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Composite midpoint rule on a uniform grid
Definition
Input real endpoints , an integer , and a function . If or , return invalid-grid without evaluating . For the remaining inputs, require a real-valued function evaluable at every point of . Put and for . The composite midpoint rule returns
The sum is the finite sum of Finite sums and finite products, by recursion. The admissibility conditions and ensure and all . The rule is an exact-real formula; floating-point evaluation and summation error are separate from its quadrature error.
Depends on
Used by
- Age-cohort quadrature along exact characteristics Algorithm
- Finite-volume size transport with positive division quadrature Algorithm
- Positive composite or capped-adaptive quadrature for renewal births Algorithm
- Standard midpoint Nyström operators need not converge in operator norm on C Counterexample
- Nyström quadrature operators and their reconstructed equations Definition
- A near-resonant Nyström equation amplifies midpoint quadrature error Example
- Convex data bracket the integral between composite midpoint and trapezoid values Example
- Direct midpoint quadrature loses order on a half-order singularity Example
- One midpoint sample destroys a quadratic stiffness coercivity bound Example
- Composite midpoint Nyström operators are collectively compact and consistent Lemma
- Midpoint quadrature defect for one-dimensional P1 stiffness Lemma
- Composite midpoint error bound for a twice continuously differentiable function Theorem
Dependency tree · two levels
13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- MATH 373 lecture notes, §8 Approximation of Integrals (standard reference, not scraped)
- Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, §5.6 (standard reference, not scraped)