{ "id": "ephws5ry4t", "slug": "quadrature-rule", "title": "Quadrature rule", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.analysis", "label": "Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.analysis" }, { "id": "physical.math.analysis.numerical_analysis_and_computation", "label": "Numerical analysis and computation", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.analysis.numerical_analysis_and_computation" }, { "id": "physical.math.analysis.numerical_analysis_and_computation.interpolation-and-approximation", "label": "Interpolation and approximation", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.analysis.numerical_analysis_and_computation.interpolation-and-approximation" } ], "geo": [ { "id": "geo.nongeo.t1000.physical", "label": "Non-geographic · 1000 to 1799: Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1000.physical", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1000", "label": "Non-geographic · 1000 to 1799", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1000" }, { "id": "geo.nongeo.t1000.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1000.physical" } ] }, { "id": "geo.nongeo.t1800.physical.math.analysis", "label": "Non-geographic · 1800 to 1945: Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.analysis", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1800", "label": "Non-geographic · 1800 to 1945", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800" }, { "id": "geo.nongeo.t1800.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical" }, { "id": "geo.nongeo.t1800.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math" }, { "id": "geo.nongeo.t1800.physical.math.analysis", "label": "Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.analysis" } ] } ], "excerpt": "A quadrature rule is a numerical method that approximates a definite integral of a function by a weighted sum of its values at selected points, or nodes.", "snippet": "A quadrature rule is a numerical method that approximates a definite integral of a function by a weighted sum of its values at selected points, or nodes.", "node": "physical.math.analysis.numerical_analysis_and_computation.interpolation-and-approximation", "markdown": "# Quadrature rule\n\nA quadrature rule is a numerical method that approximates a definite integral of a function by a weighted sum of function values at selected points, written \\( I_{n}(f) = \\sum_{j} w_{j} f(x_{j}) \\) for distinct nodes \\( x_{j} \\) and weights \\( w_{j} \\), with error \\( E_{n}(f) = I_{n}(f) - I(f) \\).[1](https://people.maths.ox.ac.uk/trefethen/exactness.pdf) Gauss-type rules are preferred when the integrand is smooth and each evaluation is costly, for example with expensive experimental data or repeated integrals in multiple integration.[2](https://encyclopediaofmath.org/wiki/Gauss_quadrature_formula)\n\n| Key fact | Detail |\n|---|---|\n| Form | \\( I_{n}(f) = \\sum_{j} w_{j} f(x_{j}) \\); nodes \\( x_{j} \\), weights \\( w_{j} \\), remainder \\( R_{n}(f) \\)[3](https://www.cs.purdue.edu/homes/wxg/selected_works/section_12/074.pdf) |\n| Degree of exactness | A rule has degree \\( d \\) if it is exact for all polynomials of degree \\( d \\) and fails for some polynomial of degree \\( d+1 \\)[4](https://www.math.unipd.it/~dottmath/corsi2012/lecturenotes/Meurant/Chap5.pdf) |\n| Gauss maximum | An \\( n \\)-point rule with free nodes reaches at most degree \\( 2n-1 \\), attained by Gauss quadrature[1](https://people.maths.ox.ac.uk/trefethen/exactness.pdf) |\n| Practical construction | Gauss nodes and weights from eigenvalues and eigenvectors of the Jacobi matrix (Golub and Welsch, 1969)[5](https://doi.org/10.1090/s0025-5718-69-99647-1) |\n| Trapezoid error bound | \\( \\lvert \\int f \\, dx - T_{n} \\rvert \\le K_{2}(b-a)^{3}/(12 n^{2}) \\), where \\( K_{2} \\) bounds \\( \\lvert f'' \\rvert \\)[6](https://kconrad.math.uconn.edu/math1132s20/handouts/numericalintegralestimates.pdf) |\n| Typical rule sizes | Fewer than 10 points in adaptive composite software; dozens for Gauss–Legendre and Gauss–Hermite; hundreds to thousands in Chebfun and ApproxFun[1](https://people.maths.ox.ac.uk/trefethen/exactness.pdf) |\n| Monte Carlo comparison | Monte Carlo error scales as \\( \\sim 1/\\sqrt{n} \\)[7](https://juliamath.github.io/QuadGK.jl/stable/gauss-kronrod/) |\n\n## How it works\n\nThe basic principle is interpolation. Given \\( n \\) nodes, the unique polynomial of degree \\( n-1 \\) agreeing with \\( f \\) at the nodes is expressed through divided differences; integrating that polynomial instead of \\( f \\) yields an interpolatory quadrature formula with degree of exactness at least \\( n-1 \\).[8](http://elib.mi.sanu.ac.rs/files/journals/bltn/33/r2008_2.pdf) Conversely, any rule with degree of exactness \\( n-1 \\) arises this way.[9](https://www.cs.purdue.edu/homes/wxg/selected_works/section_04/146.pdf) Such a rule is exact for at least degree \\( n-1 \\), and may have a higher degree of exactness.[3](https://www.cs.purdue.edu/homes/wxg/selected_works/section_12/074.pdf)\n\nExactness is the classical quality measure: a rule is of exact degree \\( d \\) if its remainder \\( R[p] \\) vanishes for all polynomials of degree \\( d \\) and is nonzero for some polynomial of degree \\( d+1 \\).[4](https://www.math.unipd.it/~dottmath/corsi2012/lecturenotes/Meurant/Chap5.pdf) With free nodes the maximum achievable degree is \\( 2M-1 \\) for an \\( M \\)-point rule, attained if and only if the nodes are the roots of the degree-\\( M \\) orthogonal polynomial for the integration weight.[10](https://sandialabs.github.io/pyapprox/gauss_quadrature.html) This is the Gauss rule; a characterization states that a rule is exact for degree \\( n-1+m \\) at least when it is interpolatory and its node polynomial is orthogonal to every polynomial of degree at most \\( m-1 \\), with exact degree \\( n-1+m \\) only if some polynomial of degree \\( m \\) is not orthogonal to it.[9](https://www.cs.purdue.edu/homes/wxg/selected_works/section_04/146.pdf) For the Legendre weight on \\( [-1,1] \\) the nodes are the roots of the degree-\\( n \\) Legendre polynomial.[1](https://people.maths.ox.ac.uk/trefethen/exactness.pdf)\n\nFor smooth integrands on a finite interval, the composite trapezoidal rule with \\( n \\) subintervals satisfies \\( \\lvert \\int_{a}^{b} f \\, dx - T_{n} \\rvert \\le K_{2}(b-a)^{3}/(12 n^{2}) \\), where \\( K_{2} \\) is a bound on \\( \\lvert f'' \\rvert \\); the bound requires \\( f'' \\) to exist and be bounded.[6](https://kconrad.math.uconn.edu/math1132s20/handouts/numericalintegralestimates.pdf) At the other extreme, Gauss quadrature converges for every continuous \\( f \\) and at an exponential rate for analytic \\( f \\), while Newton–Cotes diverges at an exponential rate for most integrands, even analytic ones.[1](https://people.maths.ox.ac.uk/trefethen/exactness.pdf) Clenshaw–Curtis, whose weights are always positive, also converges for all continuous \\( f \\) and exponentially for analytic \\( f \\).[1](https://people.maths.ox.ac.uk/trefethen/exactness.pdf) The double exponential formula achieves error \\( O(\\exp(-C \\cdot N/\\log N)) \\) as a function of the number \\( N = 2n+1 \\) of function evaluations, and no variable-transformation quadrature can decay faster.[11](https://www.kurims.kyoto-u.ac.jp/~prims/pdf/41-4/41-4-38.pdf)\n\n## How it is done\n\nConstruction and application proceed in a standard sequence: choose nodes on \\( [-1,1] \\), interpolate, transfer the rule to \\( [a,b] \\), form composite rules over subintervals, and derive error estimates used in adaptive algorithms against a user-defined tolerance.[12](https://www.math.ntnu.no/emner/TMA4130/2022h/pdf_notes/Quadrature.pdf) For Gauss rules, the practical computation is the eigenvalue algorithm of [Gene H. Golub](https://www.edgepedia.org/gene-h-golub) and John H. Welsch (Mathematics of [Computation](https://www.edgepedia.org/computation), 1969).[5](https://doi.org/10.1090/s0025-5718-69-99647-1) The nodes are the eigenvalues of the Jacobi matrix \\( J_{N} \\), and the weights are the total mass \\(\\mu_{0}\\) of the measure times the squares of the first components of the normalized eigenvectors, so for \\(\\mu_{0}=1\\) they are simply the squares of the first components.[4](https://www.math.unipd.it/~dottmath/corsi2012/lecturenotes/Meurant/Chap5.pdf) • [28](https://optmat.readthedocs.io/en/latest/generated/sources/golub_1969.html)\n\nError estimation in software relies on embedded pairs. A Gauss–Kronrod rule extends an \\( n' \\)-point Gauss rule with \\( n'+1 \\) extra points to a \\( 2n'+1 \\)-point rule, so the difference of the two weighted sums estimates the error with no additional function evaluations.[7](https://juliamath.github.io/QuadGK.jl/stable/gauss-kronrod/) The Julia `quadgk` function defaults to order \\( n'=7 \\) (15 points) and performs \\( h \\)-adaptive subdivision, splitting the interval in half and reapplying the rule until the requested tolerance is met; \\( h \\)-adaptivity is more robust than raising the rule order when the integrand has localized sharp peaks or discontinuities.[7](https://juliamath.github.io/QuadGK.jl/stable/gauss-kronrod/) Adaptive subdivision of this kind, applied recursively until convergence in each subinterval, underlies most general-purpose integration programs.[13](https://gksmyth.github.io/pubs/NumericalIntegration-Preprint.pdf) Adaptive Simpson's method is an earlier example of the same idea.[14](https://www.math.umd.edu/~mariakc/AMSC466/LectureNotes/quadrature.pdf)\n\n## Origin\n\nThe eigenvalue algorithm for computing Gauss quadrature nodes and weights was introduced by Gene H. Golub and John H. Welsch in 1969, in Mathematics of Computation.[5](https://doi.org/10.1090/s0025-5718-69-99647-1) It made practical a construction whose history is much older. [15](https://encyclopediaofmath.org/wiki/Newton%E2%80%93Cotes_quadrature_formula) The equally spaced weights are called Cotes numbers, which were computed for \\( n \\le 11 \\).[3](https://www.cs.purdue.edu/homes/wxg/selected_works/section_12/074.pdf) The formula includes the trapezoidal and Simpson formulas as special cases, both known before Newton's time.[3](https://www.cs.purdue.edu/homes/wxg/selected_works/section_12/074.pdf) The method of numerical integration asks what maximum degree of exactness is achievable when the nodes are free; the maximum \\( 2n-1 \\) is a proven upper bound, attained by Gauss quadrature; the count of \\( 2n \\) unknowns, the nodes and weights, can motivate this bound but does not prove it.[8](http://elib.mi.sanu.ac.rs/files/journals/bltn/33/r2008_2.pdf) The resulting formula, for \\( a=-1 \\), \\( b=1 \\), \\( p(x) \\equiv 1 \\), is exact for all polynomials of degree not exceeding \\( 2n-1 \\),[2](https://encyclopediaofmath.org/wiki/Gauss_quadrature_formula) and no other \\( n \\)-node rule can achieve this.[16](https://pi.math.cornell.edu/~ajt/papers/QuadratureEssay.pdf) Gauss's work was simplified by Jacobi and further developed through much of the 19th century by Mehler, Christoffel, and others.[3](https://www.cs.purdue.edu/homes/wxg/selected_works/section_12/074.pdf)\n\n## Variants\n\n**Newton–Cotes** takes equally spaced nodes.[4](https://www.math.unipd.it/~dottmath/corsi2012/lecturenotes/Meurant/Chap5.pdf) Its algebraic degree of accuracy is \\( n \\) for odd \\( n \\) and \\( n+1 \\) for even \\( n \\); all coefficients are positive for \\( n = 1 \\dots 7, 9 \\), but negative coefficients appear for \\( n = 8 \\) and \\( n \\ge 10 \\).[15](https://encyclopediaofmath.org/wiki/Newton%E2%80%93Cotes_quadrature_formula) [Simpson's rule](https://www.edgepedia.org/simpsons-rule) is the Newton–Cotes rule most often used in practice, retaining algorithmic simplicity with an adequate degree of approximation.[17](https://mml-book.github.io/book/additional_chapters/integration-methods.pdf)\n\n**Gauss–Legendre** is the \\( [-1,1] \\) Gauss rule described above.[16](https://pi.math.cornell.edu/~ajt/papers/QuadratureEssay.pdf) The **Gauss–Hermite** formula, defined on \\( (-\\infty,\\infty) \\) with weight \\( \\exp(-x^{2}) \\), was introduced by Gourier in 1883 according to Gautschi.[1](https://people.maths.ox.ac.uk/trefethen/exactness.pdf) **Gauss–Kronrod** rules estimate the error of the \\( n \\)-point Gauss formula for the Legendre weight economically, and add \\( n+1 \\) new nodes chosen with all weights so the extended rule has maximum degree of exactness.[18](https://etna.ricam.oeaw.ac.at/volumes/2011-2020/vol45/abstract.php?pages=371-404)\n\n**Fejér and Clenshaw–Curtis** rules use [Chebyshev nodes](https://www.edgepedia.org/chebyshev-nodes). Two rules were introduced, with nodes the roots of \\( T_{n} \\) and of \\( U_{n} \\), and closed formulas and positiveness of the weights were established for \\( w \\equiv 1 \\).[19](https://www.math.unipd.it/~alvise/PAPERS/fejer_v2.pdf) The rule is based on Fejér's second nodes plus the endpoints \\( -1 \\) and \\( +1 \\), equivalently integrating a finite Chebyshev expansion of the integrand; the evaluation can be performed by a discrete cosine transform, as shown by [Gentleman](https://www.edgepedia.org/gentleman).[19](https://www.math.unipd.it/~alvise/PAPERS/fejer_v2.pdf) The nodes are \\( x_{j} = \\cos(j \\cdot \\pi/(n-1)) \\), and nodes and weights are computable in \\( O(n \\log n) \\).[1](https://people.maths.ox.ac.uk/trefethen/exactness.pdf) Clenshaw–Curtis quadrature is essentially trapezoidal integration plus a coordinate transformation that removes the endpoint problem.[20](https://math.mit.edu/~stevenj/trapezoidal.pdf)\n\n**Tanh-sinh (double exponential)** quadrature applies the trapezoidal formula after the transformation \\( x = \\phi(t) = \\tanh((\\pi/2)\\sinh t) \\); the transformed integrand decays double exponentially, giving the method its name.[11](https://www.kurims.kyoto-u.ac.jp/~prims/pdf/41-4/41-4-38.pdf) [David H. Bailey](https://www.edgepedia.org/david-h-bailey-mathematician) describes tanh-sinh as the fastest currently known high-precision quadrature scheme when abscissa computation and error bounds are counted.[21](https://davidhbailey.com/dhbpapers/dhb-tanh-sinh.pdf)\n\n**Bayesian quadrature** approximates an integral with respect to a probability measure by a weighted sum, but treats the computation as statistical inference, its defining characteristic within probabilistic numerics.[17](https://mml-book.github.io/book/additional_chapters/integration-methods.pdf) It approximates the integrand \\( f \\) by a [Gaussian process](https://www.edgepedia.org/gaussian-process) with user-chosen nodes, in contrast to Newton–Cotes equidistant nodes and Gaussian roots of orthogonal polynomials.[17](https://mml-book.github.io/book/additional_chapters/integration-methods.pdf) A systematic survey of Bayesian quadrature appeared in Foundations and Trends in Machine Learning.[22](https://www.emerald.com/ftmal/article/19/2-3/121/1396422/Bayesian-quadrature)\n\n## Applications\n\nThe double exponential formula is used in molecular physics, fluid dynamics, statistics, civil engineering, financial engineering, and the boundary element method.[11](https://www.kurims.kyoto-u.ac.jp/~prims/pdf/41-4/41-4-38.pdf) Designed polynomial quadrature is used for parametric operator equations in engineering problems.[23](https://users.cs.utah.edu/~kirby/Publications/Kirby-113.pdf) In software, adaptive composite rules typically use fewer than 10 points, Gauss–Legendre and Gauss–Hermite rules use dozens, and Chebfun and ApproxFun use hundreds to thousands.[1](https://people.maths.ox.ac.uk/trefethen/exactness.pdf) High-order Newton–Cotes rules are almost never used in practice; modern integration typically applies composite low-order rules inside adaptive algorithms, while Romberg integration applies [Richardson extrapolation](https://www.edgepedia.org/richardson-extrapolation) to a sequence of composite trapezoidal approximations.[24](https://old.maa.org/press/periodicals/convergence/servois-1817-memoir-on-quadratures-the-newton-cotes-formulas)\n\n## Limitations and alternatives\n\n**Polynomial exactness is an unreliable guide.** Trefethen shows that the exactness principle does not predict actual accuracy for Newton–Cotes, Clenshaw–Curtis, and Gauss on \\( [-1,1] \\), and Gauss–Hermite, with failure extreme for Newton–Cotes and Gauss–Hermite.[1](https://people.maths.ox.ac.uk/trefethen/exactness.pdf) The mechanism behind the Newton–Cotes failure is the oscillation of polynomial interpolation at equispaced points, shown in Runge's 1901 paper and made rigorous by Pólya in 1933.[1](https://people.maths.ox.ac.uk/trefethen/exactness.pdf)\n\n**Nesting and dimension.** Gauss rules cannot be nested: the points of an \\( n' \\)-point Gauss rule are not a subset of those of an \\( n \\)-point rule for \\( 1 < n' < n \\), which is why Kronrod extension is used for reusable error estimates.[7](https://juliamath.github.io/QuadGK.jl/stable/gauss-kronrod/) In high dimension, deterministic quadrature suffers the curse of dimensionality, a problem studied since the seminal 1959 paper of Bakhvalov.[25](https://arxiv.org/abs/1409.6714) Monte Carlo integration converges at rate \\( O(N^{-1/2}) \\) independent of dimension, robust but slow, usually limiting it to about three significant figures.[26](https://www.cambridge.org/core/journals/acta-numerica/article/abs/monte-carlo-and-quasimonte-carlo-methods/FE7C779B350CFEA45DB2A4CCB2DA9B5C) • [13](https://gksmyth.github.io/pubs/NumericalIntegration-Preprint.pdf) Quasi-[Monte Carlo](https://www.edgepedia.org/monte-carlo) replaces random samples with low-discrepancy sequences and reaches approximately \\( O((\\log N)^{k} N^{-1}) \\) convergence.[26](https://www.cambridge.org/core/journals/acta-numerica/article/abs/monte-carlo-and-quasimonte-carlo-methods/FE7C779B350CFEA45DB2A4CCB2DA9B5C) A practical rule of thumb from a machine-learning reference: for settings of roughly 100 to 1000 dimensions use Monte Carlo, while Newton–Cotes, Gaussian, and Bayesian quadrature suit low and moderate dimensions.[17](https://mml-book.github.io/book/additional_chapters/integration-methods.pdf) Between Gauss and Clenshaw–Curtis, an exact comparison via Padé approximation of \\( \\log((z+1)/(z-1)) \\) at \\( z = \\infty \\) shows the Clenshaw–Curtis order of accuracy there is only half that of Gauss.[27](https://epubs.siam.org/doi/10.1137/060659831)\n\n## References\n\n1. [Exactness of Quadrature Formulas (Trefethen, SIAM Review 64, 2022)](https://people.maths.ox.ac.uk/trefethen/exactness.pdf)\n2. [Gauss quadrature formula – Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Gauss_quadrature_formula)\n3. [Gaussian quadrature (historical survey, Gautschi)](https://www.cs.purdue.edu/homes/wxg/selected_works/section_12/074.pdf)\n4. [Chapter 5: Gauss quadrature rules (Meurant lecture notes)](https://www.math.unipd.it/~dottmath/corsi2012/lecturenotes/Meurant/Chap5.pdf)\n5. [Gene H. Golub, John H. Welsch (1969). Calculation of Gauss quadrature rules. Mathematics of Computation.](https://doi.org/10.1090/s0025-5718-69-99647-1)\n6. [Making Estimates in Numerical Integration (UConn handout)](https://kconrad.math.uconn.edu/math1132s20/handouts/numericalintegralestimates.pdf)\n7. [Quadrature rules · QuadGK.jl documentation](https://juliamath.github.io/QuadGK.jl/stable/gauss-kronrod/)\n8. [Quadrature processes – development and new directions](http://elib.mi.sanu.ac.rs/files/journals/bltn/33/r2008_2.pdf)\n9. [Gautschi, work on orthogonal-polynomial quadrature (Section 04/146)](https://www.cs.purdue.edu/homes/wxg/selected_works/section_04/146.pdf)\n10. [Gauss Quadrature: Optimal Nodes from Orthogonal Polynomials – PyApprox (Sandia)](https://sandialabs.github.io/pyapprox/gauss_quadrature.html)\n11. [Publications of the Research Institute for Mathematical Sciences (PRIMS) 41-4 paper on the DE formula](https://www.kurims.kyoto-u.ac.jp/~prims/pdf/41-4/41-4-38.pdf)\n12. [Quadrature notes (TMA4130, NTNU)](https://www.math.ntnu.no/emner/TMA4130/2022h/pdf_notes/Quadrature.pdf)\n13. [Numerical Integration (Smyth, preprint)](https://gksmyth.github.io/pubs/NumericalIntegration-Preprint.pdf)\n14. [Quadrature lecture notes (AMSC466, UMD)](https://www.math.umd.edu/~mariakc/AMSC466/LectureNotes/quadrature.pdf)\n15. [Newton–Cotes quadrature formula – Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Newton%E2%80%93Cotes_quadrature_formula)\n16. [Quadrature essay (A. J. T. / Cornell)](https://pi.math.cornell.edu/~ajt/papers/QuadratureEssay.pdf)\n17. [Modern Integration Methods in Machine Learning (MML book chapter)](https://mml-book.github.io/book/additional_chapters/integration-methods.pdf)\n18. [ETNA vol. 45 abstract: survey of Gauss–Kronrod quadrature research](https://etna.ricam.oeaw.ac.at/volumes/2011-2020/vol45/abstract.php?pages=371-404)\n19. [Fast Construction of Fejér and Clenshaw-Curtis rules for general weight functions](https://www.math.unipd.it/~alvise/PAPERS/fejer_v2.pdf)\n20. [Notes on the convergence of trapezoidal-rule quadrature (Steven G. Johnson, MIT)](https://math.mit.edu/~stevenj/trapezoidal.pdf)\n21. [Tanh-sinh quadrature paper by David H. Bailey](https://davidhbailey.com/dhbpapers/dhb-tanh-sinh.pdf)\n22. [Bayesian quadrature | Foundations and Trends in Machine Learning](https://www.emerald.com/ftmal/article/19/2-3/121/1396422/Bayesian-quadrature)\n23. [Numerical Integration in Multiple Dimensions with Designed Quadrature (SIAM J. Sci. Comput., Vol. 40, No. 4)](https://users.cs.utah.edu/~kirby/Publications/Kirby-113.pdf)\n24. [Servois' 1817 'Memoir on Quadratures' – The Newton–Cotes Formulas (MAA Convergence)](https://old.maa.org/press/periodicals/convergence/servois-1817-memoir-on-quadratures-the-newton-cotes-formulas)\n25. [Some Results on the Complexity of Numerical Integration](https://arxiv.org/abs/1409.6714)\n26. [Monte Carlo and quasi-Monte Carlo methods (Acta Numerica)](https://www.cambridge.org/core/journals/acta-numerica/article/abs/monte-carlo-and-quasimonte-carlo-methods/FE7C779B350CFEA45DB2A4CCB2DA9B5C)\n27. [Is Gauss Quadrature Better than Clenshaw–Curtis? (SIAM)](https://epubs.siam.org/doi/10.1137/060659831)\n28. [golub_1969, optmat](https://optmat.readthedocs.io/en/latest/generated/sources/golub_1969.html)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Numerical analysis and computation › Interpolation and approximation*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://www.math.umd.edu/~mariakc/AMSC466/LectureNotes/quadrature.pdf", "https://pi.math.cornell.edu/~ajt/papers/QuadratureEssay.pdf", "https://math.mit.edu/~stevenj/trapezoidal.pdf", "https://users.cs.utah.edu/~kirby/Publications/Kirby-113.pdf" ], "url": "https://www.edgepedia.org/quadrature-rule", "markdown_url": "https://www.edgepedia.org/quadrature-rule.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Quadrature rule\", Edgepedia (EdgeChat), https://www.edgepedia.org/quadrature-rule. Edgepedia Community License 1.0.", "credit_md": "\"[Quadrature rule](https://www.edgepedia.org/quadrature-rule)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/quadrature-rule](https://www.edgepedia.org/quadrature-rule). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Quadrature rule\", Edgepedia (EdgeChat), https://www.edgepedia.org/quadrature-rule. Edgepedia Community License 1.0.", "speakable": "A quadrature rule is a numerical method that approximates a definite integral of a function by a weighted sum of its values at selected points, or nodes." }