# Numerical integration of related Hankel tiansforms and continued fraction expansion

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4 1674 Chave EXAMPLES AND APPLICATIONS A sequence of eight Hankel transforms of elementary functions was selected to demonstrate the features and capabilities of the algorithm; a driver program incorporating them is contained in Appendix B. The integrals are separated into four types, starting with the rapidly convergent forms s m e - p*/4n dk kcd2j,(kp) = 2a 0 (6) cc dk CkJ,(kp) = r\$. s 0 2 (7) and including the slowly convergent integrals s wdk J,(kp) = ; 0 s =dk o d&jo(b) = 7. (8) (9) all of the cases, with the two rapidly convergent integrals requiring the fewest number of partial integrations and the oscillatory kernel cases needing the most. In general, a higher order quadrature rule is required for small values of the range R; this is accomplished automatically by the code. Note that the divergent integrals are easily handled, requiring a comparable amount of computer time to the slowly convergent or oscillatory integrals. Integrals (6) and (7) were included in the test program for ZHANKS in Anderson (1979), and typically required 150 kernel function evaluations for similar accuracy to the results in Table 1. At moderate values of the parameter R, the direct method requires only slightly more computer time Table 2 shows similar computations with RERR = lo-, a far more stringent requirement with a concomitant increase in computational overhead. Convergence was achieved for all cases save the rapidly divergent form (11) at the shortest range. This is caused by a lack of sufficient numerical precision in the The algebraically divergent types a: dk kj,(kp) = 0 s m dk kj_ Jo (kp) = y (ap + 1) 0 and the oscillatory kernel forms s m J&7-1 P&-T dk cos kj,(kp) = P s 30 0 (P 2 1) 0 cos k dk - k J,(b) = 45 (p > l), P (10) (11) (12) (13) z r Y e ;,I.. L _ /,,,,,, Number of Terms (P < 1) (P 5 1) where a = (1 + n/j?. Only the first pair of integrals (6)-(7) can be handled by the digital filter method, and the divergent pair (lo)-(1 1) can only be numerically integrated using an analytic continuation algorithm such as the continued fraction expansion. The last three pairs of integrals are algebraically related, and the appropriate program feature is used in the driver of Appendix B. Results for the eight test integrals at short, moderate, and long range for two values of the relative error parameter are shown in Tables 1 and 2. The columns show the integral number corresponding~to integrais <f+ji3j, thcerrorl?agi~er~r~ returned by BESAUT, the Bessel function argument R, the NEW parameters passed to BESAUT, the relative error RERR, the real and imaginary parts of the numerical result, the real and imaginary parts of the analytic result, the number of the quadrature rule used at the last iteration, the total number of calls to FUNCT, and the number of partial integrations used in the continued fraction algorithm at the last iteration. The absolute error AERR is always taken to be 1000 times smaller than the relative error. Table 1 contains the results for RERR = 10m5, which are comparable in accuracy to ZHANKS, where that algorithm is applicable (Anderson, 1979). Convergence is easily achieved for i y q Number of Terms FIG. 1. The top plot shows the convergence behavior of the continued fraction algorithm (solid) and direct summation of the partial integrations (dashed) for the rapidly convergent integral (7) using a relative error requirement of lo-. The continued fraction expansion results in more rapid convergence by a factor of over 50 percent. The bottom plot shows similar results for the slowly convergent integral (14). The continued fraction expansion has converged after 18 terms in the series of partial integrations, while the direct sum has not converged after 100 terms.

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7 Integration of Hankel Transforms 1677 continued fraction algorithm as larger and larger terms are added to the series, and better behavior would be obtained if this portion of the algorithm (subroutines PADECF and CF) used a longer word length (e.g., REAL * 16 on the VAX or DOUBLE PRECISION on a mainframe computer). Figure I illustrates the power of the continued fraction expansion. The top plot shows the integral (7) with R = 2 and RERR = 10-r, AERR = lo- t5, The solid line shows the con- tinued fraction result, which converged after 11 terms, and the dashed line shows the direct sum of the partial integrands, which required 18 terms for identical convergence. The bottom part of the figure shows the slowly convergent integral nk J,(kp) = f. (14) 0 where the continued fraction result converges after 18 terms, but the direct sum is alternating and approaches the correct result very slowly. It is likely that numerical round-off will prevent the latter from ever reaching convergence. The application of this algorithm to a geophysical problem was discussed in Chave and Cox (1982) for a seafloor-based horizontal electric dipole (HED). The two fundamental electromagnetic modes for this type of source are nearly out of phase, necessitating high accuracy in the calculation of the electromagnetic fields. Correction for the finite length of real sources requires integration with a combination of Hankel transforms as integrands, another application with fairly stringent precision requirements. The direct integration algorithm of this paper has worked well in this application, and yields good user control of the computer time and numerical accuracy. ACKNOWLEDGMENTS This work was supported by the office of Naval Research and by the National Science Foundation under grant OCE REFERENCES Anderson, W. L., 1979, Numerical integration of related Hankel transforms of orders 0 and 1 bv adaptive digital filtering: Geophysics, v. 44. p ~ 1982, Fast Hankel transforms using related and lagged convolutions: ACM Trans. on Math. Software, v. 8, p Baker, G. A., Jr., 1975, Essentials of Pade approximants: New York, Academic Press. Bender. C. M.. and Orszag, 1978, Advanced mathematical methods for scientists and engineers: New York, McGraw-Hill. Chave. A. D., and Cox. C. S., 1982, Controlled electromagnetic sources for measuring electrical conductivity beneath the oceans, 1. forward problem and model study: J. Geophys. Res., v. 87, p Cornille, P., 1972, Computation of Hankel transforms: SIAM Rev.. v. 14, p Ghosh, P. P., 1971, The applicatton of linear filter theory to the direct interpretation of geoelectrical resistivity sounding measurements: Geophys. Prosp.. v. 19,~ Hanrei. P., Roesel, F., and Trautmann, P., 1978, Continued fraction expansions in scattering theory and statistical non-equilibrium mechanics: 2. Naturforsch., v. 33a, p ~ 1980, Evaluation of infinite series by use of continued fraction expansions: a numerical study: J. Comp. Phys., v. 37, p Lawson, C. L., Hanson, E. J., Kincaid, D. R., and Krogh, F. T., 1979, Basic linear algebra subprograms for Fortran usage: ACM Trans. on Math. Software. v. 6, p, Patterson, T. N. L., 1968, The optimum addition of points to quadrature formulae: Math. Comp., v. 22, p Ralston, A., and Rabinowitz, P., 1978, A first course in numerical analysis 2nd ed.: New York. McGraw-Hill. Watson, G. N., 1962, A treatise on the theory of Bessel functions, 2nd ed.: Cambridge, Cambridge Univ. Press.

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