Der Quotienten-Differenzen-Algorithmus

Abstract

The quotient-difference (=QD) algorithm developed by the author may be considered as an extension of Bernoulli's method for solving algebraic equations. Whereas Bernoulli's method gives the dominant root as the limit of a sequence of quotients q1(v)=s1(v+1)/s1(v) formed from a certain numerical sequence s1(v), the QD-algorithm gives (under certain conditions) all the roots λσ as the limits of similiar quotient sequences qσ(v)=sσ(v+1)/sσ(v). Close relationship exists between this method and the theory of continued fractions. In fact the QD-algorithm permits developing a function given in the form of a power series into a continued fraction in a remarkably simple manner. In this paper only the theoretical aspects of the method are discussed. Practical applications will be discussed later. © 1954 Verlag Birkhäuser AG.