COMPUTATION OF THE MODIFIED HANKEL FUNCTIONS 



Most of the computer time required to determine eigenvalues and compute depth 

 functions is spent in evaluating the modified Hankel functions of order 1/3. For this reason, 

 minimizing computer time in evaluating these functions is desirable. Gaining as many places 

 of accuracy as possible is even more important. The average normal mode computation will 

 have many modes that can be determined to far greater accuracy than is required to obtain 

 0.1 dB accuracy in the propagation loss. However, there are usually some and often many 

 modes in which many places of accuracy are lost in evaluating the determinant. Therefore, 

 maximum accuracy in the modified Hankel functions is required to extend the range of 

 cases for which computations can be carried out successfully. 



Optimization of the program is a function of the computer word length. The pro- 

 gram given in this report is for the UNIVAC 1110 with 60 bits word length in double preci- 

 sion or 18.1 decimal places. This section gives the equations and computational techniques 

 that are required to optimize this program for different computer word lengths. Complete 

 details of the functions are given in reference 1 1 . 



The Airy functions Ai(Z) and Bi(Z) can be used instead of the modified Hankel 

 functions h\ and h2. However, since hj is ideally suited to matching the boundary condi- 

 tions at great depth as formulated in this normal mode program, h^ and h2 are used here. 

 The relationship between them is as follows: 



hj (z) = k [Ai (-z) - i Bi (-z)] (46) 



h2(z) = k* [Ai(-z) + iBi(-z)] , (47) 



where 



k = (3/2)^'^ (1 - iv3/3), and k* is the complex conjugate of k. 



In this section z will be the argument of the functions h| and h2. For small values 

 of |z|, hi and h2 are computed by power series expansions. For large values, an asymptotic 

 expansion is used. In the past the asymptotic series was expanded directly. However, a 

 continued fraction expansion has been found to give both shorter running time and better 

 accuracy. 



Figure 4 shows a line in the complex plane which divides the plane into two parts. 

 For values of z within the line, the power series method is used. When z is outside the line, 

 the continued fraction method is used. This line is a function of computer word length, and 

 the method of determining it will be given after the two methods have been treated. The 

 accuracy of the methods is also treated. 



The program has a parameter called IH in the FORTRAN call statement which con- 

 trols which functions are computed. If IH is set to zero, both functions and their derivatives 

 are computed. If IH is set to 1 , only the functions are computed. If it is set to 2, only h2 

 and its derivative are computed. 



11. Tables of the Modified Hankel Functions of Order One-Third and their Derivatives, Harvard University 

 Computation Laboratory; Harvard University Press, Cambridge MA, 1945. 



22 



