Prompted by an observation about the integral of exponential functions of the form $f(x)=λe^{αx}$, we investigate the possibility to exactly integrate families of functions generated from a given function by scaling or by affine transformations of the argument using nonlinear generalizations of quadrature formulae. The main result of this paper is that such formulae can be explicitly constructed for a wide class of functions, and have the same accuracy as Newton-Cotes formulae based on the same nodes, with the latter emerging as the linear case of our general formalism. We also derive explicit bounds on the error of the nonlinear quadrature formulae, which in the linear case devolve into the well-known bounds for Newton-Cotes formulae.
On the error bounds of the Gauss-type quadrature formulae associated with spaces of parabolic and cubic spline functions with double equidistant knots
Faculty of Mathematics and Informatics, Sofia University ”St. Kliment Ohridski”（索非亚大学圣克利门特奥赫里斯基数学与信息学院）；Faculty of Pharmacy, Medical University of Sofia（索非亚医科大学药学院）
Some selected Ostrowski type inequalities and a connection with numerical integration are studied in this survey paper, which is dedicated to the memory of Professor D.S. Mitrinovic, who left us 25 years ago. His significant influence to the development of the theory of inequalities is briefly given in the first section of this paper. Beside some basic facts on quadrature formulas and an approach for estimating the error term using Ostrowski type inequalities and Peano kernel techniques, we give several examples of selected quadrature formulas and the corresponding inequalities, including the …
This paper describes a trapezoidal quadrature method for the discretization of singular and hypersingular boundary integral operators (BIOs) that arise in solving boundary value problems for elliptic partial differential equations. The quadrature is based on a uniform grid in parameter space coupled with the standard punctured Trapezoidal rule. A key observation is that the error incurred by the singularity in the kernel can be expressed exactly using generalized Euler-Maclaurin formulae that involve the Riemann zeta function in 2D and the Epstein zeta functions in 3D. These expansions are exp…
The concept of Gauss quadrature can be generalized to approximate linear functionals with complex moments. Following the existing literature, this survey will revisit such generalization. It is well known that the (classical) Gauss quadrature for positive definite linear functionals is connected with orthogonal polynomials, and with the (Hermitian) Lanczos algorithm. Analogously, the Gauss quadrature for linear functionals is connected with formal orthogonal polynomials, and with the non-Hermitian Lanczos algorithm with look-ahead strategy; moreover, it is related to the minimal partial realiz…
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…
Stochastic Lanczos Quadrature for Computational Uncertainty in Linear Algebra
University of Dodoma（多多马大学）；African Institute for Mathematical Sciences Research and Innovation Centre（非洲数学科学研究与创新中心）；University of Tübingen（蒂宾根大学）；Tübingen AI Center（蒂宾根人工智能中心）
The sub-optimality of Gauss--Hermite quadrature and the optimality of the trapezoidal rule are proved in the weighted Sobolev spaces of square integrable functions of order $α$, where the optimality is in the sense of worst-case error. For Gauss--Hermite quadrature, we obtain matching lower and upper bounds, which turn out to be merely of the order $n^{-α/2}$ with $n$ function evaluations, although the optimal rate for the best possible linear quadrature is known to be $n^{-α}$. Our proof of the lower bound exploits the structure of the Gauss--Hermite nodes; the bound is independent of the qua…
In this article, we propose an algorithm for approximating the action of $φ-$functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the $φ-$functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matr…
Convergence of the Immersed Interface Method in Linear Elasticity
University of Hasselt（哈瑟尔特大学）；Lancaster University（兰卡斯特大学）；Leiden University（莱顿大学）；University of Zaragoza（萨拉戈萨大学）；University of Johannesburg（约翰内斯堡大学）；Delft University of Technology（代尔夫特理工大学）
