Linear recurrence relations

Abstract

Walter Gautschi is a giant in the field of linear recurrence relations. His concern is with stability in computing solutions \( \{y_{n}\}_{n=0}^{\infty} \) of such equations. Suppose the recurrence relation is of the form $$\displaystyle{ y_{n+1} + a_{n}y_{n} + b_{n}y_{n-1} = 0\qquad \mbox{ for}\quad n = 1,2,3,\ldots.}$$ (21.1) It seems so deceivingly natural to start with values or expressions for y 0 and y 1, and then compute y 2, \(y_{3},\ldots\) successively from (21.1). However, this does not always work. Still, in every new generation of mathematicians or users of mathematics, along come some incorrigible optimists with a naive trust in this method. We are happy, of course, for every new optimist in the field; mathematicians do not get far without optimism, stamina, creativity, and enthusiasm. But the new ones can definitely benefit from some sensible guidance. And what they should do, is to start with Walter Gautschi’s SIAM Review paper [GA29] on three-term recurrence relations from 1967. This is what most people do, and this is what I did when I started my study of continued fractions. Continued fractions and recurrence relations indeed share a substantial intersection which, however, calls for some degree of alertness.