Quadrature and adaptive integration

The trapezoid rule replaces a function by a straight line across a panel and integrates that line. Simpson's rule instead integrates the quadratic interpolant through the endpoints and midpoint. Both are exact for certain low-degree polynomials, but exactness on a test class is not exactness for arbitrary functions. Composite rules divide the interval into panels and add their contributions. Panel widths, endpoint weights, and evaluation reuse must be handled consistently when comparing different resolutions.

For a sufficiently smooth function on a finite interval, the composite trapezoid error usually decreases quadratically with uniform step size, while composite Simpson error decreases to fourth order. Derivative bounds can turn these statements into rigorous estimates, but samples alone generally do not supply those bounds. Endpoint singularities, discontinuities, and rapidly changing derivatives can spoil the anticipated rate. Splitting at a known discontinuity or changing variables near a singular endpoint may be more effective than blindly applying a higher-order smooth formula.

Compute a coarse Simpson approximation on a panel and a refined approximation from its two halves. If the leading fourth-order asymptotic error dominates, halving the mesh reduces that error by about sixteen. Their difference therefore estimates fifteen times the refined error, up to sign. This supplies a practical local error indicator without knowing the exact integral. Reusing shared values keeps the comparison economical, since only the new quarter points require fresh function evaluations.

An adaptive strategy subdivides panels whose estimated errors are too large and combines the accepted contributions. A global absolute-error budget must be distributed across subpanels or controlled by summing their indicators; allowing every subpanel the full budget would not prove the requested global tolerance. Relative tolerances also need an absolute floor when the final integral nearly cancels. Maximum evaluation counts and minimum panel sizes provide honest failure modes when the requested target cannot be resolved.

Two rules can agree because both missed the same narrow peak. No finite set of unconstrained samples can rule out an arbitrarily large smooth bump hidden between them. Oscillatory cancellation creates another risk: many positive and negative contributions may leave a small total, making relative accuracy difficult even when local values are well resolved. If the function's structure is known, use that information through interval splitting, specialized oscillatory rules, analytic subtraction, or a transformation that exposes the relevant scale.

Validate with independent formulations when possible. Compare a changed variable, a different quadrature family, or a known limiting case, rather than merely tightening the same algorithm's tolerance. Distinguish the quadrature error from errors in the integrand itself, such as an inner nonlinear solve. Integrating an inaccurate function more carefully does not recover the intended integral. A useful result records the value, estimated error, evaluation count, assumptions, and whether the algorithm exhausted its budget. That is enough information to decide the next computation intelligently.