Taylor Forms--Use and Limits

This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 1984 by Eckmann, Koch and Wittwer, and independently studied and popularized since 1996 by Berz, Makino and Hoefkens. A highlight is their application to the verified integration of asteroid dynamics in the solar system in 2001. Apart from summarizing what Taylor forms are and do, this review puts them into the perspective of more traditional methods, in particular centered forms, discusses the major applications, and analyzes some of their elementary properties. Particular emphasis is given to overestimation properties and the wrapping effect. A deliberate attempt has been made to offer value statements with appropriate justifications; but all opinions given are my own and might be controversial.