(k, = + 2rna/d)' -jJ(k, + 2ma/d)' - k'. k' < (k, + 21najd)'. (6) The spectral domain Green's function converges rapidly for .r # x'. This is due to the exponential factor which aids in the conver- gence. However forx = x', the series converges very slowly. The Grcen's function for a one-dimensional periodic array of point sources located d units apart in the z direction is given by G=- lmexp { -,jkl(x - Y)' + ( y - y)' 4a,,I=-m C + + ( :- md)'li [(x - x')' (y - y')' + (z - 1nd)'II ' ' ' (7) IV. NUMERICAL RESULTS The convcrgcnce properties of the periodic Grccn's function se- ries given in (4), (5), and (7) are reported here for various combi- nations of source and observation points. The results obtained by O-algorithm are compared with those obtained by the application of Shanks' transform and a direct summation of the series. Since the direct summation of the Green's function series converges very slowly, the results of the direct sum that could fit within the scale chosen, are shown. A Convergence criterion defined in [ 11 is em- ployed here to terminate the summation process. A relative error measure is computed by comparing the result of the algorithm to that of summing the series to machine precision. Without loss of generality, we take k, = 0 and the reference source at the origin. defining (x', y') = (0, 0). The logarithm of the relative error magnitude versus number of terms for the spatial domain Green's function in (4) is shown in Figs. 1 and 2. The convergence factor. E,, is indicated alongside each point in the figures. It is shown in Fig. 1 that for (x. y) = (O.OlX, 0.3X), the O-algorithm gives zero relative error in 21 terms for E,. = 1 X IO-?. This indicates that the algorithm has converged to machine precision. As shown in Fig. 2 for = 1 X lo-? the 0-algorithm converges to machine precision in less than 20 terms. The Shanks' transform converges in 53 terms with the relative error approaching 1 X 10-7, The direct sum converges extremely slowly taking over 100 000 terms to arrive at three significant digit accu- racy. The computation time (on VAX 6350) as a function of I /e( for the spatial Green's function is shown in Fig. 3. As illustrated, for (x, y) = (0.01h. 0), the 0-algorithm converges in 0.06 s. Shanks' transform in 0.07 s and the direct sum in 40 s for E, = 1 x This results in a saving in computation time of the order of 600 in using the 0-algorithm over a direct summation of the series. -lr 01x10-2 -2 -1x10-2 mixlo-4 1x10-3. W tU- z -4-~1~10-3 iij -5- IT ~1x10-4 = 1x10-5 A 0-ALGORITHM (9-alaorilhml -81 4 rir -m , , , , ,,,I , , , , , , ,,, , , , , , ,,J 101 102 103 104 NUMBER OF TERMS Fig. I. Log of relative error magnitude versus number of terms for spatial domain Green's function in (4) for A = I ni. d = 0.6A. (.\\-, y) = (0.01h. 0.3A). -l c Fig. domain Green's function 2. Log of relative error magnitude versus number of terms for spatial in (4) for A = I m. d = 0.6h (x. >) = (0.1X. 0.3h). r 8 10-1: BD II I70 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. VOL 40. NO. I. JANUARY 1992 -1 1x10-4 -5/A1x;w; , , ,,,,I , , , , ,,,, , , , , , ai^ 10-4 1x10-3. -6 0 A DIRECT SUM 8-ALGORITHM SHANKS TX. -7 cI cc = 1x10-5 (8-algorithm1 101 102 103 104 NUMBER OF TERMS Fig. 4. Log of relative error magnitude versus number of terms of spectral domain Green's function in (5) for A = I m. d = I .21;. (x, y) = (0. 0.61) - U g U 01x10-3 W W 1 -4 1x10-4. - 4 U W t ~1x10-3 0 3 A 8-ALGORITHM -7 101 102 103 104 NUMBER OF TERMS Fig. 5. Log domain Green's function of relative error magnitude versus number of terms for spectral in (5) for A = I m. d = 1.2A. (.I-. y) = (0, 0.31). The logarithm of the relative error versus number of terms for the spectral domain Green's function given in (5) is shown in Figs. 4 and 5. As shown in Fig. 4, the O-algorithm converges to machine precision in 20 terms for E, = 1 X lo-'. It is shown in Fig. 5 that the direct sum converges very slowly taking several thousand terms to achieve four significant digit accuracy. On the other hand, the 0-algorithm converges to five significant digits in 17 terms for E, = 1 x The magnitude of the relative error versus the number of temis for the periodic Green's function in (7) is shown in Figs. 6 and 7. The spectacular convergence rate of the O-algorithm is illustrated in Fig. 6 for (x. j, z) = (0.2X. 0. IX, 0.3X). In this case for E, = 1 X the direct sum takes 12 000 terms. Shanks' transform takes 43 terms and the O-algorithm converges in merely 21 terms. A similar result is shown in Fig. 7 for (x, j, z) = (0.1X. 0.1X. 0.3X). For E, = I x the direct sum converges in 23 000 terms. the Shanks' transform in 40 terms and the O-algorithm in just 19 terms. Besides converging in fewer number of terms the O-algorithm has the least error. As the observation point is taken closer to the reference source point at the origin, the direct sum converges extremely slowly. Even in such cases the O-algorithm has the fastest convergence. This is illustrated in Fig. 8 in which the computation time (on VAX 6350) versus I /E< is plotted for (x, t m1~10-4 01x10-3 ~1x10-5 1x10-2 U W 01x10-4 t ~1x10-5 .1~10-5 10-5 0 /A1X10:3, , , , , , , , ,,,, A DIRECT SUM 8-ALGORITHM SHANKS' TX. 10-6 101 102 103 104 105 NUMBER OF TERMS Fig. function series 6. Relative error magnitude \\ersus number of terms for the Green's in (7) for A = I m. (/ = 0.61, (x. y. ) := (0.2A, 0.1A. 0.3A). 01x10-2 1x10-3 .1x10-~ .1x10-4 01x10-5 TI, 0 /A~~A DIRECT SUM 8-ALGORITHM , ,I , , , ~, , , , ,, ~SHANKS' TX. ~~~,ul 10-6 101 102 103 104 105 NUMBER OF TERMS Fig. function series 7. Relative error magnitude \\ersus number of in (7) for A terms for the Green's = I m. d = 0.6A, (x. y, :) = (0.11. 0.IA. 0.31). 101 f- 0 DIRECT SUM A 8-ALGORITHM SHANKS' TX. W l- 2 100: 0 2 z 0 o 0 2 10-1 0 B 5 B. 0 ,i4 481 XAA AA -0 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. VOL 10. NO. I, JANUARY 1992 171 y, I) = (0. 0, 0.3X). The saving in computation time in using the &algorithm over a direct summation of the series is of the order of 100. V. CONCLUSION The use of 0-algorithm is shown to have a dramatic impact in accelerating the convergence of slowly converging series. The al- gorithm has been applied with success to the series representing the free-space periodic Green's functions. Numerical results indi- cate that the algorithm is superior to Shanks' transform both in convergence and speed. In most cases the algorithm converges to a high degree of precision in about 20 terms. This is indeed re- markable as a direct sum of the series converges extremely slowly. The use of &algorithm results in a considerable amount of saving in computation time thereby increasing the computational e%- ciency in problem involving one-dimensional periodicity. REFERENCES celerating the convergence of series representing the free space peri- odic Green's function,\" IEEE Trans. Antentias Propagat., vol. 38. pp. 1958-1962, Dec. 1990. 12) R. E. Jorgenson and R. Mittra. \"Eficient calculation ot'the free-space periodic Green's function.\" lEEE Trutis. Antetitias Pr~p~g~t.. vol. 38. pp. 633-642, May 1990. [3] S. Singh and R. Singh, \"Application of transforms to accelerate the summation of periodic free-space Green's functions.\" lEEE Tra~r.\\. Microbtnve Theory Tcvh., vol. 38. pp. 1746-1748. Nov. 1990. 141 C. Brezinski. \"Acceleration de suites a convergence logarithmique,\" C.R. Acad. Sri. Puris Ser. A-B, vol. 273. pp. A727-A730. 1971. [SI C. Brezinski, \"Some new convergence acceleration methods.\" Moth- ematics of Computation, vol. 39. no. 159. pp. 133-145. July 1982. [6] D. Shanks. \"Non-linear transformations of divergent and slowly con- Mark. Physics, vol. 34. pp. 1-42, 1955. vergent sequences.'' .I. [I] S. Singh, W. F. Richards, J. R. Zinecker. and D. R. Wilton, \"Ac- may be accelerated by transforming the series such that the new series converges rapidly [ 11-[5]. The transformation. however, re- quires analytical work which is characteristic for each series. This in some sense limits the applicability of the method. It is our intent to demonstrate that algorithms [SI-[ I I] which can be readily ap- plied to any slowly converging series. irrespective of its functional form, are highly accurate and efficient. In particular, we report the use of the Chebyschev-Toeplitz (CT) algorithm [ 1 I] in accelerat- ing the convergence of periodic Green's function series. 11. CHEBYSCHEV-TOEPLIT7. (CT) ALGORITHM Let S,, be the partial sum of n terms of a series such that S,, S as n 03, where S is the sum of the series. The CT algorithm is defined by the following equations [I I]: TY' = /:\"'/Ui. k = 0. 1. 2. . . . . (5) The n th iterate of the CT algorithm is given by Ty\". which gives an estimate of the sum of the series. The algorithm can be illus- trated by applying it to the slowly converging Liebniz series for a: a= 4( 1)\"' c- ,,r=~~ 2m + 1 - . (6) The result of applying the CT algorithm to the sequence of par- tial sums So. SI. . . . , S8 is given in Table I. The algorithm con- verges to six significant digits. Although the even and odd order only the even iterates of the CT algorithm provide an estimate of S, orders are shown in the table. 111. FREE-SPACE PERIODIC GREEN'S FUNCTIONS On the Use of Chebyschev-Toeplitz Algorithm in Accelerating the Numerical Convergence of Infinite Series Surendra Singh and Ritu Singh Abstract-It is shown here that a simple application of the Cheby- schev-Toeplitz algorithm enhances the rate of convergence of slowlj converging series. The algorithm is applied to series representing the periodic Green's functions involving a single infinite summation. The algorithm yields highly accurate results within relatively fewer terms. A quantitative comparison is shown with methods previously reported in the literature. The spectral domain Green's function for a one-dimensional ar- ray of line source spaced d units apart in the x direction is given by G= \"vi = 1 - m J2dk,,,, where dk' - (2ma/d)'. kh = -jJ(2mr/d)' - k', k' > (2ma/d)' k' < (2mr/d)' I. INTRODUCTION The computation of electromagnetic radiation or scattering from a periodic geometry involves the summation of a Green's function series which converges very slowly. The summation of the series Manuscript received May 16, 1991; revised August 13. 1991. The authors are with the Department of Electrical Engineering, The Uni- versity of Tulsa, Tulsa, OK 74104. IEEE Log Number 9103905. k is the wave number of the medium. (x', y') locates the refer- ence source and (x, y) locates the observation point. The series in (7) converges very slowly whenever = y'. This is referred to as the \"on plane\" case. The spatial domain counterpart of the peri- odic Green's function in (7) is given by m G = ,n = -m Ht'(k[( y - y')' + (x - x' - md)']''*) (8) where Hh2' is the zeroth-order Hankel function of the second kind. The Green's function for a one-dimensional array of point sources 0018-9480/92$03.00 0 1992 IEEE因篇幅问题不能全部显示，请点此查看更多更全内容