COMPARATIVE ACCURACY 



The accuracy of the three methods — power series, asymptotic series and continued 

 fractions — has been determined on a CDC computer with 48 bits or 14.4 decimal places of 

 accuracy in the floating point word. Since this differs from the double precision word 

 length of 60 bits or 18.1 decimal places that applies to the preceding part of this report, 

 these results are for comparative and illustrative purposes only. 



Accuracy is determined by computing the functions and either comparing the an- 

 swers for the several different computing methods or computing the wronskian. The 

 wronskian is a constant given by the relationship 



hih2-h2hj = -1.457495441041 = -i96^l^lTT . (85) 



The wronskian will determine the accuracy of the functions if it can be computed without 

 loss of accuracy. If the two products in it are large, though, accuracy will be lost in the 

 subtraction. This generally happens for arguments near the negative real axis. Here accura- 

 cy must be determined by comparing answers from different methods. The accuracy of the 

 functions and their derivatives will generally be about equal. 



Figure 5 illustrates the accuracy that is obtained in different parts of the complex 

 plane of the argument, z, by using the power series method. On the inner contour, the 

 functions hj and h2 and their derivatives have 12 places of accuracy. On the outer contour, 

 the accuracy is 1 1 places. As expected, the accuracy is best for arguments of small magni- 

 tude. The accuracy remains best in directions from the origin in which the functions are 

 large in magnitude. This is because less accuracy is lost in subtraction. Accuracy must be 

 lost when individual terms of the series are large but the sum is small. 



Figure 6 shows accuracy contours for the asymptotic expansion with both the direct 

 and continued fraction evaluation of the series. Here, the best accuracy is obtained for large 

 arguments, and accuracy decreases toward the origin. As can be seen, each of the two meth- 

 ods is better in some directions from the origin. The choice of methods then depends upon 

 which directions are of most value to the normal mode program. The dots on the figure 

 show the locations at which the functions were evaluated in a typical surface duct run. 

 Although arguments can lie anywhere in the plane, most of them follow this pattern. They 

 lie just above the negative real axis and in a narrow angle above the positive real axis. The 

 continued fraction method is distinctly better on this positive side. Since computing time 

 also favors the continued fraction method, it is clearly the method to use. 



If the 12-place accuracy contour from figure 6 Ues inside that for figure 5 at some 

 angle from the origin, 12 places can be obtained at any range along this angle by using either 

 power series or asymptotic expansion in the interval of overlap. If the asymptotic expan- 

 sion contour lies outside the other, there is an interval in which 1 2 places cannot be ob- 

 tained. Only some lesser number of places can be obtained in this interval. These contours 

 apply when both functions and their derivatives are all computed by a single method. As 

 mentioned earher, increased accuracy could be obtained in some areas by computing the 

 two functions by different methods. 



34 



