General Scheme
Adaptive quadrature follows the general scheme
1. procedure integrate ( f, a, b, tau ) 2. 3. 4. if then 5. m = (a + b) / 2 6. Q = integrate(f,a,m,tau/2) + integrate(f,m,b,tau/2) 7. endif 8. return QAn approximation to the integral of over the interval is computed (line 2), as well as an error estimate (line 3). If the estimated error is larger than the required tolerance (line 4), the interval is subdivided (line 5) and the quadrature is applied on both halves separately (line 6). Either the initial estimate or the sum of the recursively computed halves is returned (line 7).
The important components are the quadrature rule itself
the error estimator
and the logic for deciding which interval to subdivide, and when to terminate.
There are, of course, several variants of this scheme. The most common will be discussed later.
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