It is to be used to evaluate a polynomial 



M 

 P(x) = ^ 

 m=0 





(75) 



This polynomial can represent any of eq (68) -(71). One of three standard forms for con- 

 tinued fractions, this form is used because it has two coefficients at each stage and therefore 

 is equivalent to an asymptotic series of twice as many terms. This reduces by half the num- 

 ber of divisions required. Since complex divisions are lengthy, requiring six real multiplica- 

 tions and two divisions, this is the only standard form of the continued fraction that can 

 compete in computer time with the asymptotic series. 



The coefficients aj and bj of eq (74) must be determined from the coefficients Cj^^. 

 The usual technique is to express P as a rational function, then use the continued fraction to 

 evaluate the rational function. The determination of the coefficients can be done in these 

 two steps or by a second method which goes directly from power series to continued frac- 

 tion coefficients. The second method is preferable because the loss of accuracy is more in 

 the first. But since the first method is more easily understood, each method will be given; a 

 computer program is included in appendix C which will determine coefficients by the sec- 

 ond method. 



Let M in eq (75) be an even number so that 2N = M. (An additional unnecessary 

 term of the series can always be used.) The rational function will have the form 



(76) 



Ux) -k^ \x^r^ fjxi , 



i=0 



i=0 



where eg = fQ - 1 and k = Cq. The coefficients ej and fj are evaluated from a set of linear 

 equations which can be described by displaying a particular case. For N = 3 they are as 

 follows: 







-1 















-1 















-1 











































k ej 



k ^2 





Cl 



C2 









k 63 





C3 



1 









C4 



2 









C5 



3_ 





_\ 





f6_ 



(77) 



With Cj and fj thus determined, R(x) is equivalent to P(x) through the first M + 1 terms. 

 R(x) can now be evaluated exactly (except for round-off error) using a continued fraction 

 of the form F(x) of eq (74). 



27 



