NDLI: Richardson Extrapolation on Some Recent Numerical Quadrature Formulas for Singular and Hypersingular Integrals and Its Study of Stability

Content Provider	Springer Nature Link
Author	Sidi, Avram
Copyright Year	2013
Abstract	Recently, we derived some new numerical quadrature formulas of trapezoidal rule type for the integrals $$I^{(1)}[g]=\int ^b_a \frac{g(x)}{x-t}\,dx$$ and $$I^{(2)}[g]=\int ^b_a \frac{g(x)}{(x-t)^2}\,dx$$ . These integrals are not defined in the regular sense; $$I^{(1)}[g]$$ is defined in the sense of Cauchy Principal Value while $$I^{(2)}[g]$$ is defined in the sense of Hadamard Finite Part. With $$h=(b-a)/n, \,n=1,2,\ldots $$ , and $$t=a+kh$$ for some $$k\in \{1,\ldots ,n-1\}, \,t$$ being fixed, the numerical quadrature formulas $${Q}^{(1)}_n[g]$$ for $$I^{(1)}[g]$$ and $$Q^{(2)}_n[g]$$ for $$I^{(2)}[g]$$ are $$\begin{aligned} {Q}^{(1)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2),\quad f(x)=\frac{g(x)}{x-t}, \end{aligned}$$ and $$\begin{aligned} Q^{(2)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2)-\pi ^2g(t)h^{-1},\quad f(x)=\frac{g(x)}{(x-t)^2}. \end{aligned}$$ We provided a complete analysis of the errors in these formulas under the assumption that $$g\in C^\infty [a,b]$$ . We actually show that $$\begin{aligned} I^{(k)}[g]-{Q}^{(k)}_n[g]\sim \sum ^\infty _{i=1} c^{(k)}_ih^{2i}\quad \text {as}\,n \rightarrow \infty , \end{aligned}$$ the constants $$c^{(k)}_i$$ being independent of $$h$$ . In this work, we apply the Richardson extrapolation to $${Q}^{(k)}_n[g]$$ to obtain approximations of very high accuracy to $$I^{(k)}[g]$$ . We also give a thorough analysis of convergence and numerical stability (in finite-precision arithmetic) for them. In our study of stability, we show that errors committed when computing the function $$g(x)$$ , which form the main source of errors in the rest of the computation, propagate in a relatively mild fashion into the extrapolation table, and we quantify their rate of propagation. We confirm our conclusions via numerical examples.
Starting Page	141
Ending Page	159
Page Count	19
File Format	PDF
ISSN	08857474
Journal	Journal of Scientific Computing
Volume Number	60
Issue Number	1
e-ISSN	15737691
Language	English
Publisher	Springer US
Publisher Date	2013-10-31
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Cauchy principal value Hadamard finite part Singular integral Hypersingular integral Numerical quadrature Trapezoidal rule Euler–Maclaurin expansion Richardson extrapolation Approximate quadratures Asymptotic approximations, asymptotic expansions Fredholm integral equations Integral equations with kernels of Cauchy type Extrapolation to the limit, deferred corrections Euler-Maclaurin formula Numerical integration Quadrature and cubature formulas Algorithms Computational Mathematics and Numerical Analysis ApplicationMathematics/Computational Methods of Engineering Theoretical, Mathematical and Computational Physics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Engineering Computational Theory and Mathematics Computational Mathematics Numerical Analysis Software

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Compact Numerical Quadrature Formulas for Hypersingular Integrals and Integral Equations

PVTSI$^{\boldmath(m)}$: A Novel Approach to Computation of Hadamard Finite Parts of Nonperiodic Singular Integrals

Unified Compact Numerical Quadrature Formulas for Hadamard Finite Parts of Singular Integrals of Periodic Functions

Hybrid Gauss-Trapezoidal Quadrature Rules

Numerical Quadratures for Singular and Hypersingular Integrals in Boundary Element Methods

High-Order Corrected Trapezoidal Quadrature Rules for Singular Functions

The Exponentially Convergent Trapezoidal Rule

Asymptotic error expansions for hypersingular integrals

Corrected Trapezoidal Rules for Boundary Integral Equations in Three Dimensions

Richardson Extrapolation on Some Recent Numerical Quadrature Formulas for Singular and Hypersingular Integrals and Its Study of Stability

Similar Documents

Compact Numerical Quadrature Formulas for Hypersingular Integrals and Integral Equations

PVTSI$^{\boldmath(m)}$: A Novel Approach to Computation of Hadamard Finite Parts of Nonperiodic Singular Integrals

Unified Compact Numerical Quadrature Formulas for Hadamard Finite Parts of Singular Integrals of Periodic Functions

Hybrid Gauss-Trapezoidal Quadrature Rules

Numerical Quadratures for Singular and Hypersingular Integrals in Boundary Element Methods

High-Order Corrected Trapezoidal Quadrature Rules for Singular Functions

The Exponentially Convergent Trapezoidal Rule

Asymptotic error expansions for hypersingular integrals

Corrected Trapezoidal Rules for Boundary Integral Equations in Three Dimensions

Richardson Extrapolation on Some Recent Numerical Quadrature Formulas for Singular and Hypersingular Integrals and Its Study of Stability