Numerical Analysis Questions Medium
Simpson's rule is a numerical method used for approximating definite integrals. It is based on the idea of approximating the curve of a function by a series of parabolic arcs.
To apply Simpson's rule, the interval of integration is divided into an even number of subintervals. The width of each subinterval, denoted as h, is determined by dividing the total interval width by the number of subintervals.
Next, the function values at the endpoints and midpoints of each subinterval are evaluated. These function values are then used to construct parabolic arcs that approximate the curve of the function within each subinterval.
The integral of each parabolic arc is then calculated using the formula:
∫[a,b] f(x) dx ≈ (h/3) * [f(a) + 4f(a+h) + 2f(a+2h) + 4f(a+3h) + ... + 2f(b-h) + 4f(b-h) + f(b)]
Finally, the integral of the entire function is approximated by summing up the integrals of all the parabolic arcs.
Simpson's rule provides a more accurate approximation of the integral compared to simpler methods like the trapezoidal rule. It is particularly effective for functions that are smooth and have a relatively simple shape. However, it may not be as accurate for functions with sharp changes or discontinuities.