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Adaptive Quadrature

Numerically integrates by subdividing intervals where the error estimate is too high.

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Quick Facts

Compact metadata for implementation and discovery.

Category: Numerical Methods
Difficulty: Intermediate
Family: Numerical Integration
Tier: Extended
Status: Verified
Review: Source Checked
Sources: 1
Formulas: 1
Relationships: 3
Code example: Ready
Visualization: Ready

Overview

Adaptive Quadrature spends more evaluations on difficult parts of a function and fewer on smooth regions.

Core Concepts

local error estimaterecursive subdivisiontolerancequadrature rule

Quick facts

Complexity

Runtime and memory behavior at a glance.

Best

O(n)

Average

Adaptive

Worst

Problem dependent

Space

O(depth)

When to Use

Use when the integrand is mostly smooth but has localized difficult regions.
Use when integrating functions with local sharp features is a central requirement.

When Not to Use

Avoid when samples are noisy enough that error estimates are unreliable.
Avoid when roundoff, conditioning, or discontinuities dominate the result.

What It Does

Estimates local integration error
Subdivides difficult intervals
Combines accepted interval estimates

Use Cases

Integrating functions with local sharp features
Scientific computing
Tolerance-controlled quadrature

Inputs

Function
Interval
Error tolerance

Outputs

Integral estimate

Key Formulas (1)

Error estimate

error ≈ |estimate_whole − (estimate_left + estimate_right)|

error ~= abs(estimate_whole - (estimate_left + estimate_right))

Compares coarse and subdivided estimates to decide whether to refine.

Pseudocode

1.Estimate the integral on an interval
2.Estimate the integral on its two halves
3.Compare the difference as an error estimate
4.Accept or subdivide recursively
5.Sum accepted estimates

Canonical Example

Input: function has a narrow peak
Output: more subintervals near the peak
Explanation: The method refines where the estimated error is largest.

Implementation Notes

Carry a per-interval tolerance so total error remains controlled.
Compare against a small hand-checkable case before scaling up.
Keep a tiny example with expected output beside the implementation.

Common Pitfalls

Highly oscillatory functions can fool simple error estimates.
Ignoring conditioning can make a correct method return unreliable numbers.
Not defining boundary, empty, or degenerate inputs before implementation.

Interactive Mechanism Walkthrough

Iterative Convergence & State Trajectory

Show an interval split more finely near a curve's sharp bend.

Step 1 of 5: Estimate the integral on an interval

numeric-iteration

Notes

A maximum recursion depth protects against pathological functions.

Sources (1)

What each reference supports for this entry.

Numerical Recipes

Book

Numerical methods reference covering adaptive quadrature and error-based subdivision.

Open source

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Alternative toRomberg Integration

Numerical Methods · Specialized

Both refine integration but with different error-control strategies.

Compare with this

Improves onSimpson's Rule

Numerical Methods · Core

Adaptive quadrature recursively allocates work where estimated error is high.