Romberg's method is a technique for estimating a definite integral over a closed (or open) range Loading...:
Loading...By applying Richardson extrapolation (see richardson.cljc
) to either the Trapezoid method or the Midpoint method.
The implementation of Richardson extrapolation in this library can be applied to any methods; many of the numerical quadrature methods (Simpson, Simpson's 3/8, Milne, Boole) involve a single step of Richardson extrapolation.
Romberg integration goes all the way. A nice way to think about this algorithm is this:
Generate a sequence of estimates of the definite integral using the Trapezoid or Midpoint methods on a geometrically increasing number of integration slices of width Loading.... This gives you a sequence of Loading... points of the form Loading..., where Loading... is the integral estimate.
Each time a new point becomes available, fit a polynomial of order Loading... to all Loading... points... and then extrapolate to Loading..., the magical area estimate where the width of each integration slice is 0.
For a wonderful reference that builds up to the ideas of Richardson extrapolation and Romberg integration, see Halfant and Sussman's "Abstraction in Numerical Methods".
References: