Simpsonregeln

Simpsonregeln, also known as Simpson's rule, is a numerical method used for approximating definite integrals. It is a composite quadrature rule, meaning it divides the integration interval into smaller subintervals and applies a specific formula to each. Simpson's rule is known for its accuracy, often providing a better approximation than simpler methods like the trapezoidal rule or midpoint rule for the same number of subintervals.

The basic form of Simpson's rule, sometimes called Simpson's 1/3 rule, approximates the integral of a function f(x) over an interval [a, b] by fitting a quadratic polynomial to the function at three points: the endpoints a and b, and the midpoint (a+b)/2. The formula for a single application of Simpson's rule over an interval [a, b] is given by (b-a)/6 * [f(a) + 4f((a+b)/2) + f(b)].

For greater accuracy, the composite Simpson's rule is typically employed. This involves dividing the entire integration interval [a, b] into an even number of subintervals, say n. Then, Simpson's rule is applied to pairs of these subintervals. The formula for the composite Simpson's rule with n subintervals (where n must be even) is h/3 * [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)], where h = (b-a)/n and x_i = a + i*h. Notice the alternating coefficients of 4 and 2 for the interior points.

Simpson's rule is particularly effective for integrating functions that can be well-approximated by quadratic polynomials. Its mathematical basis lies in approximating the function within each pair of subintervals with a parabola. The higher accuracy of Simpson's rule compared to lower-order methods stems from its use of quadratic interpolation rather than linear interpolation.