# 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 |

**How to Cite This Entry:**

Romberg method. I.P. Mysovskikh (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Romberg_method&oldid=16161