Basic Quadrature Rules
The quadrature rules generally have the form
where the nodes and weights are generally pre-computed.
In the simplest case, Newton–Cotes formulas of even degree are used, where the nodes are evenly spaced in the interval:
- .
When such rules are used, the points at which has been evaluated can be re-used upon recursion:
A similar strategy is used with Clenshaw–Curtis quadrature, where the nodes are chosen as
Or, when Fejér quadrature is used,
- .
Other quadrature rules, such as Gaussian quadrature or Gauss-Kronrod quadrature, may also be used.
An algorithm may elect to use different quadrature methods on different subintervals, for example using a high-order method only where the integrand is smooth.
Read more about this topic: Adaptive Quadrature
Famous quotes containing the words basic and/or rules:
“Southerners, whose ancestors a hundred years ago knew the horrors of a homeland devastated by war, are particularly determined that war shall never come to us again. All Americans understand the basic lessons of history: that we need to be resolute and able to protect ourselves, to prevent threats and domination by others.”
—Jimmy Carter (James Earl Carter, Jr.)
“Can rules or tutors educate
The semigod whom we await?
He must be musical,
Tremulous, impressional,
Alive to gentle influence
Of landscape and of sky
And tender to the spirit-touch
Of man’s or maiden’s eye.”
—Ralph Waldo Emerson (1803–1882)