Adaptive Simpson's method, also called adaptive Simpson's rule, is a method of
numerical integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...
proposed by G.F. Kuncir in 1962.
It is probably the first recursive adaptive algorithm for numerical integration to appear in print,
[For an earlier, non-recursive adaptive integrator more reminiscent of ODE solvers, see ] although more modern adaptive methods based on
Gauss–Kronrod quadrature and
Clenshaw–Curtis quadrature are now generally preferred. Adaptive Simpson's method uses an estimate of the error we get from calculating a definite integral using
Simpson's rule
In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761).
The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads
\int_a^b f(x) ...
. If the error exceeds a user-specified tolerance, the algorithm calls for subdividing the interval of integration in two and applying adaptive Simpson's method to each subinterval in a recursive manner. The technique is usually much more efficient than
composite Simpson's rule
In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761).
The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads
\int_a^b f(x) \ ...
since it uses fewer function evaluations in places where the function is well-approximated by a
cubic function
In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d
where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degre ...
.
Simpson's rule
In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761).
The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads
\int_a^b f(x) ...
is an interpolatory quadrature rule which is exact when the integrand is a polynomial of degree three or lower. Using
Richardson extrapolation
In numerical analysis, Richardson extrapolation is a sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value A^\ast = \lim_ A(h). In essence, given the value of A(h) for several values of h, ...
, the more accurate Simpson estimate
for six function values is combined with the less accurate estimate
for three function values by applying the correction
. So, the obtained estimate is exact for polynomials of degree five or less.
Mathematical Procedure
Defining Terms
A criterion for determining when to stop subdividing an interval, suggested by J.N. Lyness,
is
where