Lloyd N. Trefethen
The standard design principle for quadrature formulas is that they should be exact for integrands of a given class, such as polynomials of a fixed degree. We show how this principle fails to predict the actual behavior in four cases: Newton-Cotes, Clenshaw-Curtis, Gauss-Legendre, and Gauss-Hermite quadrature. Three further examples are mentioned more briefly.
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This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree $n$ requires a positive-weight quadrature rule with exactness degree $2n$. We examine the behavior of such approximation when the required exactness degree $2n$ is relaxed to $n+k$ with $0<k\leq n$. Aided by the Marcinkiewicz--Zygmund inequality, we affirm that the $L^2$ norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of $n$, and this approximation scheme is convergent as $n\rightarrow\infty$ if $k$…
This paper focuses on the approximation of continuous functions on the unit sphere by spherical polynomials of degree $n$ via hyperinterpolation. Hyperinterpolation of degree $n$ is a discrete approximation of the $L^2$-orthogonal projection of degree $n$ with its Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all spherical polynomials of degree at most $2n$. This paper aims to bypass this quadrature exactness assumption by replacing it with the Marcinkiewicz--Zygmund property proposed in a previous paper. Consequently, hyperinterpolation can be con…
Breaking quadrature exactness: A spectral method for the Allen--Cahn equation on spheres
本文提出一种球面上 Allen-Cahn 方程的谱方法，用 Marcinkiewicz-Zygmund 求积系统的受限等距关系替代传统求积精确性条件，放宽求积点选择限制并实现时间步长与扩散系数解耦，同时建立能量稳定性和几乎锐利最大值原理。
Fractional Quadrature rule and using its Exactness for the Müntz-Legendre Scaling Functions for Solving Fractional Differential Equations
针对传统求积法则计算分数幂函数积分存在误差的问题，本文提出可精确积分分数幂函数张成集内函数的分数求积法则，结合Müntz-Legendre尺度函数求解含Caputo导数的分数阶微分方程，经算例验证其精度优于Block-pulse方法。
In a recent work, we developed three new compact numerical quadrature formulas for finite-range periodic supersingular integrals $I[f]=\intBar^b_a f(x)\,dx$, where $f(x)=g(x)/(x-t)^3,$ assuming that $g\in C^\infty[a,b]$ and $f(x)$ is $T$-periodic, $T=b-a$. With $h=T/n$, these numerical quadrature formulas read \begin{align*} \widehat{T}{}^{(0)}_n[f]&=h\sum^{n-1}_{j=1}f(t+jh) -\frac{π^2}{3}\,g'(t)\,h^{-1}+\frac{1}{6}\,g'''(t)\,h, \widehat{T}{}^{(1)}_n[f]&=h\sum^n_{j=1}f(t+jh-h/2) -π^2\,g'(t)\,h^{-1}, \widehat{T}{}^{(2)}_n[f]&=2h\sum^n_{j=1}f(t+jh-h/2)- \frac{h}{2}\sum^{2n}_{j=1}f(t+jh/2-h/4). \…
Quadrature formulas from rational approximations
本文提出利用权函数Cauchy变换的有理逼近来统一推导各类求积公式，并指出求积节点区间或曲线对应近最优分支切割，从而为求积理论提供新方法和深层数学洞察。
Methods for the computation of classical Gaussian quadrature rules are described which are effective both for small and large degree. These methods are reliable because the iterative computation of the nodes has guaranteed convergence, and they are fast due to their fourth-order convergence and its asymptotic exactness for an appropriate selection of the variables. For Gauss--Hermite and Gauss--Laguerre quadratures, local Taylor series can be used for computing efficiently the orthogonal polynomials involved, with exact initial values for the Hermite case and first values computed with a conti…
The present work is devoted to extension of the trapezoidal rule in the space $W_2^{(2,1)}$. The optimal quadrature formula is obtained by minimizing the error of the formula by coefficients at values of the first derivative of a integrand. Using the discrete analog of the operator $\frac{d^2}{dx^{2}}-1$ the explicit formulas for the coefficients of the optimal quadrature formula are obtained. Furthermore, it is proved that the obtained quadrature formula is exact for any function of the set $\mathbf{F}=\mathrm{span}\{1,x,e^{x},e^{-x}\}$. Finally, in the space $W_2^{(2,1)}$ the square of the n…
The present paper is devoted to construction of an optimal quadrature formula for approximation of Fourier integrals in the Hilbert space $W_2^{(1,0)}[a,b]$ of non-periodic, complex valued functions. Here the quadrature sum consists of linear combination of the given function values on uniform grid. The difference between integral and quadrature sum is estimated by the norm of the error functional. The optimal quadrature formula is obtained by minimizing the norm of the error functional with respect to coefficients. In addition, analytic formulas for optimal coefficients are obtained using the…
This paper deals with the construction of an optimal quadrature formula for the approximation of Fourier integrals in the Sobolev space $L_2^{(1)}[a,b]$ of non-periodic, complex valued functions which are square integrable with first order derivative. Here the quadrature sum consists of linear combination of the given function values in a uniform grid. The difference between the integral and the quadrature sum is estimated by the norm of the error functional. The optimal quadrature formula is obtained by minimizing the norm of the error functional with respect to coefficients. Analytic formula…
Several new quadrature formulas for polynomial integration in the triangle
提出基于基数函数算法的三角形内多项式精确积分求积公式，点数等于低维多项式空间维数，权重为正且无外部点，最高精度达25次，其中7个公式优于已知结果。
The numerical integration of an analytical function $f(x)$ using a finite set of equidistant points can be performed by quadrature formulas like the Newton-Cotes. Unlike Gaussian quadrature formulas however, higher-order Newton-Cotes formulas are not stable, limiting the usable order of such formulas. Existing work showed that by the use of orthogonal polynomials, stable high-order quadrature formulas with equidistant points can be developed. We improve upon such work by making use of (orthogonal) Gram polynomials and deriving an iterative algorithm, together allowing us to reduce the space-co…
Quadrature formulas for the Laplace and Mellin transforms
利用 Hermite 多项式及其零点对连续 Fourier 变换进行求积，得到离散 Laplace 变换及其反演公式，并推广到 Mellin 变换。
A quadrature formula associated with a univariate quadratic spline quasi-interpolant
研究基于有界区间上C1二次样条拟插值积分的新简单求积规则，给出均匀和非均匀划分的节点与权重，提供光滑函数的误差估计，并与Simpson规则比较。
A corrected quadrature formula and applications
本文提出一个改进辛普森法则的闭型三点求积公式，利用端点导数信息实现六阶精度，并给出优于Peano界的误差界，适用于低阶导数被积函数的数值积分。
We revisit the problem of extending quadrature formulas for general weight functions, and provide a generalization of Patterson's method for the constant weight function. The method can be used to compute a nested sequence of quadrature formulas for integration with respect to any continuous probability measure on the real line with finite moments. The advantages of the method include that it works directly with the moments of the underlying distribution, and that for distributions with rational moments the existence of the formulas can be verified by exact rational arithmetic.
Error bounds of a quadrature formula with multiple nodes for the Fourier-Chebyshev coefficients for analytic functions
本文研究了在由共焦点椭圆围成的区域内解析的函数的多重节点求积公式误差界，给出了三种有效的误差界，并通过数值例子说明了计算方法。
The paper studies Sard's problem on construction of optimal quadrature formulas in the space $W_2^{(m,0)}$ by Sobolev's method. This problem consists of two parts: first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas. Here the norm of the error functional is calculated with the help of the extremal function. Then using the method of Lagrange multipliers the system of linear equations for coefficients of the optimal quadrature formulas in the space $W_2^{(m,0)}$ is obtained, moreover the existence and uniqueness of t…
Optimal quadrature formulas for the Cauchy type singular integral in the Sobolev space $L_2^{(2)}(0,1)$
本文在Sobolev空间L_2^{(2)}(0,1)中构造了用于近似计算Cauchy型奇异积分的最优求积公式，重点研究了该类积分的数值方法及最优性条件。
Approximate quadrature measures on data--defined spaces
本文研究了在数据定义空间中，如何通过求积公式估计积分误差，探讨了扩散多项式精确性与优化覆盖半径的关系。
Calculation of the Norm of the Error Functional of Optimal Quadrature Formulas in the Space $W_2^{(2,1)}(0,1)$
本文在空间 $W_2^{(2,1)}(0,1)$ 中计算了最优求积公式误差泛函的范数平方。
Optimal quadrature formulas with derivatives in Sobolev space
本文研究了在Sobolev空间L_2^{(m)}(0,1)中构造最优求积公式的Sard意义下的问题，通过S.L. Sobolev方法求得系数并计算误差泛函的范数，特别指出对于m=4,5时经典Euler-Maclaurin求积公式最优，m≥6时获得新的最优求积公式。
Optimal Quadrature Formulas with Positive Coefficients in $L_2^{(m)}(0,1)$ Space
研究 Sobolev 空间 $L_2^{(m)}(0,1)$ 中节点为 (1.5) 的最优求积公式，给出最优系数的显式形式并计算误差泛函范数，通过选择参数 $\eta_0$ 得到具有正系数的最优求积公式并与已知最优公式比较。
Optimal quadrature formulas of closed type in the space $L_2^{(m)}(0,1)$
研究 Sard 意义下节点等距时空间 $L_2^{(m)}(0,1)$ 中最优求积公式的构造问题，给出了任意自然数 $m$ 和 $N$ 下最优系数的表示。
Optimal Quadrature Formulas in the Sense of Sard in $W_2^{(m,m-1)}$ Space
本文在 $W_2^{(m,m-1)}(0,1)$ 空间中考虑 Sard 意义下的最优求积公式构造问题，利用 S.L. Sobolev 方法得到新的最优求积公式，并给出最优系数的显式公式及数值验证。
On "Upper error bounds for quadrature formulas on function classes" by K. K. Frolov
本文详述Frolov结果的完整证明，探讨函数混合导数有界的数值积分最优收敛阶。
On an optimal quadrature formula in Sobolev space $L_2^{(m)} (0,1)$
本文在Sobolev空间 $L_2^{(m)}(0,1)$ 中考虑包含节点函数值和端点一阶导数值的求积公式，对于任意固定 $N$ 和 $m\geq 2$ 找到了最优系数并计算了误差泛函的范数，证明了当 $m=2$ 和 $m=3$ 时Euler-Maclaurin求积公式是最优的。
The error bounds of Gauss quadrature formulae for the modified weight functions of Chebyshev type
本文研究了Gauss求积公式在Chebyshev型修改权重下的误差界，通过椭圆曲线分析推导了误差界，并首次提出了关于相应核在椭圆上的模行为的结果，通过数值例子验证了误差界准确性。
Quadrature formulas for integrals transforms generated by orthogonal polynomials
利用正交多项式的三项递推关系、Christoffel-Darboux型双线性生成函数及其渐近表达式，得到了经典正交多项式生成的积分变换的求积公式。
We prove lower bounds for the worst case error of quadrature formulas that use given sample points $\X_n = \{ x_1, \dots , x_n \}$. We are mainly interested in optimal point sets $\X_n$, but also prove lower bounds that hold with high probability for sets of independently and uniformly distributed points. As a tool, we use a recent result (and extensions thereof) of Vybíral on the positive semi-definiteness of certain matrices related to the product theorem of Schur. The new technique also works for spaces of analytic functions where known methods based on decomposable kernels cannot be applie…
Quadrature formulae for the positive real axis in the setting of Mellin analysis: Sharp error estimates in terms of the Mellin distance
研究梅林分析中正实轴求积公式的收敛速度，基于函数与梅林带限空间的距离，推导最佳阶误差估计，并通过数值实验验证结果。
In the present paper, optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral $\int_a^be^{2πiωx}φ(x)d x$ with $ω\in \mathbb{R}$ in the Sobolev space $L_2^{(m)}[a,b]$ of complex-valued functions which are square integrable with $m$-th order derivative. Here, using the discrete analogue of the differential operator $\frac{d^{2m}}{d x^{2m}}$, the explicit formulas for optimal coefficients are obtained. The order of convergence of the obtained optimal quadrature formula is $O(h^m)$. As an application, we implement the filtered back-projection (FBP…
The distance between the general Poisson summation formula and that for bandlimited functions; applications to quadrature formulae
研究一般泊松求和公式与带限函数公式的距离，利用经典光滑度模和适当度量，结合分数阶导数，给出余项的最佳估计。
A fast procedure for the construction of quadrature formulas for bandlimited functions
本文提出一种基于零阶长球形波函数的高效方案，用于构建带限函数的求积规则，其 CPU 时间成本在实际应用中与点数 n 成正比，并通过数值例子验证了算法性能。
On the error bounds of the Gauss-type quadrature formulae associated with spaces of parabolic and cubic spline functions with double equidistant knots
本文改进了与双等距节点三次样条空间相关的高斯型求积公式的误差常数估计，并给出了更易评估且可能更优的误差估计。
Calculation of Coefficients of the Optimal Quadrature Formulas in the $W_2^{m,m-1}(0,1)$ Space
本文利用 Sobolev 算法，在 $W_2^{(m,m-1)}(0,1)$ 空间中针对 $m=1,2$ 情形，求解形如 $\int_0^1 \phi(x)dx \cong \sum_{\beta=0}^N C_\beta \phi(x_\beta)$ 的最优求积公式的系数。
A matrix approach to the computation of quadrature formulas on the unit circle
本文通过研究单位圆上正交Laurent多项式的递推关系与矩阵表示，利用Hessenberg和五对角矩阵计算Szego求积公式的节点，并给出Rogers-Szego q-多项式的数值例子。
A note on Laguerre truncated polynomials and quadrature formula
本文研究基于权函数$w(x)= x^α e^{-x}$在区间$(0,z)$上的正交多项式的Gaussian求积公式，利用modified Chebyshev算法检验了三项递推关系系数、零点与权重的计算精度及对参数$z$的依赖性。
From an Elementary Proof of Error Representation for Hermite Quadrature to a Rediscovery of Legendre Polynomials and Rodrigues Formula
本文通过初等方法推导了Hermite求积的误差表示，重新发现Legendre多项式及其Rodrigues公式，并提出复合插值Hermite求积规则。
Error inequalities for an optimal 3-point quadrature formula of closed type
本文推导闭型最优三点求积公式，建立误差不等式并应用于数值积分。
In this paper we illustrate that paraorthogonality on the unit circle $\mathbb{T}$ is the counterpart to orthogonality on $\mathbb{R}$ when we are interested in the spectral properties. We characterize quasi-paraorthogonal polynomials on the unit circle as the analogues of the quasi-orthogonal polynomials on $\mathbb{R}$. We analyze the possibilities of preselecting some of its zeros, in order to build positive quadrature formulas with prefixed nodes and maximal domain of validity. These quadrature formulas on the unit circle are illustrated numerically.
In this work, we study a global quadrature scheme for analytic functions on compact intervals based on function values on quasi-uniform grids of quadrature nodes. In practice it is not always possible to sample functions at optimal nodes that give well-conditioned and quickly converging interpolatory quadrature rules at the same time. Therefore, we go beyond classical interpolatory quadrature by lowering the degree of the polynomial approximant and by applying auxiliary mapping functions that map the original quadrature nodes to more suitable fake nodes. More precisely, we investigate the comb…
Quadrature formulas based on rational interpolation
本文研究基于有理插值的求积公式，涵盖高斯与正交两类，证明其涉及变权重正交多项式，并给出节点渐近分布、交错与单调性结果。
Meshless moment-free quadrature formulas arising from numerical differentiation
针对Lipschitz域及其边界积分的求积权重生成问题，提出一种无需网格划分和矩计算的方法，通过求解稀疏欠定线性系统确定散乱节点权重，实验证明其在二维和三维域上鲁棒且高精度，与现有方案相比精度具竞争力。
We consider the computation of the matrix logarithm by using numerical quadrature. The efficiency of numerical quadrature depends on the integrand and the choice of quadrature formula. The Gauss--Legendre quadrature has been conventionally employed; however, the convergence could be slow for ill-conditioned matrices. This effect may stem from the rapid change of the integrand values. To avoid such situations, we focus on the double exponential formula, which has been developed to address integrands with endpoint singularity. In order to utilize the double exponential formula, we must determine…
Kernel Based Quadrature on Spheres and Other Homogeneous Spaces
本文研究球面、旋转群等紧齐次流形上基于正定和条件正定核的坐标无关求积公式，证明在拟均匀节点集下公式的精确性（最优）和稳定性，并给出球面上大规模节点权重的计算方法。
The implementation of the finite element method for linear elliptic equations requires to assemble the stiffness matrix and the load vector. In general, the entries of this matrix-vector system are not known explicitly but need to be approximated by quadrature rules. If the coefficient functions of the differential operator or the forcing term are irregular, then standard quadrature formulas, such as the barycentric quadrature rule, may not be reliable. In this paper we investigate the application of two randomized quadrature formulas to the finite element method for such elliptic boundary val…
In the present paper the optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral $\int_a^b e^{2πiωx}φ(x)d x$ with $ω\in \mathbb{R}$ in the Hilbert space $W_2^{(2,1)}[a,b]$ of complex-valued functions. Furthermore, the explicit expressions for coefficients of the constructed optimal quadrature formulas are obtained. At the end of the paper some numerical results are presented.
An Accurate Quadrature Rule on the Sphere for Fast Computation of the Radiative Transport Equation
本文提出一种球面上高效四阶规则，减少局部四阶点以提升三维辐射传输方程的计算效率，高精度方法显著降低计算资源需求。
Asymptotic Error Analysis of Gauss Quadrature for Nonsmooth Functions
本文利用余项围道积分与奇点平移逼近，推导了低正则函数 Gauss–Legendre 求积误差的一致渐近公式，可确定最优收敛速率和首项系数，并用于选点、误差校正及加速收敛。
An error estimate for the Gauss-Lobatto quadrature formula for integration over the interval $[-1, 1]$, relative to the Jacobi weight function $w^{α,β}(t)=(1-t)^α(1+t)^β$, $α,β>-1$, is obtained. This estimate holds true for functions belonging to some Sobolev-type subspaces of the weighted space $L_{w^{α,β}}^1([-1,1])$.
Optimal Quadrature Formulas for the Sobolev Space $H^1$
研究Sobolev空间H¹上任意加权积分的最优求积公式，分析节点分布对误差的影响，证明等距节点在特定条件下最优，但小n时可能非最优。
Quadrature formulas (QFs) based on radial basis functions (RBFs) have become an essential tool for multivariate numerical integration of scattered data. Although numerous works have been published on RBF-QFs, their stability theory can still be considered as underdeveloped. Here, we strive to pave the way towards a more mature stability theory for global and function-independent RBF-QFs. In particular, we prove stability of these for compactly supported RBFs under certain conditions on the shape parameter and the data points. As an alternative to changing the shape parameter, we demonstrate ho…
We consider the numerical computation of finite-range singular integrals $$I[f]=\intBar^b_a f(x)\,dx,\quad f(x)=\frac{g(x)}{(x-t)^m},\quad m=1,2,\ldots,\quad a<t<b,$$ that are defined in the sense of Hadamard Finite Part, assuming that $g\in C^\infty[a,b]$ and $f(x)\in C^\infty(\mathbb{R}_t)$ is $T$-periodic with $\mathbb{R}_t=\mathbb{R}\setminus\{t+ kT\}^\infty_{k=-\infty}$, $T=b-a$. Using a generalization of the Euler--Maclaurin expansion developed in [A. Sidi, {Euler--Maclaurin} expansions for integrals with arbitrary algebraic endpoint singularities. {\em Math. Comp.}, 81:2159--2173, 2012]…
On Weighted Generalized Gauss Quadratures for Müntz Systems
提出关于Müntz-Legendre多项式矩的递推公式，构造求解Müntz系统带幂函数权广义高斯求积的数值方法，适用于光滑及端点奇异函数，具备正权重与快速收敛特性。
Calderón Strategies for the Convolution Quadrature Time Domain Electric Field Integral Equation
本文提出了基于卷积积分法和Calderón策略结合准亥姆霍兹投影器正则化的新积分公式，有效解决了时域EFIE方法中矩阵病态和DC不稳定性问题，并利用Z变换提升计算效率。
An analysis of Ermakov-Zolotukhin quadrature using kernels
本文通过核方法分析Ermakov-Zolotukhin求积，揭示其与最优核求积的联系，并推导出期望误差公式，改进现有理论保证。
Model reduction for fractional elliptic problems using Kato's formula
本文提出了一种新的数值算法，利用模型降维计算涉及谱分数拉普拉斯算子的 stationary 偏微分方程的解。核心方法是利用已知的积分表达式，将其转化为连续和离散形式稳定的表达式，并采用降阶基方法进行积分的降维处理，最终实现多查询分数拉普拉斯问题的高效求解。
This paper is concerned with the introduction of Tikhonov regularization into least squares approximation scheme on $[-1,1]$ by orthonormal polynomials, in order to handle noisy data. This scheme includes interpolation and hyperinterpolation as special cases. With Gauss quadrature points employed as nodes, coefficients of the approximation polynomial with respect to given basis are derived in an entry-wise closed form. Under interpolatory conditions, the solution to the regularized approximation problem is rewritten in forms of two kinds of barycentric interpolation formulae, by introducing on…
Chebyshev-Type Quadrature Formulas for New Weight Classes
针对新权重类，给出了节点数为2的幂时达到最高可能精度的切比雪夫型求积公式，并构造性地描述了节点，受法拉第笼现象启发。
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