M 



f = -Az2 y c^X^ (52) 



ni=0 



M 



g' = B ^ dj^ X"i . (53) 



The coefficients of eq (50) - (53) are given by recursion relations where aQ = 1, aj = 1/3!, 

 32 = + 1 • 4/6!,a3=-l -4 • 7/9! 



^m = -an^_i/(3m)(3m-l) (54) 



bo= l,bi =-2/4!,b2 = +2 • 5/7!,b3 = -2 • 5 • 8/10! 



^m = -bn,_i/(3m)(3m+l) (55) 



co = 3/3!,Cj =-6 • 1 • 4/6! 



Cni = -Cj^_j/3m (3m+2) (56) 



dQ= l,di =-4 • 2/4! 



dm = -dj^_i/3m (3m-2) (57) 



It is important for efficient computation that the number of terms M be no larger 

 than necessary. In the current program the same value of M is used in all four sums. This is 

 done because the optimum number never differs by more than one in the four cases and the 

 determination by table look-up of four M's often would take longer than computing any 

 unnecessary terms. M for each series is determined so that adding additional terms will not 

 change the answer. Then the most stringent of the four conditions is tabulated and used. 



A precise determination of the number of terms to use requires a knowledge of the 

 size of the largest single term in the sum. When a term is smaller than this by a factor which 

 is the power of 10 equal to the number of decimal places in the computer word size, it 

 cannot affect the sum. We ignore the fact that a sum of smah terms might be significant. 

 This, then, defines the truncation point. Let m be the number of the largest term in the 

 sum, k the number of terms to be used, and h the number of decimal digits in the machine 

 word. Then for a given k, the largest absolute value of the argument z that can be used to 

 compute g' is given as 



k 



Iz^l dj„ = Iz-'l dj,- 10'^ . (58) 



The power of ten can be replaced by 2 raised to a power of the number of binary bits in the 

 computer word if preferred. The coefficient d of eq (57) is used. Each of the other three 

 should also be tried, to find the smallest number of the group for a given k. Equation (58) 

 can be solved for |z|, giving 



log |z| = (log dk - log dj„ + h)/(3m - 3k) . (59) 



24 



