Romberg method

Romberg rule

A method for calculating a definite integral based on Richardson extrapolation. Suppose a value of some functional is to be calculated; also, let a calculated approximate value depend on a parameter so that as a result of the computations one obtains an approximate equality . Let some information be known concerning the behaviour of the difference as a function of , namely,

(1)

where is a positive integer and depends on the functional to be approximated, on the function on which this functional is calculated, on the approximating method, and (weakly) on . If simultaneously with , is calculated, then by Richardson's method one obtains for the approximation

(2)

This approximation is the better, the weaker the dependence of in (1) on . In particular, if is independent of , then (2) becomes an exact equality.

Romberg's method is used to calculate an integral

The interval is chosen to facilitate the writing; it can be any finite interval, however. Let

(3)

Calculations by Romberg's method reduce to writing down the following table:

where in the first column one finds the quadrature sums (3) of the trapezium formula. The elements of the -nd column are obtained from the elements of the -st column by the formula

(4)

When writing down the table, the main calculating effort is concerned with calculating the elements of the first column. The calculation of the elements of the following columns is a bit more complicated than the calculation of finite differences.

Each element in the table is a quadrature sum approximating the integral:

(5)

The nodes of the quadrature sum are the points , , and its coefficients are positive numbers. The quadrature formula (5) is exact for all polynomials of degree not exceeding .

Under the assumption that the integrand has a continuous derivative on of order , the difference can be represented in the form (1), where . Hence it follows that the elements of the -nd column, calculated by formula (4), are better Richardson approximations than the elements of the -st column. In particular, the following representation is valid for the error of the quadrature trapezium formula

and the Richardson method provides a better approximation to :

turns out to be a quadrature sum of the Simpson formula, and since for the error of this formula the following representation holds:

one can again use the Richardson method, etc.

In Romberg's method, to approximate one takes ; also, one assumes the continuous derivative on to exist. A tentative idea of the precision of the approximation can be obtained by comparing to .

This method was for the first time described by W. Romberg [1].

References

[1]	W. Romberg, "Vereinfachte numerische Integration" Norske Vid. Sels. Forh. , 28 : 7 (1955) pp. 30–36
[2]	F.L. Bauer, H. Rutishauser, E. Stiefel, "New aspects in numerical quadrature" N.C. Metropolis (ed.) et al. (ed.) , Experimental Arithmetic, high-speed computing and mathematics , Proc. Symp. Appl. Math. , 15 , Amer. Math. Soc. (1963) pp. 199–218