h) + ... + 2f(b-h) + 4f(b-h) + f(b)]
where h is the step size, given by (b-a)/n, and n is the number of subintervals. Simpson's rule provides a more accurate approximation compared to other numerical integration methods, such as the trapezoidal rule, especially for functions with higher order derivatives.