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.
Read more about this topic: Adaptive Quadrature
Famous quotes containing the words general and/or scheme:
“[The Republican Party] consists of those who, believing in the doctrine that mankind are capable of governing themselves and hating hereditary power as an insult to the reason and an outrage to the rights of men, are naturally offended at every public measure that does not appeal to the understanding and to the general interest of the community.”
—James Madison (1751–1836)
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)