Calculation of Radau-Kronrod and Lobatto-Kronrod quadrature formulas

D. Laurie, University of Stellenbosch, South Africa

Monday, June 30
at 16.30
in HG D1.2

The Kronrod extension of an n-point quadrature formula Q of degree d is a quadrature formula K such that

• K has m=d+2 nodes, including all the nodes of Q;
• the degree of K is at least 2m-1-n.

When Q is a Gauss (all nodes free), Radau (one node preassigned) or Lobatto (two nodes preassigned) formula, these three properties define a unique quadrature formula (which does not necessary exist) and we call K respectively a Gauss-Kronrod, Radau-Kronrod or Lobatto-Kronrod formula.

If enough three-term recursion coefficients of the underlying weight function are available, the the Gauss formula has the same recursion coefficients, and there are known efficient (O(n²)) and stable algorithms for modifying these recursion coefficients to yield the recursion coefficients for the Radau, Lobatto and Gauss-Kronrod formulas. But if we modify the recursion coefficients for the Gauss-Kronrod formula in the same way that produces a Radau formula from the recursion coefficients for a Gauss formula, we do not general obtain the Radau-Kronrod formula, etc.

In this talk, I discuss the algorithms refered to above, and finish with a new algorithm for computing n-point Radau-Kronrod and Lobatto-Kronrod formulas in O(ⁿ) operations. The algorithm is applicable to any weight function for which enough three-term recursion coefficients are known.