The most popular methods for the solution of stiff initial value problems for ordinary differential equations are the backward differentiation formulas (BDFs). Because the stability of these formulas deteriorates rapidly as the order increases within the family, a great deal of effort has been devoted to finding formulas of moderate to high order with better stability. Stability is not the only issue, however: the formulas also have to be computationally efficient.

In two earlier papers, Cash proposed and developed some formulas known as modified extended backward differentiation formulas (MEBDFs). They have good properties, but developing a production-grade code based on the formulas that could compete fairly with the highly polished BDF codes in use is a task of considerable magnitude. This  paper  presents such a code. The authors describe some of the algorithmic developments that are so important to quality software. They present substantial experiments comparing the code to the popular code LSODE, based on the BDFs, and to SECDER, based on second derivative methods.

No way of solving stiff initial value problems is best in general. The evidence presented in this paper makes it clear that the MEBDFs as implemented in this code compete well with the BDFs and are superior for certain kinds of problems.