Empirical mathematics: the first patterson extension of gauss-kronrod rules

Abstract

The Kronrod extension (KE) of an n-point Gauss integration rule with respect to some weight w is a (2n+l)-point rule of optimal precision which consists of the n Gauss points and n+1 additional distinct real points contained in the integration interval. KE's may or may not exist depending on w and n. For a KE which exists, we define the first Patterson extension (PE) to be a similar optimal extension of the KE. It contains at least 3n+3 points, in which case it is said to be minimal. We give here some experimental results on the computation of PE's and make some conjectures about the existence of minimal and nonminimal PE's when w is a Jacobi weight.

Keywords:

• Gauss integration rule
• Kronrod extension
• Radau rule
• Lobatto rule
• Patterson extension
• Stieltjes polynomial
• Gegenbauer weight
• Jacobi weight