Equivalence Theorems in Numerical Analysis: Integration, Differentiation and Interpolation

Jossey, John; Hirani, Anil N.

Permalink

https://hdl.handle.net/2142/11426 Copy

Description

Title

Equivalence Theorems in Numerical Analysis: Integration, Differentiation and Interpolation

Author(s)

• Jossey, John
• Hirani, Anil N.

Issue Date

2007-09

Keyword(s)

numerical analysis

Abstract

We show that if a numerical method is posed as a sequence of operators acting on data and depending on a parameter, typically a measure of the size of discretization, then consistency, convergence and stability can be related by a Lax-Richtmyer type equivalence theorem �a consistent method is convergent if and only if it is stable. We de?ne consistency as convergence on a dense subspace and stability as discrete well-posedness. In some applications convergence is harder to prove than consistency or stability since convergence requires knowledge of the solution. An equivalence theorem can be useful in such settings. We give concrete instances of equivalence theorems for polynomial interpolation, numerical di?erentiation, numerical integration using quadrature rules and Monte Carlo integration.

Type of Resource

text

Permalink

http://hdl.handle.net/2142/11426

Copyright and License Information

You are granted permission for the non-commercial reproduction, distribution, display, and performance of this technical report in any format, BUT this permission is only for a period of 45 (forty-five) days from the most recent time that you verified that this technical report is still available from the University of Illinois at Urbana-Champaign Computer Science Department under terms that include this permission. All other rights are reserved by the author(s).

Owning Collections