Classes of semidefinite Peano-kernels

Abstract

For applications to the error analysis of quadrature formulas, or of multistep-methods, associated with the form L, the basic question is whether the Peano-kernel of L has a fixed sign. We are going to prove that, loosely speaking, a stable form of degree p = m (k + 1)+ 1 has almost always a semidefinite Peanokernel and that for degree p=m(k+ 1) this is true in many cases. Actually, all (generalized) Newton-Cotes-forms with even k and odd m and all (generalized) Adams-Moulton-forms turn out to have such a kernel. For details see Theorems 4.2 and 5.2. The proof is essentially based on several theorems concerning the location of the roots of certain polynomials. These results are the topic of Section 2.