If each rule in a family is embedded in another rule in the same family, then the rules of that family are called progressive. (For example, trapezoidal rule of degree 4 might be referred to as. Īn integration rule of degree that is a member of a family of rules with a common derivation and properties but different degrees will be denoted as, where might be chosen to identify the family. If the set of sampling points of a rule of degree contains the set of sampling points of a rule of a lower degree, that is,, then is said to be embedded in. Each null rule may be thought of as the difference between a basic integration rule and an appropriate integration rule of a lower degree. Ī null rule of degree will integrate to zero all monomials of degree and will fail to do so for at least one monomial of degree. Ī multidimensional integration rule is said to be of degree if it integrates exactly all monomials of degree or less, and will fail to do so for at least one monomial of degree, that is, the rule is exact for all monomials of the form, where is the dimension,, and. "Ī one-dimensional integration rule is said to be of degree if it integrates exactly all polynomials of degree or less, and will fail to do so for at least one polynomial of degree. The application of an integration rule to a function will be referred as an integration of, for example, "when is integrated by, we get. The integration rule is said to be exact for the function if. When these rules are applied to other regions, their abscissas and estimates need to be scaled accordingly. So if is one of these regions, will be used instead of. The sampling points of the rules considered below are chosen to compute estimates for integrals either over the interval, or the unit cube, or the "centered" unit cube, where is the dimension of the integral. If a rule is applied over the region this will be denoted as, where is the integrand. An integration rule is a functional, that is, it maps functions over the interval (or a more general region) into real numbers. An integration rule estimates the integral with the weighted sum. Corresponding to each sampling point there is a weight number. In the literature these are also called abscissas. A separate discussion applies to other types of rules such as "LevinRule".Īn integration rule samples the integrand at a set of points, called sampling points. The following general notes pertain to weighted sum type integration rules such as "GaussKronrodRule" and "MultidimensionalRule". In the context of NIntegrate usage, an integration rule object provides both an integral estimate and an error estimate as a measure of the integral estimate's accuracy. An integration rule computes an estimate of an integral over a region, typically using a weighted sum.
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