Suppose we have equispaced samples of a smooth function f defined on [1,1]. If f is periodic, then the obvious way to interpolate the data is by a trigonometric polynomial. The integral of the interpolant gives the trapezoidal rule quadrature formula. What if f is not periodic? The trigonometric interpolant and the trapezoidal rule are not much good, because of the Gibbs phenomenon. However, we can improve the trapezoidal rule to any order by endpoint corrections. The EulerMaclaurin formula uses corrections involving derivatives, and NewtonGregory quadrature uses onesided finite differences. These ideas go back 300 years. What about using endpoint corrections to improve the interpolant, not just the quadrature formula? Amazingly, so far as we can tell, in 300 years this question has never been considered. In this talk we present such corrections, which we call EulerMaclaurin and NewtonGregory interpolants. Each takes the form of an algebraic polynomial plus a trigonometric polynomial. They approximate f to any order, and their integrals are exactly the results of the EulerMaclaurin and NewtonGregory quadrature formulas. EulerMaclaurin and NewtonGregory Interpolants Tuesday, April 21, 2015 4: 00 PM – 5:00 PM 127 HayesHealy Center Colloquium Tea 3:30 PM to 4:00 PM 154 Hurley Hall Nick Trefethen Mathematical Institute Oxford University Department of Applied and Computational Mathematics and Statistics Colloquium Nick Trefethen is Professor of Numerical Analysis and head of the Numerical Analysis Group in the Mathematical Institute at Oxford University. He was educated at Harvard and Stanford and held professorial positions at NYU, MIT, and Cornell before moving to Oxford in 1997. He is a Fellow of the Royal Society and a member of the US National Academy of Engineering.

4'21'15 Abstract Nick Trefethen - University of Notre Dame€¦ · · 2016-06-02Nick%Trefethen! Mathematical*Institute ... Microsoft Word - 4'21'15 Abstract Nick Trefethen.docx

Download PDF Report

Upload
dinhhanh
View
216
Download
3

Embed Size (px)

Citation preview

Page 1: 4'21'15 Abstract Nick Trefethen - University of Notre Dame€¦ · · 2016-06-02Nick%Trefethen! Mathematical*Institute ... Microsoft Word - 4'21'15 Abstract Nick Trefethen.docx

Suppose we have equispaced samples of a smooth function f defined on [-‐1,1]. If f is periodic, then the obvious way to interpolate the data is by a trigonometric polynomial. The integral of the interpolant gives the trapezoidal rule quadrature formula.

What if f is not periodic? The trigonometric interpolant and the trapezoidal rule are not much good, because of the Gibbs phenomenon. However, we can improve the trapezoidal rule to any order by endpoint corrections. The Euler-‐Maclaurin formula uses corrections involving derivatives, and Newton-‐Gregory quadrature uses one-‐sided finite differences. These ideas go back 300 years.

What about using endpoint corrections to improve the interpolant, not just the quadrature formula? Amazingly, so far as we can tell, in 300 years this question has never been considered. In this talk we present such corrections, which we call Euler-‐Maclaurin and Newton-‐Gregory interpolants. Each takes the form of an algebraic polynomial plus a trigonometric polynomial. They approximate f to any order, and their integrals are exactly the results of the Euler-‐Maclaurin and Newton-‐Gregory quadrature formulas.

Euler-‐Maclaurin and Newton-‐Gregory Interpolants

Tuesday, April 21, 2015 4:00 PM – 5:00 PM

127 Hayes-‐Healy Center Colloquium Tea 3:30 PM to 4:00 PM 154 Hurley Hall

Nick Trefethen Mathematical Institute

Oxford University

Department of Applied and Computational Mathematics and Statistics Colloquium

Nick Trefethen is Professor of Numerical Analysis and head of the Numerical Analysis Group in the Mathematical Institute at Oxford University. He was educated at Harvard and Stanford and held professorial positions at NYU, MIT, and Cornell before moving to Oxford in 1997. He is a Fellow of the Royal Society and a member of the US National Academy of Engineering.