Composite trapezoidal quadrature

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

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 — Provide reference integrals and orientation.
• Numerical approximation and error — Separate signed and absolute errors.
• Arithmetic operations — Calculate each endpoint average.
• Usual order on the real numbers — Require strict partition increase.
• 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.