Simpson formula

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A special case of the Newton–Cotes quadrature formula, in which three nodes are specified:


Let the interval be broken up into an even number of subintervals , , of length , and calculate the integral over the interval by the quadrature formula (1):

Summation over from 0 to on both sides leads to the composite Simpson formula


where , . The quadrature formula (2) is also called Simpson's formula (that is, the word composite is dropped). The algebraic degree of accuracy of (2), and of (1), is equal to 3.

If the integrand has a continuous derivative of the fourth order on , then the error of the quadrature formula (2) — the difference between the left-hand and right-hand members of the approximate equation (2) — can be written as

where is some point in the interval .

Simpson's formula was named after Th. Simpson, who obtained it in 1743, although the formula was already known, for example to J. Gregory, in 1668.


Simpson's formula is also called Simpson's rule.


[a1] R. Courant, "Vorlesungen über Differential- und Integralrechnung" , 1 , Springer (1971)
[a2] D.M. Young, R.T. Gregory, "A survey of numerical mathematics" , Dover, reprint (1988) pp. §7.4
How to Cite This Entry:
Simpson formula. I.P. Mysovskikh (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098