Rather than R(x), however, a similar expression in y = 1/x is the form that is well 

 suited to evaluating asymptotic series. This expression is obtained by dividing each term of 

 R(x) by X^. The order of the coefficients can now be reversed and a simple algebraic oper- 

 ation can yield a value of 1 for each of the two initial coefficients and a new value for k. We 

 will call this new rational function with renamed coefficients R(y). It will have the form of 

 eq (76) but different coefficients, say e and f instead of e and f. 



The coefficients aj and bj are determined from ej and fj by a recursive formula which 

 involves constructing an n X n triangular matrix Q with elements qj ; as follows: 



^0 = % 



Ql^i = (ej-eofp/aj i=l,2, ...,N, 



where q j j = 1 , giving a j , and 



bi = fi-qi^2 • 

 The second row: 



^2,1 = ^fi-qi,i+l-biqi,i)/a2 i = 2,3,...,N , 



where q2 2 ~ ' ' S^^i^S ^2' ^""^ 



^2 " qi,2"q2,3 • 



Elements outside the matrix are assigned a value of zero. The remaining rows for m = 3 to 

 N are as follows: 



^m,! = (lm-2,i-qm-l,i+l-bm-iqm-l,i)/am ' = m,m+l,...,N , 



where q^^j^ = 1 , giving a^^, and 



^m " ^m-Um'^^m.m+l • 



The second method determines the continued fraction coefficients aj and bj directly 

 from the asymptotic series coefficients Cj. This method is preferable to the first because the 

 loss of accuracy in inverting the matrix in eq (77) can be more than the loss in this second 

 method. 



It has been pointed out that the second method is probably a variant of the 

 Viskovakoff algorithm described by Khovanskii (ref 12) and as such is unstable — subject to 

 accumulation of errors. However, it is sufficiently stable to obtain the required coefficients. 



*Private communication with AN Stokes, CSIRO, Wembley, Western Australia. 



12. The Application of Continued Fractions and their Generalizations to Problems of Approximate Analy- 

 sis, by AN Khovanskii; a monograph in Russian, 1956. 



28 



