Der Quotienten-Differenzen-Algorithmus

Summary The quotient-difference (=QD) algorithm developed by the author may be considered as an extension ofBernoulli's method for solving algebraic equations. WhereasBernoulli's method gives the dominant root as the limit of a sequence of quotientsq ₁ ^(v) =s ₁ ^(v+1) /s ₁ ^(v) formed from a certain numerical sequences ₁ ^(v) , the QD-algorithm gives (under certain conditions) all the roots as the limits of similiar quotient sequencesq ^(v) =s ^(v+1) /s ^(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.