For many years the author has been engaged in studies of those
fields of mathematical analysis that are of primary concern to the
engineer and the physicist- That this area of "workable mathematics"
did not receive the same attention during the 19th century as did the
classical fields of analysis is perhaps the result of a historical
misunderstanding. Until the time of Gauss and Legendre the "workable"
methods of analysis received the closest attention of the best
mathematicians. The brilliant discovery of the theory of limits changed the
emphasis. Thenceforth it was considered satisfactory to design
infinite approximation processes by which the validity of certain
analytical results could be established, irrespective of whether the
process used were feasible or not for a given problem-
It was then that the gradual separation of "pure" and "applied"
mathematics occurred until we now have the "pure analyst," who
pursues his ideas in a world of purely theoretical constructions, and
the "numerical analyst," who translates the processes of analysis into
machine operations.
In actual fact there is a large area between the two, which is not less
analytical than the analysis of infinite processes but devoted to a
different branch of analysis, namely, the analysis of'finite algorithms.
Here our objective is the analysis and design of finite processes which
approximate the solution of an analytical problem. To design
procedures which will effectively minimize the error in a small number
of steps and which will estimate the error with sufficient accuracy
is not a matter of practical interest only, but a matter of scientific
interest as well. This book is largely devoted to such problems.