# Composite trapezoidal quadrature

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

On a strictly increasing partition, add each subinterval width times its two-endpoint average to approximate the definite integral.

mko-trapezoidal-quadrature · algorithm · v0.10.0



## Mathematical statement

Given a=x_0<x_1<...<x_n=b and every endpoint value y_i=f(x_i), the composite trapezoid value is sum_(i=1)^n (x_i−x_(i−1))*(y_(i−1)+y_i)/2. Nonuniform partitions are allowed: use each interval own width and exact rational arithmetic. There are n+1 nodes for n subintervals. Affine functions are exact on every qualified partition. A full quadratic polynomial permits an exact reference integral on the specified partition, signed error I−T and separate absolute error; this is not a general second-derivative error bound or convergence-order theorem. A sample table unbound to a complete function specifies only this quadrature sum, not a unique integrand or integral.





## Representative formula (exact TeX)

T(P,f)=q





## Assumptions


- Node/value arrays are aligned and have at least two entries; nodes are strictly increasing exact rationals.
- The quadratic reference family declares the whole polynomial so its exact definite integral can be computed separately.
- Signed error is exact integral minus trapezoid value; absolute error is reported separately.





## Symbol scope


- T — composite-trapezoid operator
- P — strict partition
- f — integrand or endpoint table
- q — exact quadrature value





## Intuition, definition, and misconceptions

Qualify the partition and aligned node/value arity first.

Every nonuniform subinterval keeps its own width rather than a single h.

Affine exactness and one quadratic reference error do not constitute a general error theorem.

n nodes do not make n subintervals; a nonuniform partition cannot use one global h, and affine exactness does not imply exactness for all functions.





## Adopted derivation guide

[Nonuniform trapezoids and signed error](https://ocme.evemisslab.com/en/derivations/derivation-mko-trapezoidal-quadrature-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Signed polynomial definite integrals on finite intervals](https://ocme.evemisslab.com/en/mko/mko-definite-integral/) — Provide reference integrals and orientation.
- [Numerical approximation and error](https://ocme.evemisslab.com/en/mko/mko-numerical-approximation-error/) — Separate signed and absolute errors.
- [Arithmetic operations](https://ocme.evemisslab.com/en/mko/mko-arithmetic-operations/) — Calculate each endpoint average.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — Require strict partition increase.
- [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) — Keep ordered aligned node/value arrays.





## Verification boundary

No general error bound requiring second derivatives and a supremum is supplied, and no floating-point result is claimed.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add a composite-trapezoid error theorem with explicit smoothness and supremum bounds.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/trapezoidal-quadrature.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-trapezoidal-quadrature/index.md)