Abstract
 Top of page
 Abstract
 1. Introduction
 2. Spectral, Spatial, and Ewald Green's Function Representations
 3. Splitting Parameter (E)
 4. Choice of the Splitting Parameter
 5. Number of Terms Needed for Convergence
 6. Results
 7. Conclusions
 Appendix A
 References
 Supporting Information
[1] Accurate and efficient computation of periodic freespace Green's functions using the Ewald method is considered for three cases: a 1D array of line sources, a 1D array of point sources, and a 2D array of point sources. A limitation on the numerical accuracy when using the “optimum” E parameter (which gives optimum asymptotic convergence) at high frequency is discussed. A “best” E parameter is then derived to overcome these limitations. This choice allows for the fastest convergence while maintaining a specific level of accuracy (loss of significant figures) in the final result. Formulas for the number of terms needed for convergence are also derived for both the spectral and the spatial series that appear in the Ewald method, and these are found to be accurate in almost all cases.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Spectral, Spatial, and Ewald Green's Function Representations
 3. Splitting Parameter (E)
 4. Choice of the Splitting Parameter
 5. Number of Terms Needed for Convergence
 6. Results
 7. Conclusions
 Appendix A
 References
 Supporting Information
[2] In applying numerical full wave methods like the Method of Moments (MoM) or Boundary Integral Equations (BIE) to periodic structures involving conducting or dielectric electromagnetic scatterers, fast and accurate means for evaluating the freespace periodic Green's function (FSPGF) are often needed. This type of Green's function arises in a wide variety of applications, ranging from microwaves to optics, to the study of metamaterials and nanostructures. The Ewald method is one of the fastest methods for calculating the FSPGF. In the Ewald method, the FSPGF is expressed as the sum of a “modified spectral” and a “modified spatial” series. The terms of both series possess Gaussian decay, leading to an overall series representation that exhibits a very rapid convergence rate. The convergence rate is optimum when the “optimum” value of the Ewald splitting parameter is used [Jordan et al., 1986], denoted here as E_{opt}. (For some applications, such as when using a periodic Green's function in a MoM solution with fulldomain basis functions, one may not wish to have a balanced convergence between the two series, as explained in Mathis and Peterson [1996] and Mathis and Peterson [1998]. However, when using subdomain basis functions and performing the integrations over the basis and testing functions in the spatial domain, the objective is to minimize the computation time of the periodic Green's function, and this is accomplished by using E_{opt}. In this case the singular integrals involved in evaluating the matrix elements can be handled by specially designed numerical quadrature rules [Khayat and Wilton, 2005].)
[3] However, the numerical accuracy of the Ewald method degrades very quickly [Kustepeli and Martin, 2000] with increasing frequency (i.e., when the periodicity becomes large relative to a wavelength). This is due to a catastrophic loss of significant figures in combining the contributions of the two series, wherein the leading terms (and to a lesser extent, other nearby terms) in each series become very large but nearly equal and opposite in sign.
[4] The method proposed and studied here limits the size of the largest terms in the series relative to that of the total Green's function by modifying the value of the splitting parameter E to avoid undue loss of accuracy. Increasing the E parameter limits the size of the largest terms in both series at the expense of decreasing the convergence rate. Hence, there is a tradeoff between the size of the largest term allowed, which determines the number of significant figures lost, and the series convergence rate. A value E_{L} of the Ewald splitting parameter is then obtained based on the number of significant figures L that may be lost. This “best” value, E_{L}, then yields the fastest convergence of the Ewald series while limiting the loss of significant figures to the userdefined value L.
[5] A preliminary and intuitive analysis for the “best” choice of the Ewald splitting parameter was performed in Capolino et al. [2005] for the case of 1D array of line sources, and in Capolino et al. [2007] for the case of a 1D array of point sources. The case of a 2Darray of point sources has been treated in detail in Oroskar et al. [2006]. Here we extend the analysis of the last paper to the two other cases, and provide a unified formalism for the choice of the Ewald splitting parameter and the summation limits that is valid for all three cases.
2. Spectral, Spatial, and Ewald Green's Function Representations
 Top of page
 Abstract
 1. Introduction
 2. Spectral, Spatial, and Ewald Green's Function Representations
 3. Splitting Parameter (E)
 4. Choice of the Splitting Parameter
 5. Number of Terms Needed for Convergence
 6. Results
 7. Conclusions
 Appendix A
 References
 Supporting Information
[6] Three different FSPGF cases are considered: a 1D array of line sources with interelement period d, a 2D array of point sources on a general skewed lattice, and a 1D array of point sources with interelement spacing d. The geometries are depicted along with relevant coordinate systems and geometrical definitions in Figures 1, 2, and 3, respectively. An interelement phase shift along the array direction(s) is assumed. Both spatial and spectral representations of each Green's function exist in the general form
where the terms of the spatial representations are
whereas those of the spectral representations are
where Δz ≡ z − z′ and an e^{jωt} time dependence is assumed and suppressed. In the above, (x, y, z) and (x′, y′, z′) denote the observation and referenceelement source points, respectively. For the 1D cases, the periodicity is along x with a period of d (see Figures 1 and 2), and
For the 2D array with lattice vectors s_{1}, s_{2} (see Figure 3), the parameters are defined as
The transverse phasing wave vector k_{t00} = k sin θ_{0} cos ϕ_{0} + k sin θ_{0} sin ϕ_{0} defines the interelement phasing for the 2D array in terms of the propagation angles θ_{0}, ϕ_{0} of the (0, 0) Floquet mode. For 1D arrays, this quantity becomes the scalar phasing k_{x0} = k cos θ_{0}, where θ_{0} is measured with respect to the xaxis. Physically the FSPGF is the timeharmonic scalar potential produced by the associated array of phased sources.
[7] For a lossless medium the square root for the wave numbers k_{ρp} and k_{zpq} is that which gives a positive real number or a negative imaginary number, corresponding to waves that propagate outward or decay away from the sources, respectively. For a lossy medium the wave numbers are complex and it suffices to require the imaginary part of the wave numbers to be negative. Henceforth, it will be assumed that the medium is lossless, but the formulas can be extended to the lossy case.
[8] When employing the Ewald method for the evaluation of the FSPGF, the Green's function is expressed as a sum of two series [Ewald, 1921] of the form
The terms that appear in the three different cases may be found in Capolino et al. [2005], Capolino et al. [2007], and Oroskar et al. [2006], for the 1D array of line sources, the 1D array of point sources, and the 2D array of point sources, respectively. Summarizing, the terms of the modified spatial representations are
where erfc(z) is the complementary error function and E_{q}(z) denotes the exponential integral function of order q. The terms of the modified spectral representations are
For the 1D array of point sources, the exponential integral function that appears may have a negative argument, depending on the frequency. In this case the argument is interpreted as being infinitesimally above the branch cut of the exponential integral function on the negative real axis, corresponding to an infinitesimal amount of loss.
3. Splitting Parameter (E)
 Top of page
 Abstract
 1. Introduction
 2. Spectral, Spatial, and Ewald Green's Function Representations
 3. Splitting Parameter (E)
 4. Choice of the Splitting Parameter
 5. Number of Terms Needed for Convergence
 6. Results
 7. Conclusions
 Appendix A
 References
 Supporting Information
[9] In equations (5) and (6) the spatial and spectral series both involve a “splitting” parameter E. The “optimum value” E_{opt} for the splitting parameter [Jordan et al., 1986] balances the asymptotic rate of convergence of the spatial and spectral series, and consequently minimizes the overall number of terms needed to calculate the total FSPGF. The optimum value is found to be
However, numerical difficulties are encountered when the lattice separations (periods) become large relative to a wavelength. This was first discovered in Kustepeli and Martin [2000], and subsequently also discussed in Capolino et al. [2005], Oroskar et al. [2006], and Capolino et al. [2007]. This happens because, for large arguments, both the complementary error function and the exponential integrals contribute terms of the form exp[(k/(2E))^{2}] that produce extremely large initial terms (and to a lesser extent, large nearby terms) in both the spatial and the spectral series. The resulting series then each converge to very large, but nearly equal and oppositely signed values, resulting in a total sum of moderate value but with a catastrophic loss of significant figures when the two series are combined. Exponential overflow is another potential concern as well, due to the large initial values in each series.
[10] To circumvent the problem, it is desirable to limit the size of the largest terms of each series by choosing an E value larger than the “optimum” value, which reduces the maximum values of both the complementary error function and the exponential integrals. As a result, one avoids loss of accuracy in adding the two series, and a more accurate result for the total Green's function is obtained [Kustepeli, and Martin, 2000] at the expense of slower convergence.
[11] In the following sections, a recipe for finding the best choice for E, called E_{L}, that achieves the fastest convergence under the constraint of limiting the loss of significant figures to L digits, is obtained for a general source and observation point, for all three cases.
4. Choice of the Splitting Parameter
 Top of page
 Abstract
 1. Introduction
 2. Spectral, Spatial, and Ewald Green's Function Representations
 3. Splitting Parameter (E)
 4. Choice of the Splitting Parameter
 5. Number of Terms Needed for Convergence
 6. Results
 7. Conclusions
 Appendix A
 References
 Supporting Information
[12] The strategy is to limit the size of the largest terms relative to the value of the total Green's function, with the largest terms in each series being the initial (0) or (0, 0) terms in 1D or 2D, respectively. The value of E = E_{L} is obtained by enforcing the following conditions:
where G^{est} is a closedform estimate of the FSPGF, and the integer parameter L indicates (roughly) the number of significant figures one is willing to sacrifice in the calculation. For a strict bound, the factor α in (8) should be chosen as 1/2 to ensure that each of the two initial terms in (8) (one from the spatial series and one from the spectral series) contributes no more than half the total error limit. However, it is found in each of the three cases that one of the initial terms (either the spatial one or the spectral one, depending on the case) is significantly larger than the other, and a factor of α = 1 therefore becomes more appropriate, and this is adopted here. It suffices to use a rough estimate of the Green's function on the righthand side of (8), which may be obtained by examining the most nearly singular terms from both the spatial and the spectral series. (This is an improvement over the previous derivations, as in Oroskar et al. [2006], where only the spatial term was used.) This yields the following magnitude estimate for the overall Green's function:
In (9), the indices p, q, m, n that produce the smallest values for R_{m}, R_{mn}, k_{ρp}, k_{zp}, and k_{zpq} must be determined; although this may be done analytically, it is also very easy to find these values from a simple numerical search.
[13] To proceed with the derivation of E_{L}, we replace the terms on the left hand side of (8) by their highfrequency asymptotic estimates. The complementary error function terms are estimated by using the asymptotic relation
valid for large arguments. In addition, in (5) and (6) series of exponential integral functions appear. The simplest way to asymptotically evaluate these series is to recast them back into their integral forms. The integral forms are equation (15) in Capolino et al. [2007] and equation (11) in Capolino et al. [2005], which are reproduced below:
The integrals that appear in the above identities may be asymptotically evaluated for k∞ via integration by parts [Felsen and Marcuvitz, 1994; Bleistein and Handelsman, 1986] (see case 1a and case 2a of Appendix A for further details), yielding
Applying (9)–(14) in (8), as detailed in Oroskar et al. [2006] yields in each case (spectral and spatial) a transcendental equation of the form
where the constant K and the function F in (15) are defined in Table 1 (in each case the function F contains a factor c_{1} that is also shown). Solving the quadratic form for on the lefthand side of (15) puts the equation into the following form that may be efficiently solved iteratively, due to the slow variation of the ln function:
where F^{i} = F(^{i}).
Table 1. Definition of Parameters Appearing in the Iterative Equation for Determining E_{L}    K  c_{1}  F() 

1D array of line sources  Spatial    10^{L}4πG^{est}  c_{1}^{2} 
Spectral    10^{L}2k_{z0}dG^{est}  
1D array of point sources  Spatial    10^{L}4(π)R_{0}G^{est}  
Spectral    10^{L}4πdG^{est}  c_{1}^{2} 
2D array of point sources  Spatial    10^{L}4(π)R_{00}G^{est}  
Spectral    10^{L}2k_{z00}AG^{est}  
[14] The solution of (15) yields the parameter , which (see Table 1) is inversely proportional to the desired Ewald parameter E. Two E values, E_{spectral} and E_{spatial}, result from this procedure. To properly bound both series, E should then be chosen as
The resulting value is the smallest value of E that ensures that the largest term in both the spectral and the spatial series (see equation (8)) is limited in magnitude to avoid losing more than L significant figures when the two series are combined. At low or moderate frequency, where the initial terms of the spectral and spatial series are not large, E_{L} = E_{opt} since E_{opt} will be the largest of the three terms in (17).
5. Number of Terms Needed for Convergence
 Top of page
 Abstract
 1. Introduction
 2. Spectral, Spatial, and Ewald Green's Function Representations
 3. Splitting Parameter (E)
 4. Choice of the Splitting Parameter
 5. Number of Terms Needed for Convergence
 6. Results
 7. Conclusions
 Appendix A
 References
 Supporting Information
[15] Having determined the “best” value of the E parameter E_{L} as a function of frequency, the next goal is to determine how many terms should be summed in each series (modified spatial and modified spectral) in (4) to achieve convergence. (One could, of course, check convergence as the series are summed, but we eventually want to select between the Ewald and alternative methods for computing the FSPGF that may be more efficient in some situations; for that an a priori estimate of the number of series terms and their relative computational cost is required.) Recall that a given value of L, the number of significant figures sacrificed in the calculation, has been assumed. For the resulting value of E = E_{L} we must then determine how many terms in each series are needed to guarantee convergence of the Green's function to S significant figures. A method is developed here to calculate the series index limits ±P, ±Q, ±M, and ±N for the series indices p, q, m, and n, respectively (Q and N only apply for the 2D geometry). If the accuracy of the arithmetic is limited to T significant figures due to the machine precision (or, more likely in practice, limited by the accuracy of the complementary error function and exponential integrals), the value of S specified should be limited to S < T − L in order to avoid unnecessary computation.
[16] Owing to the Gaussian convergence, a rough estimate of the truncation error for both the spectral and spatial series is obtained by using the sum of the largest of the first neglected terms along each principal sum index in the series. Limiting the convergence error in each series to one half of the total, we thus require that
The error in stopping the summations is approximated in the above equations as the sum of the two (four) values that give the largest contributions to the summed values for 1D (2D) arrays. The factor β is introduced to allow for an empirical adjustment of the summation limits. Using β = 1 corresponds to a strict error bound based on the assumed asymptotic approximations, and therefore usually represents a worstcase error bound. However, factors of β = 2 and β = 4 have been found to work well for the 1D and 2D cases, respectively. If we assume that the contributions of the positively indexed terms on the LHS of inequalities (18) and (19) are dominant, the choice of the limits amounts to choosing M, N, P, Q such that
(The dominance of the positively indexed terms may be assumed without loss of generality, by placing absolute values on the spatial displacements and phasing wave numbers, as seen in (22)–(25).) Using the asymptotic estimates of Table A1, case 1b and case 2b, which assume large values of M, N and P, Q for the terms in (20) above, yields a transcendental equation of the form
where Δ = ξ^{2} + c_{2}^{2}, and the constants c_{2} and W in (21) are defined in Table 2.
Table 2. Definition of Parameters Appearing in the Summation Limit Equation   ξ  c_{2}  W 

1D array of line sources  Spatial  R_{m}E  0  10πβ 
Spectral   jEΔz  10^{S} 
1D array of point sources  Spatial  R_{m}E   10^{S} (π) 
Spectral   0  10 
2D array of point sources  Spatial  R_{mn}E   10^{S} (π) 
Spectral   jEΔz  10^{S} 
[18] Once Δ and thus ξ are determined, the indexed quantities defined in the ξ column of Table 2 may be used to determine the index limits m = M + 1, p = P + 1 for 1D arrays, and the index pairs (m, n) = (M + 1, 0), (0, N + 1) and (p, q) = (P + 1, 0), (0, Q + 1) for the 2D array.
[19] The results for the case of a 1D line source array are
For the 1D point source array the results are
For the 2D point source array the result is (restricting the result to the case of the rectangular lattice _{1} = a, _{2} = b for simplicity):
In these final results, noninteger solutions of (21) for the summation limits should be rounded up to the next larger integer to obtain M + 1, etc., or equivalently, rounded down to obtain M, etc., as assumed in (22)–(25), in which the Int function truncates to the next lower integer.
[20] One note regarding (22)–(25) should be made in connection with the square roots. Depending on the geometry of the problem and the specified convergence accuracy, it may occur that the argument of one of the square roots is negative, yielding a complex value for the summation limit. This occurs because the asymptotic approximation used to estimate term magnitudes becomes invalid when the series actually needs only a few terms to converge, corresponding to a summation limit of zero or one. The problem is circumvented by always using a summation limit that is at least unity.
6. Results
 Top of page
 Abstract
 1. Introduction
 2. Spectral, Spatial, and Ewald Green's Function Representations
 3. Splitting Parameter (E)
 4. Choice of the Splitting Parameter
 5. Number of Terms Needed for Convergence
 6. Results
 7. Conclusions
 Appendix A
 References
 Supporting Information
[21] In this section results are presented for the three cases: 1D line sources, 1D point sources, and 2D point sources. For all results, freespace conditions are assumed (k = k_{0} and λ = λ_{0}). For the 1D cases, d = 0.5 m. The 2D results are shown for a square lattice (a = b = 0.5 m). As a result, the optimum splitting parameter is E_{opt} = 3.5449 for all three cases. A zero progressive phase shift is assumed (all source elements are in phase). The reference source element is located at the origin and the observation point is located at (0, 0, Δz) with the vertical distance from the source plane set at Δz = 0.05 m. For brevity's sake, only the magnitude of the Green's function terms are shown in the results.
[22] For each case, three tables are shown. For the 1D array of line sources, Tables 3a3b–3c are shown; for the 1D array of point sources, Tables 4a4b–4c are shown; and for the 2D array of point sources, Tables 5a5b–5c are shown. The first tables in each case (part (a)) illustrate that the values of _{0,0}^{E} and G_{0,0}^{E} become enormous at high frequency. For these tables the number of lost significant digits is set to L = 3. (The calculations of the special functions were performed with sufficient accuracy to ensure that T > S + L in all cases.) The value of E_{L} limits the size of the largest of the (0, 0) terms, namely G_{0,0}^{E}, to a value on the order of 10^{3} times the exact Green's function G (denoted “G Exact” in the tables), as expected. The numerically exact Green's function has been calculated using a pure spectral method, shown in (3).
Table 3b. OneDimensional Line Source Array: G_{0}^{E} and _{0}^{E} for Various Values of L, Keeping the Frequency Fixed at d = 5.5λ_{0}L  E_{opt}  E_{L}  G_{0}^{E} Using E_{L}  _{0}^{E} Using E_{L}  G Exact 

1  3.544906  15.795788  1.0881778  0.2137888  0.147732 
2  3.544906  12.793900  11.564017  2.9467840  0.147732 
3  3.544906  11.045671  115.39839  34.849919  0.147732 
4  3.544906  9.8719462  1142.9574  391.50175  0.147732 
5  3.544906  9.0139223  11330.598  4287.8952  0.147732 
6  3.544906  8.3508864  112514.12  46237.312  0.147732 
Table 3c. OneDimensional Line Source Array: Calculated and Actual Values of the Summation Limits for Various Values of S_{spec}, Keeping the Frequency Fixed at d = 5.5λ_{0}, Using β = 2 as a Factor^{a}S_{spec}  P_{cal}  P_{act}  M_{cal}  M_{act}  S_{act} 


1  5  4  0  0  2.01 
2  6  5  0  0  4.32 
3  6  6  0  0  4.32 
4  7  6  0  0  6.60 
5  7  7  0  0  6.60 
6  7  7  0  0  6.60 
Table 4b. OneDimensional Point Source Array: G_{0}^{E} and _{0}^{E} for Various Values of L, Keeping the Frequency Fixed at d = 5.5λ_{0}L  E_{opt}  E_{L}  G_{0}^{E} Using E_{L}  _{0}^{E} Using E_{L}  G Exact 

1  3.544906  14.70097  20.70892  4.545456  1.8099522 
2  3.544906  12.07276  195.6083  53.04047  1.8099522 
3  3.544906  10.51319  1863.048  592.5034  1.8099522 
4  3.544906  9.452852  18043.82  6467.390  1.8099522 
5  3.544906  8.670248  176616.1  69600.95  1.8099522 
6  3.544906  8.060855  1739616.1  741610.1  1.8099522 
Table 4c. OneDimensional Point Source Array: Calculated and Actual Values of the Summation Limits for Various Values of S_{spec}, Keeping the Frequency Fixed at d = 5.5λ_{0}, Using β = 2 as a Factor^{a}S_{spec}  P_{cal}  P_{act}  M_{cal}  M_{act}  S_{act} 


1  5  5  0  0  0.21 
2  5  5  0  0  2.21 
3  6  6  0  0  4.67 
4  6  6  0  0  4.67 
5  7  7  0  0  7.23 
6  7  7  0  0  7.23 
Table 5b. TwoDimensional Point Source Array: G_{0,0}^{E} and _{0,0}^{E} for Various Values of L, Keeping the Frequency Fixed at a = 5.5 λ_{0}L  E_{opt}  E_{L}  G_{0,0}^{E} Using E_{L}  _{0,0}^{E} Using E_{L}  Exact 

1  3.544906  14.70096  20.70891  0.956796  2.6124583 
2  3.544906  12.07275  195.6083  14.43552  2.6124583 
3  3.544906  10.51319  1863.047  189.1693  2.6124583 
4  3.544906  9.452851  18043.82  2323.698  2.6124583 
5  3.544906  8.670248  176616.1  27471.14  2.6124583 
6  3.544906  8.060855  1739616.  316496.5  2.6124583 
Table 5c. TwoDimensional Point Source Array: Calculated and Actual Values of the Summation Limits for Various Values of S_{spec}, Keeping the Frequency Fixed at a = 5.5 λ_{0}, Using β = 4 as a Factor^{a}S_{spec}  P_{cal}, Q_{cal}  P_{act}, Q_{act}  M_{cal}, N_{cal}  M_{act}, N_{act}  S_{act} 


1  5, 5  5, 5  0, 0  0, 0  2.07 
2  5, 5  5, 5  0, 0  0, 0  2.07 
3  6, 6  6, 6  0, 0  0, 0  4.53 
4  6, 6  6, 6  0, 0  0, 0  4.53 
5  6, 6  7, 7  0, 0  0, 0  4.53 
6  7, 7  7, 7  0, 0  0, 0  7.09 
[23] The second tables shown for each case (part (b)) show the values of G_{0,0}^{E} and _{0,0}^{E} using E_{L} obtained for different values of L, keeping the frequency fixed fairly high such that d = a = 5.5 λ_{0}. It can be seen that the largest of the (0, 0) terms, G_{0,0}^{E}, has a magnitude on the order of 10^{L} times the magnitude of the total Green's function, as expected.
[24] If the number of significant digits desired for convergence S_{spec} is specified, the calculated summation limits for the spectral series, P_{cal} and Q_{cal}, and the calculated summation limits for the spatial series, M_{cal} and N_{cal}, can be calculated using the formulae derived previously. These values are shown in part (c) of the tables for each case, along with the actual values P_{act}, Q_{act} and M_{act}, N_{act} needed to achieve convergence to S_{spec} significant figures, obtained numerically. Also shown is S_{act}, the actual number of significant digits to which the Ewald method has converged using the formula
where G_{tot}^{E} = G^{E} + ^{E} is the value obtained from the Ewald method after summing the two series using P_{cal}, Q_{cal}, M_{cal} and N_{cal}. The frequency is again fixed such that d = a = 5.5 λ_{0}. For the spectral and the spatial series, the adjustment factor β = 1 works in all cases but is excessively conservative. A factor of β = 2 was assumed for the 1D cases, and β = 4 for the 2D cases. The agreement between the actual and specified values of S is good, especially for larger values of S, with S_{act} > S_{spec} in all cases except one (the 2D case with S_{spec} = 5).
7. Conclusions
 Top of page
 Abstract
 1. Introduction
 2. Spectral, Spatial, and Ewald Green's Function Representations
 3. Splitting Parameter (E)
 4. Choice of the Splitting Parameter
 5. Number of Terms Needed for Convergence
 6. Results
 7. Conclusions
 Appendix A
 References
 Supporting Information
[25] The Ewald method is a very efficient method for calculating the periodic freespace Green's function for three different cases: a 1D array of line sources, a 1D array of point sources, and a 2D array of point sources. However, as noted in Kustepeli and Martin [2000] the method suffers from accuracy problems at high frequency due to a loss of significant figures that occurs from a cancellation when combining the spectral and spatial series that appear in the method. The method proposed here determines the “best” value of the splitting parameter E that appears in the method to yield the fastest convergence of the Ewald sums while limiting the number of lost significant digits to a specified level L. The derivation has been presented in a unified fashion so that all three cases are treated together, with Table 1 giving the parameters needed for the different cases. The predicted loss of significant digits is verified through numerical simulations and the results illustrate the accuracy of the proposed formula.
[26] Approximate expressions for the summation limits required to achieve a specified convergence accuracy to S significant figures for the Green's function were also formulated and tested for the three different cases. In these expressions the value of S is arbitrary, except that it should satisfy the constraint that S < T − L, where T is the number of digits in the arithmetic. Again, a unified derivation has been presented, with Table 2 summarizing the parameters to be used for each of the three different cases. The specified number of significant digits S desired for convergence was compared to the actual number of significant digits of the resulting series and found to be in good agreement in almost all cases, thereby validating the formulas.