On the efficient evaluation of the azimuthal Fourier components of the Green’s function for Helmholtz’s equation in cylindrical coordinates

Abstract

In this paper, we develop an efficient algorithm to evaluate the azimuthal Fourier components of the Green’s function for the Helmholtz equation in cylindrical coordinates. A computationally efficient algorithm for this modal Green’s function is essential for solvers for electromagnetic scattering from bodies of revolution (e.g., radar cross sections, antennas). Current algorithms to evaluate this modal Green’s function become computationally intractable when the source and target are close or when the wavenumber is large or complex. Furthermore, most state-of-the-art methods cannot be easily parallelized. In this paper, we present an algorithm for evaluating the modal Green’s function that has performance independent of both source-to-target proximity and wavenumber, and whose cost grows as O(m), where m is the Fourier mode. Our algorithm’s performance is independent of whether the wavenumber is real or complex. Furthermore, our algorithm is embarrassingly parallelizable.

1. Introduction

This paper details how to efficiently compute the azimuthal Fourier components of the Green’s function (i.e., the modal Green’s function) for the Helmholtz equation in three dimensions, given by the formula

$$G_m(\mathit{x},{\mathit{x}}^{\prime })=G_m(|\mathit{x}-{\mathit{x}}^{\prime }|)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\frac{1}{4\pi }\frac{e^{-ik|\mathit{x}-{\mathit{x}}^{\prime }|}}{|\mathit{x}-{\mathit{x}}^{\prime }|}{e}^{-im\theta }d\theta ,$$ (1.1)

where x, ${\mathit{x}}^{\prime }\in {ℝ}^3$, k is the wavenumber, and m is the azimuthal Fourier mode. Rewriting this equation in cylindrical coordinates, with x = (r, θ, z) and ${\mathit{x}}^{\prime }=(r^{\prime },{\theta }^{\prime },z^{\prime })$, and letting $\varphi =\theta -{\theta }^{\prime }$, the formula for the mth Fourier coefficient becomes

$$G_m(r,z,r^{\prime },z^{\prime })=\frac{1}{8{\pi }^2{R}_0}{\int }_{-\pi }^{\pi }\frac{e^{-i\kappa \sqrt{1-\alpha \cos \varphi }}}{\sqrt{1-\alpha \cos \varphi }}{e}^{-im\varphi }d\varphi ,$$ (1.2)

where κ = kR₀, $\alpha =2r{r}^{\prime }/R_0^2$, and $R_0^2=r^2+{r^{\prime }}^2+{(z-z^{\prime })}^2$.

This integral has two features which make numeric integration difficult: the integrand is oscillatory, and it is near-singular when the distance between (r, z) and $(r^{\prime },z^{\prime })$ is small (i.e., as α is close to one). However, the integrand vanishes for sufficiently large imaginary values of κ, suggesting that Cauchy’s theorem can be used to construct a contour on which all the oscillations occur where the integrand is negligible.

When devising an appropriate contour, it is helpful to consider three cases: 1) when κ is zero and $m\ge 0,2$) when κ is arbitrary and m is small, and 3) when both κ and m are large.

Determining the appropriate contour when κ = 0 and $m\ge 0$ (when the Helmholtz equation becomes the Laplace equation) is trivial, because, on any vertical contour in the lower half-plane the integrand of (1.2) monotonically decays. When κ > 0 and m = 1, the appropriate contours were solved by Gustafsson [9] via the method of steepest descent. However, Gustafsson did not analyze cases where both κ > 0 and m > 1. It would appear that steepest descent contours for the entire integrand are indeed possible (see Figure 13), suggesting that an O(1) evaluator for an arbitrary Fourier mode is possible (see Section 6.5). We observe that the resulting contours are defined implicitly as the solution to a transcendental equation, and depend on every parameter appearing in the integrand. Consequently, the relationship between the contours and the parameters is fairly complicated, and an evaluator based on these contours is challenging to implement. However, in practice, a single mode is rarely of interest, and instead the user requires a collection of M modes, where M scales with the wavenumber and the source-to-target distance (see Section 1.2.1). As it turns out, an O(1) evaluator is not necessary to evaluate M modes in O(M) time. If a single Fourier mode can be evaluated in O(m) time, using a procedure described in Section 6.3, a collection of M modes may be computed in O(M) time. Therefore, any O(m) evaluator for G_m can be used to achieve an amortized cost of O(1).

Fig. 13: Phase-amplitude plot of the oscillatory part of the integrand and the associated steepest-descent contour.

Shown is the phase-amplitude plot of the oscillatory part of the integrand of formula (1.2) (i.e., the product of the numerator of the spherical-wave term and the Fourier exponent) for κ = 85, β₋ = 0.49, and m = 10. The red contour beginning at −π is a steepest descent contour on which all oscillations occur where the integrand is negligible.

In this paper, we describe such an O(m) evaluator. First, observe that the integral (1.2) can be written in the form

$$G_m(r,z,r^{\prime },z^{\prime })=\frac{1}{4{\pi }^2{R}_0}{\int }_0^{\pi }\frac{e^{-i\kappa \sqrt{1-\alpha \cos \varphi }}}{\sqrt{1-\alpha \cos \varphi }}\cos m\varphi d\varphi .$$ (1.3)

We develop on Gustafsson’s work by integrating along the contour on which the numerator of the spherical-wave term,

$$\frac{e^{-i\kappa \sqrt{1-\alpha \cos \varphi }}}{\sqrt{1-\alpha \cos \varphi }},$$ (1.4)

monotonically decays. The part of the integrand of (1.3) dependent on azimuthal frequency, cos(mϕ), behaves poorly and grows on this contour. To circumvent this behavior, we replace the term cos(mϕ) with a rational function approximation which does not grow in the complex plane. The growth of cos(mϕ) along the contour is subsumed in a collection of residues which must be added to the resulting integral. The resulting algorithm’s cost scales as O(m) and is completely independent of the source-to-target distance and wavenumber.

1.1. Motivation for a Fast Evaluator of the Modal Green’s Function

A boundary integral equation defined on a curve in ${ℝ}^2$ is easier to compute than one defined on a surface in ${ℝ}^3$. For the case of a body of revolution and a Green’s function which is rotationally invariant about the axis of symmetry, by use of the decomposition described below, we can convert a problem in ${ℝ}^3$ to a series of decoupled problems in ${ℝ}^2$, each of which utilize the azimuthal Fourier expansion of the Green’s function evaluated at points along the boundary generating curve. Let Γ be the body of revolution generated by rotating the boundary-generating curve γ(t) about the z-axis. Consider the second-kind integral equation

$$\sigma (\mathit{x})+{\int }_{\text{Γ}}G(\mathit{x},{\mathit{x}}^{\prime })\sigma (\mathit{x})da({\mathit{x}}^{\prime })=f\left(\mathit{x}\right),$$ (1.5)

where $\mathit{x}\in {ℝ}^3$, G is a rotationally invariant Green’s function, f is the given function, and σ is the solution. We first express x in cylindrical coordinates as (r, θ, z), then expand the solution and the right-hand side in terms of their respective Fourier series in the azimuthal direction, given by

$$\sigma \left(r,z\right)=\sum _{-\infty }^{\infty }{\sigma }_m\left(r,z\right)e^{im\theta },f\left(r,z\right)=\sum _{-\infty }^{\infty }{f}_m\left(r,z\right)e^{im\theta },$$ (1.6)

where the Fourier coefficients σ_m and f_m are given by

$${\sigma }_m(r,z)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\sigma (r,z,\theta )e^{-im\theta }d\theta ,f_m(r,z)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(r,z,\theta )e^{-im\theta }d\theta .$$ (1.7)

Substituting the Fourier expansions for σ and f into (1.5) and collecting terms mode-by-mode, we arrive at the decoupled integral equations

$${\sigma }_m(r,z)+{\int }_{\gamma }{\sigma }_m(r,z){\int }_{-\pi }^{\pi }G(r,z,r^{\prime },z^{\prime },\theta -{\theta }^{\prime })e^{-im{\theta }^{\prime }}{r}^{\prime }d{\theta }^{\prime }d{r}^{\prime }d{z}^{\prime }=f_m(r,z),$$ (1.8)

which simplifies to

$${\sigma }_m(r,z)+2\pi {\int }_{\gamma }{G}_m(r,z,r^{\prime },z^{\prime }){\sigma }_m(r,r^{\prime })r^{\prime }d{r}^{\prime }d{z}^{\prime }=f_m(r,z),$$ (1.9)

where for all m, G_m are the Fourier modes of G, given by

$$G_m(r,r^{\prime },z,z^{\prime })=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }G(r,r^{\prime },z,z^{\prime },\varphi )e^{-im\varphi }d\varphi .$$ (1.10)

Observe that solving (1.9) requires numerous evaluations of $G_m(r,z,r^{\prime },z^{\prime })$ for points along γ. Therefore, converting this problem from ${ℝ}^3$ to the m decoupled problems in ${ℝ}^2$ is only computationally efficient if the evaluation of $G_m(r,z,r^{\prime },z^{\prime })$ is fast.

1.2. The Modal Green’s Function for the Helmholtz Equation

The Green’s function for the Helmholtz equation in three dimensions satisfies

$$({\nabla }^2+k^2)G_k(\mathit{x},{\mathit{x}}^{\prime })=\delta (\mathit{x}-{\mathit{x}}^{\prime }),$$ (1.11)

where k is the wave number and x, ${\mathit{x}}^{\prime }\in {ℝ}^3$. There are three basic solutions,

$$G_{0,k}(\mathit{x},{\mathit{x}}^{\prime })=\frac{1}{4\pi }\frac{\cos k|\mathit{x}-{\mathit{x}}^{\prime }|}{|\mathit{x}-{\mathit{x}}^{\prime }|},G_{+,k}(\mathit{x},{\mathit{x}}^{\prime })=\frac{1}{4\pi }\frac{e^{ik|\mathit{x}-{\mathit{x}}^{\prime }|}}{|\mathit{x}-{\mathit{x}}^{\prime }|},G_{-,k}(\mathit{x},{\mathit{x}}^{\prime })=\frac{1}{4\pi }\frac{e^{-ik|\mathit{x}-{\mathit{x}}^{\prime }|}}{|\mathit{x}-{\mathit{x}}^{\prime }|}.$$ (1.12)

The solution to the Helmholtz equation, when viewed as the solution to the time-harmonic wave equation, has two time-harmonic conventions, $e^{-i\omega t}$ and $e^{+i\omega t}$. With the $e^{-i\omega t}$ time-harmonic convention, these Green’s functions correspond to the stationary, outgoing, and incoming spherical waves, respectively. For a given application, the choice of the Green’s function is driven by convenience, such that it has the desired asymptomatic behavior away from a volume of interest given the practitioner’s time-harmonic convention. When the function grows with distance from the origin, it is referred to as the advanced Green’s function, and when the function decays with distance from the origin, it is referred to as the retarded Green’s function. In this work, we examine the retarded Green’s function with a negative time-harmonic convention. This function corresponds to the form G_−,k, which requires adopting the convention that Re $k\ge 0$; for attenuating media this convention also requires that Im $k\le 0$. In other words, we require k to be in quadrant IV of the complex plane. To use our algorithm to evaluate the retarded Green’s function with the positive time convention, observe that

$$G_{+,k}=\overline{\overline{G_{+,k}}}=\overline{\frac{1}{4\pi }\frac{\overline{e^{ik|\mathit{x}-{\mathit{x}}^{\prime }|}}}{|\mathit{x}-{\mathit{x}}^{\prime }|}}=\overline{\frac{1}{4\pi }\frac{e^{-\overline{ik}|\mathit{x}-{\mathit{x}}^{\prime }|}}{|\mathit{x}-{\mathit{x}}^{\prime }|}}=\overline{G_{-,\overline{k}}}.$$ (1.13)

We consider a problem with rotational symmetry (i.e., a body of revolution). Switching to cylindrical coordinates and expanding G_−,k in its Fourier series, we have

$$G_{-,k}(r,z,r^{\prime },z^{\prime })=\sum _{m=-\infty }^{\infty }{G}_{-,m,k}(r,z,r^{\prime },z^{\prime })e^{-im(\theta -{\theta }^{\prime })},$$ (1.14)

where x = (r, θ, z), ${\mathit{x}}^{\prime }=(r^{\prime },{\theta }^{\prime },z^{\prime })$. Let $\varphi =\theta -{\theta }^{\prime }$ denote the difference in azimuthal angles. The formula for the mth coefficient is given by (1.10). We adopt notation consistent with the literature (see, for example, [5, 7, 21]) and omit the subscripts specifying the choice of the Green’s function and wavenumber (i.e., we omit the subscripts – and k). Expanding the representation for the mth Fourier coefficient, we have

$$G_m(r,z,r^{\prime },z^{\prime })=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\frac{e^{-ik}\sqrt{r^2+{r^{\prime }}^2-2r{r}^{\prime }\cos \varphi +{(z-z^{\prime })}^2}}{4\pi \sqrt{r^2+{r^{\prime }}^2-2r{r}^{\prime }\cos \varphi +{(z-z^{\prime })}^2}}{e}^{-im\varphi }d\varphi .$$ (1.15)

We then use the parameter R₀ (introduced in Section 1) to rewrite (1.15), by defining $\kappa =k{R}_0$ and $\alpha =2r{r}^{\prime }/R_0^2$, and apply the formula for the Fourier coefficient of an even function to obtain

$$G_m(r,z,r^{\prime },z^{\prime })=\frac{1}{4{\pi }^2{R}_0}{\int }_0^{\pi }\frac{e^{-i\kappa }\sqrt{1-\alpha \cos \varphi }}{\sqrt{1-\alpha \cos \varphi }}\cos m\varphi d\varphi .$$ (1.16)

When solving problems involving boundary integrals for vector-valued σ, there are two additional necessary modal Green’s functions which must be evaluated,

$$G_{c,m}=\frac{1}{4{\pi }^2{R}_0}{\int }_0^{\pi }\frac{e^{-i\kappa }\sqrt{1-\alpha \cos \varphi }}{\sqrt{1-\alpha \cos \varphi }}\cos m\varphi \cos \varphi d\varphi ,G_{s,m}=\frac{1}{4{\pi }^2{R}_0}{\int }_0^{\pi }\frac{e^{-i\kappa }\sqrt{1-\alpha \cos \varphi }}{\sqrt{1-\alpha \cos \varphi }}\sin m\varphi \sin \varphi d\varphi .$$ (1.17)

We refer the reader to [12] for a derivation of the Fourier decomposition associated with problems with vector-valued σ and the resulting decoupled equations in terms of G_m, G_c,m, G_s,m. This work presents an algorithm for the evaluation of G_m; a straight-forward substitution of the integrand results in an algorithm for G_c,m. Likewise, substitution of the integrand results in an algorithm for G_s,m, with a minor difference related to removing a singularity (see Remark 4.1).

For notational convenience, we introduce a scaled modal Green’s function,

$$G_m^s=4{\pi }^2{R}_0{G}_m={\int }_0^{\pi }\frac{e^{-i\kappa \sqrt{1-\alpha \cos \varphi }}}{\sqrt{1-\alpha \cos \varphi }}\cos m\varphi d\varphi ,$$ (1.18)

where G_m is understood to be a function of r, $r^{\prime }$, z, and $z^{\prime }$. In a slight abuse of notation, we will denote $G_m^s$ by G_m for the remainder of the paper. Any numerical scheme for evaluating G_m must depend on four parameters: κ, α, R₀, and m. Notably, α is bounded by 0 ≤ α < 1, and determines the growth of the integrand near ϕ = 0. In Section 3.1.1, we will follow the notation of Gustafsson [9] and introduce the parameters β₋ and β₊, defined to be

$${\beta }_{-}=\sqrt{1/\alpha -1},{\beta }_{+}=\sqrt{1/\alpha +1}.$$ (1.19)

We also introduce the parameters Δ and ρ₀, defined as

$$\text{Δ}=\sqrt{{(r-r^{\prime })}^2+{(z-z^{\prime })}^2},$$ (1.20)

$${\rho }_0=2r{r}^{\prime }.$$ (1.21)

Note that Δ is the minimum distance between the source and the target, R₀ is the maximum distance between the source and the target, and that ${\text{Δ}}^2=R_0^2-{\rho }_0$. Lastly, we observe that β₋ and β₊ are also given by the formulae

$${\beta }_{-}=\frac{\text{Δ}}{{\rho }_0},{\beta }_{+}=\sqrt{\frac{R_0^2+{\rho }_0}{{\rho }_0}}.$$ (1.22)

We note that numerically computing β₋ from α using (1.19) will result in cancellation error when α ≈ 1, so it is better to compute β₋ directly from formula (1.22), and α from β₋ using (1.19).

A representative sample of the literature related to the evaluation of the modal Green’s function can be found in [2, 6, 7, 8, 9, 10, 12, 15, 19].

1.2.1. Number of Fourier Coefficients Needed

Matviyenko in [15] derived an upper bound, r₊, such that all Fourier modes m > r₊ geometrically decay as m increases, with r₊ given by

$$r_{+}=\frac{\kappa }{\sqrt{2}}\sqrt{1+\sqrt{1-{\alpha }^2}},$$ (1.23)

where $\alpha ={\rho }_0/R_0^2$ and κ = kR₀ (see [15], formulae (37) and (38)). When $\alpha \approx 1$, formula (1.23) simplifies to

$$r_{+}\approx \frac{\kappa }{\sqrt{2}}.$$ (1.24)

Using Matviyenko’s formula for the decay of the modal Green’s functions (see [15], formula (40)), it can be shown that the magnitude of any Fourier coefficient m > r₊ is bounded by

$$|G_m|<|G_{\lfloor r_{+}\rfloor }|{\left(\frac{\sqrt{1-\sqrt{1-{\alpha }^2}}}{\sqrt{1+\sqrt{1-{\alpha }^2}}}\right)}^{m-\lfloor r_{+}\rfloor }.$$ (1.25)

Substituting $\alpha =1/({\beta }_{-}^2+1)$ into (1.25), this bound can be simplified to

$$|G_m|<|G_{\lfloor r_{+}\rfloor }|{\left(1-\frac{{\beta }_{-}\sqrt{{\beta }_{-}^2+2}}{1+{\beta }_{-}^2}\right)}^{\frac{m-\lfloor r_{+}\rfloor }{2}}{\left(1+\frac{{\beta }_{-}\sqrt{{\beta }_{-}^2+2}}{1+{\beta }_{-}^2}\right)}^{-\frac{m-\lfloor r_{+}\rfloor }{2}},$$ (1.26)

where β₋ is the scaled source-to-target distance. When β₋ is small, $1+{\beta }_{-}^2\approx 1$, and (1.26) can be approximated as

$$|G_m|\lesssim |G_{\lfloor r_{+}\rfloor }|{\left(\frac{1-{\beta }_{-}\sqrt{2}}{1+{\beta }_{-}\sqrt{2}}\right)}^{\frac{m-\lfloor r_{+}\rfloor }{2}}\approx |G_{\lfloor r_{+}\rfloor }|{\left(1-{\beta }_{-}2\sqrt{2}\right)}^{\frac{m-\lfloor r_{+}\rfloor }{2}},$$ (1.27)

where we have replaced the exponentiated term with its truncated Taylor expansion in β₋. Formula (1.27) can be used to determine the Fourier mode M such that, for m > M, $|G_m|<ϵ$, where M is given by

$$M\approx \frac{-2\log (ϵ)+2\log (|G_{\lfloor r_{+}\rfloor }|)}{\log (1-2\sqrt{2}{\beta }_{-})}+\lfloor r_{+}\rfloor .$$ (1.28)

By substituting (1.23) into (1.28), we can characterize M as a function of β₋ and κ when the source and target are close (i.e., $\alpha \gtrsim 0.99$ or, equivalently, ${\beta }_{-}\lesssim {10}^{-2}$) as

$$\begin{array}{l}M\approx \frac{-2\log (ϵ)+2\log (|G_{\lfloor r_{+}\rfloor }|)}{\log (1-2\sqrt{2}{\beta }_{-})}+\frac{\kappa }{\sqrt{2}}\\ =\left(1-\frac{1}{\sqrt{2}{\beta }_{-}}\right)(-\log (ϵ)+\log (|G_{\lfloor r_{+}\rfloor }|)+\frac{\kappa }{\sqrt{2}}+O({\beta }_{-}^2)\\ =O\left(\frac{1}{{\beta }_{-}}+\kappa \right),\end{array}$$ (1.29)

where we have replaced the denominator of (1.28) with its Taylor expansion in β₋.

Remark 1.1.

For complex wavenumber, the rate of decay of the Fourier coefficients also decreases as $|\mathrm{Im}\kappa |$ grows.

Consider the modal Green’s function expressed as the product

$$\frac{e^{-i\kappa \sqrt{1-\alpha \cos \varphi }}}{\sqrt{1-\alpha \cos \varphi }}=\frac{e^{-i\mathrm{Re}\kappa \sqrt{1-\alpha \cos \varphi }}}{\sqrt{1-\alpha \cos \varphi }}{e}^{\text{Im}\kappa \sqrt{1-\alpha \cos \varphi }}.$$ (1.30)

The decay of the Fourier coefficients of the left-hand term in the product is characterized in the preceding section. When Im κ < 0, the right-hand term monotonically increases with ϕ on [0, π], resembling a scaled dirac function centered at π (i.e., δ(ϕ − π) when Im $\kappa \ll 0$). Therefore, by the convolution theorem, the total number of required Fourier components increases with $|\mathrm{Im}\left(\kappa \right)|$.

1.3. Review of the Literature

Recall from Section 1.2 that the modal Green’s function is a function of three parameters: κ, m, and α. We divide the literature on fast algorithms for evaluating the modal Green’s function into two categories: those that evaluate the general case for any combination of input parameters, and those that evaluate special cases of input parameters (e.g., when the source and target are well-separated, when m = 1, etc.).

Almost all modern fast general-case algorithms are based on the application of the fast Fourier transform (FFT) (see, for example, [7, 8, 10, 11, 12, 19, 20, 21]). In contrast, the special-case algorithms have a diverse set of methodologies which cannot easily be summarized. Because this paper’s topic is a general-case algorithm which works for all input parameters, we do not review the literature of special-case algorithms, with the exception of Gustfasson’s contour integration technique [9], which we develop on extensively in this paper.

Because the FFT is inefficient for non-smooth functions, and the modal Green’s function is not very smooth for α ≈ 1, modern FFT-based algorithms utilize kernel-splitting, a technique in which the integral is split into a smooth part and a non-smooth part. The smooth part’s Fourier coefficients are evaluated with the FFT, and the non-smooth part is handled separately, often with a purpose-made recurrence. Two splittings are used in the literature, the splitting of Helsing [11] and the splitting of Gedney [8]. It can be shown that the fastest algorithm using the splitting of Gedney (presented in [19]) actually also used the splitting of Helsing (i.e., the algorithm uses both splittings), and is computationally equivalent to the fastest implementation of Helsing’s splitting (presented in [7]). Therefore, we only present the method of Epstein et al., which utilizes Helsing’s splitting.

1.3.1. Method of Epstein et al.

In Epstein et al. [7], the modal Green’s functions are computed using a fast-Fourier-transform-based method, with the kernel splitting of Helsing [11]. In the following, the definitions for m, κ, α, and R₀ are identical to those used in Section 1.2.

The authors divide the evaluation of the modal Green’s functions for the Fourier modes −M, −M + 1, … , M − 1, M into two cases: one where the source and target are well-separated (0 ≤ α < 1/1.005), and one where the source and target are close $\left(1/1.005\le \alpha <1\right)$.

In the former case, the integrand is relatively smooth, and the modal Green’s functions are computed using an L-point FFT, obtaining near double precision accuracy when $L\ge 4\left|\kappa \right|$.

For the near-singular case, $\alpha \approx 1$, the authors follow [11] by first rewriting (1.16) as

$$G_m(\mathit{x},{\mathit{x}}^{\prime })=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\frac{\cos (\kappa \sqrt{1-\alpha \cos \varphi })+i\sin (\kappa \sqrt{1-\alpha \cos \varphi })}{4\pi R_0\sqrt{1-\alpha \cos \varphi }}{e}^{-im\varphi }d\varphi$$ (1.31)

(see [11], Section 3, formula (9)). The integrand of (1.31) is split into a smooth sine term, H^s, and a near-singular cosine term, H^c, where H^s and H^c are given by

$$H^s(\varphi ;\kappa ,\alpha )=\frac{\sin (\kappa \sqrt{1-\alpha \cos \varphi })}{\sqrt{1-\alpha \cos \varphi }},H^c(\varphi ;\kappa ,\alpha )=\frac{\cos (\kappa \sqrt{1-\alpha \cos \varphi })}{\sqrt{1-\alpha \cos \varphi }}.$$ (1.32)

The Fourier modes of H^c are computed as the linear convolution of the Fourier modes of $\cos (\kappa \sqrt{1-\alpha \cos \varphi })$ and the Fourier modes of $1/\sqrt{1-\alpha \cos \varphi }$. The Fourier modes of $\cos (\kappa \sqrt{1-\alpha \cos \varphi })$ are computed via the FFT, while the Fourier modes of $1/\sqrt{1-\alpha \cos \varphi }$ are known to be proportional to $Q_{m-1/2}\left(\chi \right)$ (see [5]), where $Q_{m-1/2}$ is the Legendre function of the second kind of half-integer order, with χ given by

$$\chi =\frac{r^2+{r^{\prime }}^2+{(z-z^{\prime })}^2}{2r{r}^{\prime }}=\frac{1}{\alpha }.$$ (1.33)

Note that $\chi \approx 1$ when $\alpha \approx 1$ (i.e., when the minimum distance between the source and target is very small). The authors complete their algorithm by computing $Q_{m-1/2}\left(\chi \right)$ via a recurrence, which has a cost that grows as $O\left(1/{\beta }_{-}\right)$, where ${\beta }_{-}=\sqrt{1/\alpha -1}$. Thus, their recurrence has poor performance for $\chi \approx 1$ (i.e., when the target and source are close). We note that a fast algorithm was recently introduced by Bremer in [3], which evaluates $Q_{m-1/2}\left(\chi \right)$ in constant run-time independent of m. Bremer’s algorithm for evaluating the Legendre function of the second kind of half-integer order [3] is useful, not only as an improvement to [7], but as an ingredient in a potential O(1) evaluator for an arbitrary mode of the Green’s function for the Laplace equation (see also Section 6.2 for an alternative algorithm). The total computational cost of Epstein et al.’s algorithm (and of kernel splitting techniques) is summarized as follows. Recall that $R=\sqrt{1-\alpha \cos \varphi }$. After performing the splitting of Helsing, the sin(R)/R term is evaluated in O(L log L) time with the FFT, where L is the maximum of 4κ and M. The cos(R)/R term is evaluated as the convolution of the Fourier coefficients of 1/R (the Laplace term) and the Fourier coefficients of cos(R). The Fourier coefficients of cos(R) are evaluated in O(L log L) time, and the coefficients of the 1/R term are evaluated in $O\left(1/{\beta }_{-}\right)$ time, where ${\beta }_{-}=\sqrt{1/\alpha -1}$. Lastly, the convolution of the coefficients of cos(R) and the coefficients of 1/R is evaluated in O(κM) time. Finally, we summarize Epstein et al.’s algorithm for the modal Green’s function and its cost as

$$\underset{O\left(L\log L\right)}{\underbrace{ℱ\left(\cos (\kappa \sqrt{1-\alpha \cos \varphi })\right)}}\underset{O\left(\kappa M\right)}{\underbrace{★}}\underset{O\left(1/{\beta }_{-}\right)}{\underbrace{ℱ\left(\frac{1}{\sqrt{1-\alpha \cos \varphi }}\right)}}+\underset{O\left(L\log L\right)}{\underbrace{ℱ\left(\frac{\sin (\kappa \sqrt{1-\alpha \cos \varphi })}{\sqrt{1-\alpha \cos \varphi }}\right),}}$$ (1.34)

where ★ is the discrete convolution operator, $ℱ$ is the discrete Fourier transform (with its cost denoted by its implementation via the FFT), L = max(4κ, M), and β₋ is the scaled minimum source-to-target distance given by ${\beta }_{-}=\sqrt{1/\alpha -1}$. Hence, the cost of Epstein et al.’s algorithm for the modal Green’s function is

$$O(L\log L)+O(\kappa M)+O\left(1/{\beta }_{-}\right),$$ (1.35)

where L = max(4κ, M) and $M=O\left(1/{\beta }_{-}+\kappa \right)$. Epstein et al.’s algorithm can be improved by the application of an O(1) evaluator for the modal Green’s function for the Laplace equation, resulting in a cost of

$$O(L\log L)+O(\kappa M).$$ (1.36)

Remark 1.2.

For complex κ, both the cos(κ) term and the i sin(κ) grow exponentially with $\left|\mathrm{Im}\kappa \right|$. However, their sum is bounded by the value of the integrand at ϕ = 0, which decays exponentially with Im κ < 0. Because the sum of two exponentially growing terms is bounded by an exponentially decaying term, for complex κ with Im $\kappa \not\approx 0$, kernel splitting techniques incur catastrophic cancellation error.

2. Preliminaries

In this section, we review formulae necessary to evaluate the modal Green’s function via contour integration.

2.1. Chebyshev Polynomials

The Chebyshev polynomials are a collection of polynomials on the unit interval [−1, 1], denoted by T_n(x), which are orthogonal with respect to the weight function $1/\sqrt{1-x^2}$. The nth Chebyshev polynomial is given by the formula

$$T_n(x)=\cos (n\arccos (x))$$ (2.1)

(see [1]).

2.2. The Joukowski Transformation

The Joukowski transformation

$$J(v)=\frac{1}{2}(v+\frac{1}{v}),$$ (2.2)

is both a bijection from the deleted disc $D\backslash \left\{0\right\}$, where $D=\{v\in ℂ:\left|v\right|<1\}$, to the region $ℂ\backslash [-1,1]$, and a bijection from $ℂ\backslash \overline{D}$ to $ℂ\backslash [-1,1]$, with the point at v = 0 mapped to the point at ∞ and the unit circle mapped to the interval [−1, 1]. The inverse transformation from $ℂ\backslash [-1,1]\to D\backslash \left\{0\right\}$ is given by the formula

$$J_1^{-1}(z)=z-\sqrt{z+1}\sqrt{z-1},$$ (2.3)

and the inverse transformation from $ℂ\backslash [-1,1]\to ℂ\backslash \overline{D}$ is given by the formula

$$J_2^{-1}(z)=z+\sqrt{z+1}\sqrt{z-1},$$ (2.4)

for all $z\in ℂ\backslash [-1,1]$, where the $\sqrt{\cdot }$ functions are all taken with respect to the principal branch.

The forward mapping is analytic on $ℂ\backslash \left\{0\right\}$, while the inverse mappings are analytic on $ℂ\backslash \left[-1,1\right]$, with branch cuts along [−1, 1] and square root singularities at z = ±1.

2.3. The Chebyshev Polynomials Evaluated on the Bernstein Ellipse

Recall that the mth order Chebyshev polynomial with complex argument, T_m(z), is given by

$$T_m(z)=\cos m\theta ,$$ (2.5)

where θ = arccos(z). An equivalent form of (2.5), often used for applications in the complex plane (see, for example, [18]), is given by

$$T_m(w)=\frac{z^m+z^{-m}}{2},$$ (2.6)

where $w=\frac{1}{2}(z+z^{-1})$ and $z=\exp \left(i\theta \right)$. This form can be conveniently rewritten in terms of the Joukouwski transformation (2.2), so that (2.6) becomes

$$T_m(J(z))=\frac{z^m+z^{-m}}{2},$$ (2.7)

for all $z\in ℂ$.

Let C_ρ denote a circle of radius ρ. The Joukowski transformations of the circles C_ρ with ρ ≠ 1 have special significance in approximation theory, and are named the Bernstein ellipses, denoted E_ρ, given by

$$\begin{array}{l}{E}_{\rho }(\theta )=J(C_{\rho }(\theta ))=J(\rho e^{i\theta })=\frac{1}{2}(\rho e^{i\theta }+{\rho }^{-1}{e}^{i\theta })\\ =\frac{1}{2}(\rho \cos \theta +i\rho \sin \theta +{\rho }^{-1}\cos \theta -i{\rho }^{-1}\sin \theta ),\end{array}$$ (2.8)

where we have used the standard parametrization of the circle, $C_{\rho }\left(\theta \right)=\rho e^{i\theta }$. Note that both C_ρ and C_1/ρ under the Joukowski transformation yield the same Bernstein ellipse, that is, $E_{\rho }=E_{1/\rho }$. We adopt the convention in the literature (see, for example, [14, 18]) of parameterizing the Bernstein ellipses by ρ > 1. Formula (2.8) can be simplified into the familiar form of an ellipse,

$$E_{\rho }\left(\theta \right)=a\cos \theta +ib\sin \theta ,$$ (2.9)

where

$$a=\frac{1}{2}(\rho +{\rho }^{-1}),b=\frac{1}{2}(\rho -{\rho }^{-1}).$$ (2.10)

Because the Bernstein ellipses are the Joukowski transformations of circles, formula (2.7) yields a formula for the composition of a Chebyshev polynomial and the parameterization of the Bernstein ellipse, given by

$$T_m(E_{\rho }(\theta ))=T_m(J(C_{\rho }(\theta )))=\frac{{\rho }^m{e}^{im\theta }+{\rho }^{-m}{e}^{-im\theta }}{2}.$$ (2.11)

Formula (2.11) leads to a useful inequality,

$$\frac{1}{2}({\rho }^m-{\rho }^{-m})\le |T_m(E_{\rho }(\theta ))|\le \frac{1}{2}({\rho }^m+{\rho }^{-m}),$$ (2.12)

for ρ > 1.

2.4. The Decay of Chebyshev Expansion Coefficients of Analytic functions

The following theorem states that, if a function f(z) can be analytically continued to the Bernstein ellipse E_ρ, then the decay of the coefficients of its Chebyshev expansion can be nicely bounded. A discussion of this theorem can be found in, for example, Chapter 8 of [17], and a proof can be found in, for example, Chapter 5, §5 of [13].

Theorem 2.1.

Suppose that f(z) is an analytic function on a neighborhood of the interior of the Bernstein ellipse E_ρ, where it satisfies $\left|f\left(z\right)\right|\le L$ for all $z\in E_{\rho }^o$, for some constant L > 0. Suppose further that

$$f(z)=\sum _{k=0}^{\infty }{a}_k{T}_k(z),$$ (2.13)

for all z ∈ [−1, 1], where T_k(z) is the Chebyshev polynomial of order k. Then its Chebyshev expansion coefficients a_k satisfy

$$|a_k|\le 2L{\rho }^{-k},$$ (2.14)

for all $k\ge 1$.

2.5. The Number of Terms in the Chebyshev Expansions of Analytic Functions

The following corollary of Theorem 2.1 bounds the number of Chebyshev polynomials required to represent a function f(z), that is analytic and bounded in absolute value by L on the interior of E_ρ, with ρ = M^1/m, in terms of M, L, and m.

Corollary 2.2.

Suppose that M > 1, and let $\rho =M^{1/m}$, for some integer m > 1. Suppose further that f(z) is an analytic function on the interior of the Bernstein ellipse E_ρ, where it satisfies $|f(z)|\le L$ for all $z\in E_{\rho }^o$ for some constant L > 0. Suppose further that

$$f(z)=\sum _{k=0}^{\infty }{a}_k{T}_k(z),$$ (2.15)

for all z ∈ [−1, 1], where T_k(z) is the Chebyshev polynomial of order k. Finally, let $0<ϵ\ll 1$ be some small real number. Then, if

$$k_0=m(\log (2L)-\log (ϵ))/\log (M),$$ (2.16)

then $|a_k|\le ϵ$ for all positive $k\ge k_0$.

Proof.

The proof follows in a straightforward way from Theorem 2.1. ■

2.6. Recurrence for a Certain Integral Involving a Monomial Divided by $\sqrt{a{\tau }^2+b}$

In Gustafsson (see [9], equations (25) and (26)), a recurrence relation is given for the integral of an nth degree monomial divided by the square root of a pure quadratic,

$$\int \frac{{\tau }^n}{\sqrt{a{\tau }^2+b}}d\tau =\frac{{\tau }^{n-1}\sqrt{a{\tau }^2+b}}{na}-\frac{(n-1)b}{na}\int \frac{{\tau }^{n-2}}{\sqrt{a{\tau }^2+b}}d\tau ,$$ (2.17)

for $n\ge 2$, with the base cases given by the formulae

$$\int \frac{1}{\sqrt{a{\tau }^2+b}}d\tau =\frac{1}{\sqrt{a}}\ln \left(\tau \sqrt{a}+\sqrt{a{\tau }^2+b}\right)$$ (2.18)

and

$$\int \frac{\tau }{\sqrt{a{\tau }^2+b}}d\tau =\frac{\sqrt{a{\tau }^2+b}-\sqrt{b}}{a}.$$ (2.19)

This recurrence can be evaluated stably when $|b|<|a|$.

2.7. Recurrence for a Certain Integral Involving a Monomial Times $\sqrt{a{\tau }^2+b}$

The recurrence relation for the integral of a monomial multiplied by the square root of a pure quadratic is given by the formula

$$\int {\tau }^n\sqrt{a{\tau }^2+b}d\tau =\frac{{\tau }^{n-1}{(a{\tau }^2+b)}^{\frac{3}{2}}}{(n+2)a}-\frac{(n-1)b}{(n+2)a}\int {\tau }^{n-2}\sqrt{a{\tau }^2+b}d\tau ,$$ (2.20)

with the base cases given by the formulae

$$\int \sqrt{a{\tau }^2+b}d\tau =\frac{\tau \sqrt{a{\tau }^2+b}}{2}-\frac{b}{2\sqrt{a}}\log (\sqrt{a{\tau }^2+b}-\tau \sqrt{a})$$ (2.21)

and

$$\int \tau \sqrt{a{\tau }^2+b}d\tau =\frac{{(a{\tau }^2+b)}^{\frac{3}{2}}}{3a}.$$ (2.22)

This recurrence can be evaluated stably when $|b|<|a|$.

2.8. The Mapping Between a Legendre Expansion and a Taylor Series

The following standard formula relates the Legendre polynomials and their derivatives (see, for example, [1]).

Theorem 2.3.

Suppose that $n\ge 1$ is an integer. Then

$$(2n+1)P_n(x)=P_{n+1}^{\prime }(x)-P_{n-1}^{\prime }(x).$$ (2.23)

This formula can be used to spectrally differentiate a Legendre expansion, as follows. Suppose that

$$p(x)=\sum _{i=0}^n{c}_i{P}_i(x),$$ (2.24)

and that

$$p^{\prime }(x)=\sum _{i=0}^{n-1}{c^{\prime }}_i{P}_i(x).$$ (2.25)

The coefficients $c_i^{\prime }$ can be computed from c_i by iterating from from k = n, n − 1, … , 2 and, at each iteration, assigning $c_{k-1}^{\prime }$ the value $(2k-1)c_k$, and assigning c_k−2 the value $c_{k-2}+c_k$.

To compute the Taylor series of an n-term Legendre expansion at the point x₀, it is sufficient to compute the expansion coefficients of its first n derivatives by spectral differentiation, and then to evaluate each successive derivative at x₀. Since each derivative will require O(n) operations to compute, the Taylor series can be computed in O(n²) operations.

2.9. Contour Integral of a Monomial Divided by a First Degree Polynomial

For any $k\ge 0$, note the elementary indefinite integral

$$\int \frac{z^k}{z-x}dz=\sum _{i=0}^{k-1}\frac{z^{k-i}{x}^i}{k-i}+x^k\log (z-x),$$ (2.26)

for all $x\in ℂ$.

3. Analytical Apparatus

In this section, we review the contour integration method of Gustafsson [9] to evaluate the modal Green’s function. We demonstrate that Gustafsson’s variable substitution requires special treatment of the resulting Chebyshev polynomial, and then propose a technique to replace the Chebyshev polynomial with an approximation, based on applying a quadrature rule to Cauchy’s integral formula. Lastly, we present a geometric argument which demonstrates that the domain of integration necessary for our approximation is well-separated from Gustafsson’s contours for all m.

3.1. Steepest Descent Contour

Recall that the modal Green’s function (1.18) is the mth Fourier coefficient of the spherical wave

$$H^w(\cos \varphi )=\frac{e^{-i\kappa \sqrt{1-\alpha \cos \varphi }}}{\sqrt{1-\alpha \cos \varphi }}.$$ (3.1)

Rewriting (1.18) using (3.1), we have

$$G_m={\int }_0^{\pi }{H}^w(\cos \varphi )\cos (m\varphi )d\varphi ,$$ (3.2)

where we have omitted rewriting the variables r, $r^{\prime }$, z, $z^{\prime }$. In formula (3.2), G_m is understood to be a function of four parameters: R₀, α, κ, and m. Lastly, we denote the integrand of (3.2) by H_m, where

$$H_m(\cos \varphi )=H^w(\cos \varphi )\cos (m\varphi ).$$ (3.3)

This leads to an abbreviated form of G_m, given by

$$G_m={\int }_0^{\pi }{H}_m(\cos \varphi )d\varphi .$$ (3.4)

When κ or m are large, $H_m(\cos \varphi )$ is highly oscillatory along the real axis. However, $H^w(\cos \varphi )$ decays to zero in quadrant IV of the complex plane for complex arguments with sufficiently large negative imaginary components, provided that $0<\mathrm{Re}(\varphi )<\pi$. This suggests that contour integration may be used to avoid evaluating the oscillatory segment along the real axis. The integrand is analytic on a neighborhood of [0, π], so Cauchy’s integral theorem can be used to deform the integration contour to complex-valued ϕ.

Applying Cauchy’s integral theorem, we have

$${\oint }_{\text{Γ}}{H}_m(\cos \varphi )d\varphi =0,$$ (3.5)

where Γ is some closed contour passing along the interval [0, π] on the real axis, and extending into quadrant IV in the complex plane. We rearrange (3.5) into an expression for G_m, given by

$$G_m=-{\int }_{\text{Γ}\backslash \left[0,\pi \right]}{H}_m(\cos \varphi )d\varphi ,$$ (3.6)

where the contour is traversed in the counterclockwise direction. Determining an appropriate contour Γ \ [0, π] is the subject of the subsequent section. Ideally, one would construct a contour on which $H_m(\cos \varphi )$ undergoes a finite number of oscillations where $H_m(\cos \varphi )$ is not negligible, such that the number of oscillations is independent of both κ and m. Although it is difficult to construct a contour on which $H_m(\cos \varphi )$ (given by formula (3.3)) has a finite number of such oscillations (see Figure 13 and Section 6.5), it is straightforward to construct a contour on which the numerator of the spherical-wave term (i.e., the numerator of $H^w(\cos \varphi )$ does not oscillate, regardless of κ, α, or R₀.

3.1.1. Gustafsson’s Contours

Gustafsson [9] proposed using contour integration to evaluate the modal Green’s function by constructing a contour on which the numerator of the spherical-wave term (see formula (3.1)) is non-oscillatory. In this section, we present an alternative construction of Gustafsson’s contours which, unlike Gustafsson’s construction, permits complex-valued κ. Recall that our goal is to construct a contour Γ \ [0, π] which begins at the point ϕ = 0, travels down into the complex plane sufficiently far, traverses right, and then travels up to the point ϕ = π. An adequate contour has the property that the numerator of the spherical-wave term decays (or grows) monotonically on the first and last segments, which we denote γ₁ and γ₂, respectively. The contour which connects γ₁ and γ₂, corresponds to an integral which, by design, is negligible. We denote this segment by γ_c for “connecting.” Because it is noncontributory, we do not derive an expression for it. We split the integral in (3.5) into

$${\int }_{\left[0,\pi \right]}{H}_m(\cos \varphi )d\varphi +{\int }_{{\gamma }_1}{H}_m(\cos \varphi )d\varphi +{\int }_{{\gamma }_c}{H}_m(\cos \varphi )d\varphi +{\int }_{{\gamma }_2}{H}_m(\cos \varphi )d\varphi =0,$$ (3.7)

where the contours are traversed in the counterclockwise direction. We then substitute formula (3.4) into (3.7) to obtain the formula

$$G_m=-{\int }_{{\gamma }_1}{H}_m(\cos \varphi )d\varphi -{\int }_{{\gamma }_c}{H}_m(\cos \varphi )d\varphi -{\int }_{{\gamma }_2}{H}_m(\cos \varphi )d\varphi .$$ (3.8)

We construct the contours γ₁ and γ₂ as follows. To construct γ₁, we choose a curve which intersects the point ϕ = 0, and on which the numerator of $H^w(\cos \varphi )$ does not oscillate. From formula (3.1), it is easy to see that this occurs when $\mathrm{Re}(\kappa \sqrt{1-\alpha \cos \varphi })$ is constant. Because γ₁ must intersect ϕ = 0, this contour is defined by

$${\gamma }_1=\{\tilde{\varphi }:\mathrm{Re}(\kappa \sqrt{1-\alpha \cos \tilde{\varphi }})=\mathrm{Re}(\kappa )\sqrt{1-\alpha }\}.$$ (3.9)

Representing κ in polar form $\kappa =r{e}^{i\varphi }$, formula (3.9) simplifies to

$${\gamma }_1=\{\tilde{\varphi }:\mathrm{Re}(e^{i\varphi }\sqrt{1-\alpha \cos \tilde{\varphi }})=\mathrm{Re}(e^{i\varphi })\sqrt{1-\alpha }\}.$$ (3.10)

To represent this curve in parametric form, we first perform a change of variables cos ϕ = z. The curve, γ₁(s), which is the solution to

$$e^{i\varphi }\sqrt{1-\alpha {\gamma }_1(s)}=-\sqrt{\alpha }is+e^{i\varphi }\sqrt{1-\alpha },$$ (3.11)

satisfies (3.10). With the negative time-harmonic convention, the spherical-wave term decays on γ₁(s) in the s > 0 direction. Note that the constant associated with the is term in equation (3.11) is arbitrary; our choice of $\sqrt{\alpha }$ is for convenience. We solve for γ₁(s) by dividing both sides by $e^{i\varphi }$, squaring both sides, subtracting one from each side, and dividing both sides by −α, arriving at the formula

$${\gamma }_1(s)=e^{-2i\varphi }{s}^2+2{\beta }_{-}i{e}^{-i\varphi }\frac{\sqrt{1-\alpha }}{\sqrt{\alpha }}s+1,$$ (3.12)

where $s\ge 0$. Introducing the parameters ${\beta }_{-}=\sqrt{1/\alpha -1}$ and ${\beta }_{+}=\sqrt{1/\alpha -1}$ (see Section 1.2), we rewrite (3.12) as

$${\gamma }_1(s)=e^{-2i\varphi }{s}^2+2i{\beta }_{-}{e}^{-i\varphi }s+1,$$ (3.13)

where s ≥ 0. To construct γ₂, we use the same method, except that we require γ₂ to intersect ϕ = π. A similar procedure used to arrive at (3.13) results in a formula for γ₂(s) in the cos ϕ-plane, given by

$${\gamma }_2(s)=e^{-2i\varphi }{s}^2+2i{\beta }_{+}{e}^{-i\varphi }s-1,$$ (3.14)

where $s\ge 0$. We note that our formulae for γ₁(s) and γ₂(s) differ from the formulae in the literature in that they contain the coefficients $e^{-i\varphi }$ and $e^{-2i\varphi }$. Observe that for real κ, ϕ = 0. Lastly, for the case of real κ, making the substitution $s=t/{\beta }_{-}$ for γ₁ and $s=t/{\beta }_{+}$ for γ₂ yields the formulae in [9] (see [9], formula (9)), given by

$${\gamma }_1(t)=\frac{t^2}{4{\beta }_{-}^2}+it+1,$$ (3.15)

$${\gamma }_2(t)=\frac{t^2}{4{\beta }_{+}^2}+it-1.$$ (3.16)

Integration on these contours requires the change of variables $z=\cos \varphi$. Thus, $dz=-\sin \varphi$ and $d\varphi =-\sqrt{1-z^2}$. Recalling that the Chebyshev polynomials of the first kind are given by

$$T_m(z)=\cos (m\arccos (z)),$$ (3.17)

the cos mϕ term with the above substitution becomes T_m(z). Thus, the integral in formula (3.6) becomes

$$G_m=-{\int }_{\text{Γ}\backslash \left[0,\pi \right]}{H}_m(\cos \varphi )d\varphi ={\int }_{\text{Γ}\backslash \left[0,1\right]}\frac{H_m(z)}{\sqrt{1-z^2}}dz,$$ (3.18)

where the contour on the right-hand side is traversed in the counterclockwise direction.

It is not difficult to show that, for κ in quadrant IV of the complex plane, H_m(z) vanishes as $\mathrm{Im}(z)\to +\infty$ provided that $0\le \mathrm{Re}(z)\le 1$ or, equivalently, that $0\le \mathrm{Re}(\varphi )\le \pi$. Thus, if we construct γ₁ and γ₂ to travel sufficiently high into the complex plane, we have that

$${\int }_{{\gamma }_c}\frac{H_m(z)}{\sqrt{1-z^2}}dz\to 0,$$ (3.19)

where γ_c is the contour connecting γ₁ and γ₂. After this change of variables, we arrive at a formula for G_m where the integrand has a non-oscillatory term corresponding to the numerator of the spherical-wave term, given by

$$G_m=-{\int }_{{\gamma }_1}\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{T}_m(z)dz-{\int }_{{\gamma }_2}\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{T}_m(z)dz,$$ (3.20)

where we have used (3.19) to omit the integral corresponding to γ_c.

Our formula (3.20) departs from the form given in [9] (see [9], formula (19)) in that (3.20) is a formula for all m, while the formula appearing in [9] is for the special case where m = 1. Although the integrand in (3.20) has a spherical-wave term which monotonically decays on γ₁ and γ₂, the rest of the integrand oscillates and grows along γ₁ and γ₂. In the subsequent section, we characterize the growth, oscillation, and sign behavior of the integrand on these contours. We then demonstrate that this results in concomitant cancellation error when evaluating formula (3.20).

3.1.2. Cancellation Error on Gustafsson’s Contours

We consider the integrand in (3.20) as the product of two terms, H^w(z), and T_m(z), with H^w(z) given as

$$H^w(z)=\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}}.$$ (3.21)

On both contours γ₁ and γ₂, for points distant from the real axis (i.e., points with large imaginary component), the exponential term in H^w(z) decays far faster than T_m(z) grows, meaning the integrand decays to zero as $\mathrm{Im}(z)\to +\infty$. However, for points on γ₁ and γ₂ near the real axis, T_m(z) can be far larger than 1/H^w(z), meaning that the integrand takes on values with large magnitude, particularly when evaluating the modal Green’s function for large values of m and small values of κ.

Being the Fourier coefficient of an analytic function, G_m exhibits geometric decay in m, but is expressed as the sum of two integrals, each of which exhibits geometric growth in m. We summarize this behavior with the formula

$$O(a^{-m})\approx G_m=-{\int }_{{\gamma }_1}\frac{H^w(z)T_m(z)}{\sqrt{1-z^2}}dz={\int }_{{\gamma }_2}\frac{H^w(z)T_m(z)}{\sqrt{1-z^2}}dz\approx O(a^m)+O(a^m),$$ (3.22)

which is only possible if the integrals have opposite sign. Therefore, integrating the form in (3.20) incurs cancellation error which grows geometrically with m.

3.2. Rational Function Approximation of the Chebyshev Polynomial

Evaluation of (3.20) incurs cancellation error which grows geometrically in m, due to the growth of the Chebyshev polynomial away from the real axis. In this section, we characterize its growth, and then propose a rational function approximation which approximately equals the Chebyshev polynomial on the interval [−1, 1], but instead decays in the complex plane.

3.2.1. The Growth of the Chebyshev Polynomial in the Complex Plane

It is helpful to characterize the growth of the Chebyshev polynomial in the complex plane. Recall that the formula for the Bernstein ellipse indexed by the parameter ρ > 0 is

$$E_{\rho }(\theta )=a\cos \theta +ib\sin \theta ,$$ (3.23)

where

$$a=\frac{1}{2}(\rho +{\rho }^{-1}),b=\frac{1}{2}(\rho -{\rho }^{-1}).$$ (3.24)

Recall that (2.12) provides the bound

$$\frac{1}{2}({\rho }^m-{\rho }^{-m})\le |T_m(E_{\rho }(\theta ))|\le \frac{1}{2}({\rho }^m+{\rho }^{-m}),$$ (3.25)

characterizing the growth of $T_m(E_{\rho }(\theta ))$. Note that (3.25) can be immediately extended to any point z in the interior of the Bernstein ellipse E_ρ by the maximum principle. Thus,

$$\frac{1}{2}({\rho }^m-{\rho }^{-m})\le |T_m(z)|\le \frac{1}{2}({\rho }^m+{\rho }^{-m}),$$ (3.26)

for all $z\in E_{\rho }^o$, where $E_{\rho }^o\in ℂ$ denotes the interior of the region bounded by E_ρ.

3.2.2. Choice of the Bernstein Ellipse Parameter ρ for an mth Order Chebyshev Polynomial

Recall that, by convention, the parameter ρ > 1. Hence, by (3.26),

$$|T_m(z)|\le \frac{1}{2}({\rho }^m+{\rho }^{-m})<{\rho }^m,$$ (3.27)

for all $z\in E_{\rho }^o$. Thus, to bound the mth order Chebyshev polynomial by an arbitrary constant M, we pick ρ by the formula,

$$\rho =M^{\frac{1}{m}},$$ (3.28)

which by (3.27) bounds T_m(z) by $|T_m(z)|<M$ for $z\in E_{\rho }^o$, where $E_{\rho }^o$ denotes the interior of the region bounded by E_ρ.

3.2.3. Rational Function Approximation of the Chebyshev Polynomial via the Cauchy Integral Formula

In this section, we construct a rational function approximation which is approximately equal to T_m(z) on the interval [−1, 1], but, instead of exhibiting polynomial growth in the complex plane, decays.

The Chebyshev polynomial T_m(z) is analytic everywhere in the complex plane. Thus, by Cauchy’s integral formula

$$T_m(z)=\frac{1}{2\pi i}\underset{\text{Γ}}{\oint }\frac{T_m(v)}{v-z}dv,$$ (3.29)

where Γ is any simple closed contour, and z is a point in the interior of Γ. Let Γ be a Bernstein ellipse with parameter ρ, denoted by E_ρ. Then (3.29) is given by

$$T_m(z)=\frac{1}{2\pi i}\underset{0}{\overset{2\pi }{\int }}\frac{T_m(E_{\rho }(\theta ))E_{\rho }^{\prime }(\theta )}{E_{\rho }(\theta )-z}d\theta .$$ (3.30)

Suppose that the integral in (3.30) can be efficiently estimated with a quadrature rule, given by the nodes θ₁, θ₂, … , θ_n and weights w₁, w₂, … , w_n. Then, $T_m(z)\approx R_m(z)$, where

$$R_m(z)=\frac{1}{2\pi i}\sum _{i=1}^n\frac{T_m(E_{\rho }({\theta }_i))E_{\rho }^{\prime }({\theta }_i)}{E_{\rho }({\theta }_i)-z}{w}_i.$$ (3.31)

Recall from Section 2.3 that

$$T_m(E_{\rho }(\theta ))=T_m(J(C_{\rho }(\theta )))=\frac{{\rho }^m{e}^{im\theta }+{\rho }^{-m}{e}^{-im\theta }}{2}.$$ (3.32)

Thus, we rewrite (3.31) as

$$R_m(z)=\frac{1}{2\pi i}\sum _{i=1}^n\frac{a_i}{v_i-z},$$ (3.33)

where

$$a_i=(\frac{{\rho }^m{e}^{im{\theta }_i}+{\rho }^{-m}{e}^{-im{\theta }_i}}{2})E_{\rho }^{\prime }({\theta }_i)w_i,v_i=E_{\rho }({\theta }_i),$$ (3.34)

for i = 1, 2, … , n.

3.3. An Analytic Mapping Exchanging the Bernstein Ellipse with the Interval [–1, 1]

In order to estimate the number of quadrature nodes required for the construction of the approximation R_m(z) described in the previous section, we will make use of the following map. Let $\mathscr{H}\subset ℂ$ denote the upper half-plane, and let E_ρ denote the Bernstein ellipse with parameter ρ > 1. From the discussions of the Joukowski transformation in Section 2.2 and the Bernstein ellipse in Section 2.3, we can construct a conformal mapping that exchanges the upper half of the Bernstein ellipse $E_{\rho }\cap \mathscr{H}$ and the unit interval [−1, 1] (see Figure 2).

Fig. 2: The mapping ϕ exchanging the upper half of the Bernstein ellipse with the interval [−1, 1].

The mapping ϕ given by formula (3.35) is a conformal mapping from $E_{{\rho }^2}^o\cap \mathscr{H}\text{ to }{E}_{\rho }^o\backslash ((-\infty ,-1]\cup [1,\infty ))$, where $\mathscr{H}$ denotes the upper half-plane, such that the upper half of the Bernstein ellipse $E_{\rho }\cap \mathscr{H}$ and the unit interval [−1, 1] are exchanged.

Lemma 3.1.

Let J(v) denote the Joukowski transformation, given by formula (2.2), and let $J_2^{-1}(z)$ denote the inverse of the Joukowski transformation, given by formula (2.4). Then the mapping

$$\varphi (z)=J(\frac{1}{\rho }\cdot J_2^{-1}(z))$$ (3.35)

is a conformal mapping from $E_{{\rho }^2}^o\cap \mathscr{H}to{E}_{\rho }^o\backslash ((-\infty ,-1]\cup [1,\infty ))$, and exchanges the unit interval [−1, 1] with the upper half of the Bernstein ellipse E_ρ, in the sense that $\varphi ([-1,1])=E_{\rho }\cap \mathscr{H}$ and $\varphi (E_{\rho }\cap \mathscr{H})=[-1,1]$.

Note that the mapping ϕ described above approaches the identity map as ρ approaches 1.

3.4. The Decay of Chebyshev Expansion Coefficients of p(z)/(z − w) for z ∈ [−1, 1], where w ∈ E_ρ

The following theorem bounds the decay of the coefficients of the Chebyshev expansion of p(z)/(z − w) for z ∈ [−1, 1], where p(z) is a polynomial of order m and w ∈ E_ρ. Note the similarities between this theorem and Theorem 2.1.

Theorem 3.2.

Suppose that p(z) is a polynomial of order m which satisfies $|p(z)|\le L$ for all $z\in E_{\rho }^o$, where $E_{\rho }^o$ denotes the interior of the Bernstein ellipse E_ρ, for some constant L > 0. Suppose further that w ∈ E_ρ, and that

$$\frac{p(z)}{z-w}=\sum _{k=0}^{\infty }{a}_k{T}_k(z),$$ (3.36)

for all z ∈ [−1, 1], where T_k(z) is the Chebyshev polynomial of order k. Finally, let v ∈ C_ρ be the point on the circle C_ρ of radius ρ > 1 such that $w=J(v)=\frac{1}{2}(v+v^{-1})$. Then the Chebyshev expansion coefficients a_k satisfy

$$|a_k|\le \frac{4L\rho }{|v^2-1|}{\rho }^{-k},$$ (3.37)

for all positive $k\ge m$.

Proof.

We begin by observing that

$$a_k=\frac{2}{\pi }{\int }_{-1}^1\frac{p(s)}{s-w}\cdot \frac{T_k(s)}{\sqrt{1-s^2}}ds,$$ (3.38)

for all $k\ge 1$. Making the change of variables $s=\frac{1}{2}(z+z^{-1})$ and using identity (2.7), we have that

$$a_k=\frac{i}{\pi }{\int }_C\frac{p(\frac{1}{2}(z+z^{-1}))}{\frac{1}{2}(z+z^{-1})-w}\cdot \frac{z^k+z^{-k}}{2}\cdot \frac{dz}{z},$$ (3.39)

where C is the circle of radius one with the usual counterclockwise orientation. Since w ∈ E_ρ, where E_ρ is the Bernstein ellipse with parameter ρ, there exists some v ∈ C_ρ, where C_ρ is the circle of radius ρ, such that $w=\frac{1}{2}(v+v^{-1})$ (see Section 2.3). Expressing w in terms of v, formula (3.39) becomes

$$a_k=\frac{i}{\pi }{\int }_C\frac{p(\frac{1}{2}(z+z^{-1}))}{\frac{1}{2}(z+z^{-1})-\frac{1}{2}(v+v^{-1})}\cdot \frac{z^k+z^{-k}}{2}\cdot \frac{dz}{z},$$ (3.40)

where $v\in C_{\rho }$, which simplifies to

$$a_k=\frac{i}{\pi }{\int }_C\frac{p(\frac{1}{2}(z+z^{-1}))}{(z+z^{-1})-(v+v^{-1})}\cdot (z^k+z^{-k})\cdot \frac{dz}{z},$$ (3.41)

where $v\in C_{\rho }$. Since

$$\frac{1}{(z+z^{-1})-(v+v^{-1})}=\frac{1}{z-v}\cdot \frac{1}{1-{(zv)}^{-1}}=\frac{1}{z-v^{-1}}\cdot \frac{1}{1-z^{-1}v},$$ (3.42)

we observe that the integrand of (3.41) has two simple poles, one at v and the other at v⁻¹. We assume that ρ > 1 (recall that $E_{\rho }=E_{1/\rho }$), so that $|v|=\rho >1$ and $|v^{-1}|=1/\rho <1$. Splitting the integral in (3.41) into two parts, we write $a_k=a_k^{(1)}+a_k^{(2)}$, where

$$a_k^{(1)}=\frac{i}{\pi }{\int }_C\frac{1}{z-v}\cdot \frac{p(\frac{1}{2}(z+z^{-1}))}{1-{(zv)}^{-1}}\cdot z^{-k}\cdot \frac{dz}{z},$$ (3.43)

and

$$a_k^{(2)}=\frac{i}{\pi }{\int }_C\frac{1}{z-v^{-1}}\cdot \frac{p(\frac{1}{2}(z+z^{-1}))}{1-z^{-1}v}\cdot z^k\cdot \frac{dz}{z}.$$ (3.44)

We first consider $a_k^{(1)}$. Since the integrand has a simple pole at $v\in C_{\rho }$, by the residue theorem we can express the contour integral over C as the sum of the residue at v and a contour integral over C_R, where R > ρ > 1, so that

$$a_k^{(1)}=\frac{2p(\frac{1}{2}(v+v^{-1}))}{v-v^{-1}}{v}^{-k}+\frac{i}{\pi }{\int }_{C_R}\frac{1}{z-v}\cdot \frac{p(\frac{1}{2}(z+z^{-1}))}{1-{(zv)}^{-1}}\cdot z^{-k}\cdot \frac{dz}{z}.$$ (3.45)

Since $p(z)=O(z^m)$ as $|z|\to \infty$, we see that $p(\frac{1}{2}(z+z^{-1}))=O(z^m)$ as $|z|\to \infty$. Furthermore, since

$$\frac{1}{z-v}\cdot \frac{1}{1-{(zv)}^{-1}}\sim \frac{1}{z}$$ (3.46)

as $|z|\to \infty$, we have that, when $k\ge m$, the integral over C_R in (3.45) vanishes as $R\to \infty$. Thus,

$$a_k^{(1)}=\frac{2p(\frac{1}{2}(v+v^{-1}))}{v-v^{-1}}{v}^{-k},$$ (3.47)

for all $k\ge m$.

Next, we consider $a_k^{(2)}$. Since the integrand has a simple pole at $v^{-1}\in C_{{\rho }^{-1}}$, by the residue theorem we have that

$$a_k^{(2)}=\frac{2p(\frac{1}{2}(v+v^{-1}))}{v-v^{-1}}{v}^{-k}+\frac{i}{\pi }{\int }_{C_r}\frac{1}{z-v^{-1}}\cdot \frac{p(\frac{1}{2}(z+z^{-1}))}{1-z^{-1}v}\cdot z^k\cdot \frac{dz}{z},$$ (3.48)

where $r<{\rho }^{-1}<1$. Like before, we observe that, since $p(z)=O(z^m)$ as $|z|\to \infty$, $p(\frac{1}{2}(z+z^{-1}))=O(z^{-m})$ as $|z|\to 0$. Since

$$\frac{1}{z-v^{-1}}\cdot \frac{1}{1-z^{-1}v}\sim z$$ (3.49)

as $|z|\to 0$, we have that, when $k\ge m$, the integral over C_r in (3.48) vanishes as $r\to 0$. Thus,

$$a_k^{(2)}=\frac{2p(\frac{1}{2}(v+v^{-1}))}{v-v^{-1}}{v}^{-k},$$ (3.50)

for all $k\ge m$.

Combining (3.47) and (3.50), we have

$$a_k=\frac{4p(\frac{1}{2}(v+v^{-1}))}{v-v^{-1}}{v}^{-k},$$ (3.51)

for all $k\ge m$. Since $v\in C_{\rho }$ and $|p(z)|\le L$ for $z\in E_{\rho }^o$, it is easy to see that

$$|a_k|\le \frac{4L\rho }{|v^2-1|}{\rho }^{-k},$$ (3.52)

for all $k\ge m$, and we are done. ■

3.5. The Number of Terms in the Chebyshev Expansions of p(z)/(z − w) for z ∈ [−1, 1], where w ∈ E_ρ

The following corollary of Theorem 3.2 bounds the number of Chebyshev polynomials required to represent a function p(z)/(z − w) for z ∈ [−1, 1], where p(z) is a polynomial of order m that is bounded by L in $E_{\rho }^o$, with w ∈ E_ρ and $\rho =M^{1/m}$, in terms of M, L, and m.

Corollary 3.3.

Suppose that M > 1, and let $\rho =M^{1/m}$, for some integer $m\gg 1$. Suppose further that p(z) is a polynomial of order m which satisfies $|p(z)|\le L$ for all $z\in E_{\rho }^o$, where $E_{\rho }^o$ denotes the interior of the Bernstein ellipse E_ρ, for some constant L > 0. Suppose that w ∈ E_ρ, and that

$$\frac{p(z)}{z-w}=\sum _{k=0}^{\infty }{a}_k{T}_k(z),$$ (3.53)

for all z ∈ [−1, 1], where T_k(z) is the Chebyshev polynomial of order k. Finally, let $0<ϵ\ll 1$ be some small real number, and let

$$k_0=m(\log (2L)-\log (ϵ))/\log (M).$$ (3.54)

Then, when w is well-separated from ±1,

$$|a_k|\lesssim ϵ$$ (3.55)

for all $k\ge \max (k_0,m)$, and when $w\approx \pm 1$,

$$|a_k|\lesssim \frac{ϵ}{|\sqrt{w^2-1}|},$$ (3.56)

for all $k\ge \max (k_0,m)$.

Proof.

By Theorem 3.2, the Chebyshev expansion coefficients a_k satisfy

$$|a_k|\le \frac{4L\rho }{|v^2-1|}{\rho }^{-k},$$ (3.57)

for all positive $k\ge m$. Let $v\in C_{\rho }$ be the point of the circle of radius ρ > 1 such that $w=J(v)=\frac{1}{2}(v+v^{-1})$. Since $\rho =M^{1/m}$, it follows that, when m is large, $\rho \approx 1$. Suppose that w is well-separated from ±1. In this case, it is easy to see that $|v^2-1|\approx 2$ (observe that $|v^2-1|=2$ exactly when $v=\pm i$). Thus,

$$\frac{4L\rho }{|v^2-1|}\approx 2L$$ (3.58)

when w is well-separated from ±1.

Suppose now that $w\approx \pm 1$. Since $w=J(v)=\frac{1}{2}(v+v^{-1})$, it follows from formula (2.4) that,

$$v=J_2^{-1}(w)=w+\sqrt{w+1}\sqrt{w-1}.$$ (3.59)

We observe that, if $w\approx 1$, then $v\approx 1$. Subtracting one from both sides of (3.59), we have that

$$v-1=w-1+\sqrt{w+1}\sqrt{w-1}.$$ (3.60)

As $w\to 1$, w −1 becomes much smaller than $\sqrt{w-1}$, so we have that $|v-1|\sim |\sqrt{w^2-1}|$ as $w\to 1$. Since $|v+1|\to 2$ as $w\to 1$, it also follows that

$$|v^2-1|\sim 2|\sqrt{w^2-1}|,$$ (3.61)

as $w\to 1$. By essentially the same argument, we can show that (3.61) also holds as $w\to -1$. Putting all of this together, we have that, when $w\approx \pm 1$,

$$\frac{4L\rho }{|v^2-1|}\approx \frac{2L}{|\sqrt{w^2-1}|}.$$ (3.62)

Letting

$$k_0=m(\log (2L)-\log (ϵ))/\log (M),$$ (3.63)

we observe that

$$2L{\rho }^{-k}\le ϵ$$ (3.64)

for for positive $k\ge k_0$. Combining this with (3.57), (3.58), and (3.62), we find that, when w is well-separated from ±1,

$$|a_k|\lesssim ϵ$$ (3.65)

for all $k\ge \max (k_0,m)$, and when $w\approx \pm 1$,

$$|a_k|\lesssim \frac{ϵ}{|\sqrt{w^2-1}|},$$ (3.66)

for all $k\ge \max (k_0,m)$, and we are done. ■

3.6. The Geometry of the Bernstein Ellipse

Our approach is to compute the modal Green’s function by integrating along Gutsafson’s contours, where the spherical-wave term is non-oscillatory. However, on these contours, the Chebyshev polynomial term oscillates. To avoid the cancellation error which occurs from integrating the Chebyhsev polynomial term along Gustafson’s contours, we instead replace the Chebyshev polynomial with a rational function approximation described in Section 3.2.3. Such a function, by design, vanishes outside the Bernstein ellipse. Therefore, the behavior of our algorithm is defined entirely by the contour within the Bernstein ellipse. In this section, we describe several properties of the Bernstein ellipse in relation to the contours γ₁ and γ₂, which we will require when we describe our algorithm.

3.6.1. Approximations for the Major and Minor Axes as Functions of m

Recall from Section 3.2.2 that, when constructing the rational function approximation for the mth order Chebyshev polynomial, we choose the Bernstein ellipse parameter ρ using the formula

$$\rho =M^{\frac{1}{m}},$$ (3.67)

where M > 1 is an arbitrary constant. Recall also from Section 2.3 that the axes of the Bernstein ellipse E_ρ, where ρ > 1, are given by

$$a=\frac{1}{2}(\rho +\frac{1}{\rho }),b=\frac{1}{2}(\rho -\frac{1}{\rho }),$$ (3.68)

where a is the semi-major axis (along the real axis) and b is the semi-minor axis (along the imaginary axis). For convenience we analyze the case where M = e, so that $\rho =e^{1/m}$. It can be shown that, by taking the Taylor expansion of ρ and 1/ρ, the semi-major axis of the Bernstein ellipse in terms of m is

$$a=\frac{1}{2}(\rho +\frac{1}{\rho })=1+\frac{1}{m^2}+O(\frac{1}{m^4}).$$ (3.69)

Likewise, the semi-minor axis is

$$b=\frac{1}{2}(\rho -\frac{1}{\rho })=\frac{1}{m}+O(\frac{1}{m^3}).$$ (3.70)

3.6.2. The Distances from the Points z = 1 and z = −1 to the Bernstein Ellipse as Functions of m

Recall that Gustafsson’s contours γ₁ and γ₂ begin at the points z = 1 and z = −1, respectively, which are the foci of the Bernstein ellipses (see Section 3.1.1). For each focus, we are interested in two quantities: the vertical distance from the focus to the Bernstein ellipse, and the horizontal distance from the focus to the Bernstein ellipse. We examine the distances for z = 1, knowing that, by symmetry, they are the same for z = −1. Observe that the horizontal distance from z = 1 to the Bernstein ellipse is equal to a − 1. Formula (3.69) immediately yields

$$a-1=\frac{1}{m^2}+O(\frac{1}{m^4}).$$ (3.71)

Observe that the vertical distance from z = 1 to the Bernstein ellipse is the same as the y-coordinate of the intersection of the vertical line x = 1 with the Bernstein ellipse E_ρ in the upper half-plane. This intersection’s y-coordinate is approximated by substituting the Taylor series expansions of the semi-major and semi-minor axes into the formula for the Bernstein ellipse. Solving for the resulting y, it can be shown that

$$y=\sqrt{\frac{2}{m^4}}\sqrt{1+O(\frac{1}{m^2})}.$$ (3.72)

Recall that the Taylor series of $\sqrt{1+x}$ is

$$\sqrt{1+x}=1+\frac{x}{2}-\frac{x^2}{8}+\cdots .$$ (3.73)

Substituting (3.73) into (3.72), we have that

$$y=\frac{\sqrt{2}}{m^2}+O(\frac{1}{m^4}).$$ (3.74)

Hence, the vertical distance from both z = 1 and z = −1 to the Bernstein ellipse is on the order of $\sqrt{2}/m^2$ (see Figure 3).

Fig. 3: The distances a − 1, b, and the distance from the focus z = 1 to the intersection of the line x = 1 with the Bernstein ellipse, as functions of m.

The distance from z = 1 to the intersection of E_ρ with the y = 0 axis is $\approx 1/m^2$, and is equal to a − 1, where a is the semi-major axis of E_ρ. The intersection of x = 1 with E_ρ has a distance of $\approx \sqrt{2}/m^2$ to the point z = 1. The vertical distance from z = 0 to E_ρ is ≈ 1/m, and is equal to b, the semi-minor axis of E_ρ.

3.6.3. Geometry of the Angles of Intersection Between Gustafsson’s Contours and the Bernstein Ellipse

Recall from Section 3.1.1 that Gustafsson’s contours γ₁ and γ₂ can be parameterized as

$${\gamma }_1(t)=\frac{t^1}{4{\beta }_{-}^2}+it+1,$$ (3.75)

$${\gamma }_2(t)=\frac{t^2}{4{\beta }_{+}^2}+it-1,$$ (3.76)

for t > 0, where ${\beta }_{-}=\sqrt{1/\alpha -1}$ and ${\beta }_{+}=\sqrt{1/\alpha +1}$, for 0 < α < 1. Consider the sets Γ₁ and Γ₂, consisting of all possible γ₁ and γ₂, respectively, defined as

$${\text{Γ}}_1=\left\{{\gamma }_1:0<{\beta }_{-}<\infty \right\},{\text{Γ}}_2=\left\{{\gamma }_2:1<{\beta }_{+}<\infty \right\}.$$ (3.77)

The boundary of Γ₁, denoted by ∂Γ₁, is given by the union of the contours γ₁ associated with the limits ${\beta }_{-}\to 0$ and ${\beta }_{-}\to \infty$. Likewise, the boundary of Γ₂, denoted by ∂Γ₂, is given by the union of the contours γ₂ associated with the value ${\beta }_{+}=1$ and the limit ${\beta }_{+}\to \infty$ (see Figure 4). We observe that, in the limit as ${\beta }_{-}\to \infty$ and ${\beta }_{+}\to \infty$, the contours γ₁ and γ₂ become vertical lines. From the sets ∂Γ₁ and ∂Γ₂, together with the bounds from Section 3.6.2, it is easy to see that the angles that γ₁ and γ₂ make with E_ρ are bounded from below. In other words, Gustafsson’s contours are never close to being tangent to the Bernstein ellipse.

Fig. 4: The set of all possible Gustafsson contours together with the Bernstein ellipse E_ρ in the z = cos ϕ-plane.

Recall that Gustafsson’s contours are denoted γ₁ and γ₂, where γ₁ begins at the point z = 1 and γ₂ begins at the point z = −1. The set of all possible γ₁ is denoted by Γ₁. The set of all possible γ₂ is denoted by Γ₂.

Finally, we bound the length of Gustafsson’s contours within the Bernstein ellipse. Because the signs of the quadratic term and the linear term in Gustafson’s contours (see (3.75) and (3.76)) are the same, the lengths of the contours within the ellipse are well-approximated by the distance from the foci to the intersections of the contours with the Bernstein ellipse. For the contour γ₁ (associated with the point z = 1), because the Bernstein ellipse is convex, the length is at most O(1/m). For the contour γ₂ (associated with the point z = −1), because the angle of intersection is nicely bounded, the length is also at most O(1/m). Therefore, on either contour, the mth-order Chebyshev polynomial term oscillates at most once within the Bernstein ellipse, as does the rational approximation.

3.7. Evaluating the Modal Green’s Function

In this section, we use the rational function approximation R_m(z), described in Section 3.2, to express the modal Green’s function in terms of integrals over Gustafsson’s contours, in such a way that the integrals do not incur the large cancellation errors described in Section 3.1.2.

Recall from Section 3.1.1 that, after the variable substitution of z = cos ϕ, dz = − sin ϕ dϕ, the formula for the modal Green’s function, G_m, is

$$G_m=-\underset{\left[-1,1\right]}{\int }\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{T}_m(z)dz.$$ (3.78)

Our rational function approximation, R_m(z), is approximately equal to T_m(z) on the interval [−1, 1]. Therefore, substituting R_m(z) for T_m(z), we arrive at a formula for G_m,

$$G_m\approx -\underset{\left[-1,1\right]}{\int }\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{R}_m(z)dz.$$ (3.79)

The integrand of (3.79) is analytic everywhere in the complex plane except for a finite number of poles, so the integral can be deformed. By Cauchy’s residue theorem,

$${\oint }_{\text{Γ}}\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{R}_m(z)dz=2\pi i\sum _{k=1}^n\underset{z=z_k}{\text{Res}}\left(\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{R}_m(z)\right),$$ (3.80)

where z₁, …, z_n are the poles inside Γ. Thus, if Γ is a closed contour which includes the interval [−1, 1], we have that

$$G_m\approx -\underset{\text{Γ}\backslash \left[-1,1\right]}{\int }\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{R}_m(z)dz+2\pi i\sum _{k=1}^n\underset{z=z_k}{\text{Res}}\left(\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{R}_m(z)\right),$$ (3.81)

where Γ is a contour starting at z = 1 and ending at z = −1. We select Γ \ [−1, 1] to be the Gustafsson contour γ₁ + γ_c + γ₂, which we describe in Section 3.1.1. Since the integrand vanishes over γ_c, we have that

$$\begin{array}{l}{G}_m\approx -{\int }_{{\gamma }_1}\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{R}_m(z)dz-{\int }_{{\gamma }_2}\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{R}_m(z)dz\\ +2\pi \text{i}\sum _{k=1}^n\underset{z=z_k}{\text{Res}}\left(\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{R}_m(z)\right).\end{array}$$ (3.82)

Since R_m(z), unlike T_m(z), does not grow as $\mathrm{Im}(z)\to +\infty$, it follows that formula (3.82) can be evaluated without cancellation error, provided that the nodes and weights of the quadrature formula used to construct R_m(z) are chosen correctly, which is described in the sequel.

3.8. Removing the Singularity

Recall that the integral in (3.20) corresponding to the γ₁ contour has the formula

$$-{\int }_{{\gamma }_1}\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{T}_m(z)dz=-{\int }_{{\gamma }_1}\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z}\sqrt{1+z}}{T}_m(z)dz.$$ (3.83)

Observe that the integrand in (3.83) has square-root singularities at z = 1 and z = −1. Furthermore, when $\alpha \approx 1$, the product of the terms,

$$\frac{1}{\sqrt{1-\alpha z}\sqrt{1-z}}\approx \frac{1}{1-z},$$ (3.84)

meaning that the integrand will have a 1/z-type singularity at z = 1. By careful reparameterization of the contour γ₁, the singularities in (3.83) can be removed. The variable substitutions and analysis of the singularities in this section are unchanged when R_m(z) is substituted for T_m(z). Recall from Section 3.1.1 that the contour γ₁ can be parameterized as

$${\gamma }_1(t)=\frac{t^2}{4{\beta }_{-}^2}+it+1,$$ (3.85)

for t > 0. We then follow [9] and perform the substitution $t=2{\beta }_{-}{\tau }^2$ and reparameterize the contour γ₁ as ${\tilde{\gamma }}_1$, given by

$${\tilde{\gamma }}_1(\tau )={\gamma }_1(2{\beta }_{-}{\tau }^2)={\tau }^4+2i{\beta }_{-}{\tau }^2+1.$$ (3.86)

Gustafsson showed (see [9], equations (15) and (16)) that, after substituting $z={\tilde{\gamma }}_1(\tau ),dz={\tilde{\gamma }}_1^{\prime }(\tau )d\tau$,

$$dz=4\tau ({\tau }^2+i{\beta }_{-})d\tau ,$$ (3.87)

$$\sqrt{1-\alpha z}=\frac{\sqrt{\alpha }}{i}({\tau }^2+i{\beta }_{-}).$$ (3.88)

Thus, with the parameterization $z={\tilde{\gamma }}_1(\tau )$, formula (3.83) becomes

$$\frac{-4i}{\sqrt{\alpha }}{\int }_0^{\infty }\frac{e^{-i\kappa \sqrt{1-\alpha {\tilde{\gamma }}_1(\tau )}}}{\sqrt{1-{\tilde{\gamma }}_1(\tau )}\sqrt{1+{\tilde{\gamma }}_1(\tau )}}{T}_m(\tilde{\gamma }(\tau ))\tau d\tau ,$$ (3.89)

where we have used (3.87) and (3.88) to cancel the $\sqrt{1-\alpha z}$ term. The integrand in (3.89) has a square-root singularity near z = 1. After substituting (3.86) into the $\sqrt{(}1-{\tilde{\gamma }}_1(\tau ))$ term in the denominator of (3.89) and factoring the radical, formula (3.89) becomes

$$\frac{-4}{\sqrt{\alpha }}{\int }_0^{\infty }\frac{e^{-i\kappa \sqrt{1-\alpha {\tilde{\gamma }}_1(\tau )}}}{\sqrt{{\tau }^2+2i{\beta }_{-}}\sqrt{1+{\tilde{\gamma }}_1(\tau )}}{T}_m({\tilde{\gamma }}_1(\tau ))d\tau .$$ (3.90)

Note that the integrand of (3.90) is the product of a smooth function and the function $1/\sqrt{{\tau }^2+2i{\beta }_{-}}$. Let F₁(τ) be the smooth term, given by the formula

$$F_1(\tau )=\frac{e^{-i\kappa \sqrt{1-\alpha {\tilde{\gamma }}_1(\tau )}}}{\sqrt{1+{\tilde{\gamma }}_1(\tau )}}{T}_m({\tilde{\gamma }}_1(\tau )).$$ (3.91)

We now rewrite (3.90) using (3.91), so that the integral in (3.20) corresponding to the γ₁ contour is given by

$$\frac{-4}{\sqrt{\alpha }}{\int }_0^{\infty }\frac{F_1(\tau )}{\sqrt{{\tau }^2+i{\beta }_{-}}}d\tau .$$ (3.92)

The variable substitutions for the integral corresponding to the γ₂ contour are similar. Recall that γ₂ can be parameterized as

$${\gamma }_2(t)=\frac{t^2}{4{\beta }_{+}^2}+it-1.$$ (3.93)

We reparameterize γ₂(t) as ${\tilde{\gamma }}_2(\tau )$, given by the formula

$${\tilde{\gamma }}_2(\tau )={\tau }^4+2i{\beta }_{+}{\tau }^2-1.$$ (3.94)

By proceeding as before, we arrive at the formula for F₂(τ),

$$F_2(\tau )=\frac{e^{-i\kappa \sqrt{1-\alpha {\tilde{\gamma }}_2(\tau )}}}{\sqrt{1-{\tilde{\gamma }}_2(\tau )}}{T}_m({\tilde{\gamma }}_2(\tau )),$$ (3.95)

such that the formula for the integral in (3.20) corresponding to the γ₂ contour is given by

$$\frac{4i}{\sqrt{\alpha }}{\int }_0^{\infty }\frac{F_2(\tau )}{\sqrt{{\tau }^2+2i{\beta }_{+}}}d\tau .$$ (3.96)

We combine (3.92) and (3.96) to write a formula for the mth modal Green’s function,

$$G_m=\frac{4}{\sqrt{\alpha }}{\int }_0^{\infty }\frac{F_1(\tau )}{\sqrt{{\tau }^2+2i{\beta }_{-}}}d\tau -\frac{4i}{\sqrt{\alpha }}{\int }_0^{\infty }\frac{F_2(\tau )}{\sqrt{{\tau }^2+2i{\beta }_{+}}}d\tau .$$ (3.97)

Because β₊ is bounded from below by 1, the denominator in (3.96) is always greater than 1. In contrast, when $\alpha \approx 1$, we have that ${\beta }_{-}\approx 0$, which means that the denominator in (3.92) $\approx \sqrt{{\tau }^2}=\tau$.

4. Algorithm

Recall that kernel splitting methods (e.g., [7]) have computational cost which scales with both |κ| and 1/β₋, and cannot be easily parallelized (see Section 1.3.1). Furthermore, kernel splitting techniques experience catastrophic cancellation error for modest $|\mathrm{Im}\kappa |$. In contrast, the method of Gustafsson [9] has computational cost independent of κ and β₋, but incurs cancellation error which grows geometrically in m (see Section 3.1.2).

Our technique is to compute the modal Green’s function by integrating along Gustafsson’s contours using a rational function approximation in place of the Chebyshev polynomial. Because the spherical-wave term in the integrand monotonically decays, our algorithm’s cost is completely independent of κ. Unlike the method of Gustafsson, because the size our rational function approximation R_m(z) is bounded by our choice of Bernstein ellipse E_ρ, our approach does not have cancellation error which geometrically grows in m. This comes at the price of having to evaluate the residues of R_m(z) on the boundary of the corresponding Bernstein ellipse E_ρ, with a cost which scales with m. We also use the same technique as Gustafsson to evaluate the modal Green’s function when ${\beta }_{-}\approx 0$ (i.e., when the source and target are close), in time independent of of β₋. Consequently, our algorithm’s computational cost depends only on m and is independent of both κ and β₋, and scales as O(m).

4.1. Choice of the Rational Function Approximation

Recall from Section 3.2.3 that the Chebyshev polynomial T_m(z) can be approximated on the interval [−1, 1] with a rational function, R_m(z), constructed via an application of Cauchy’s integral formula followed by the application of a quadrature rule. This rational function approximation decays quickly in the complex plane. In this section, we introduce a different approximation, also denoted R_m(z), which is the sum of a Cauchy integral and a rational function.

By Cauchy’s integral formula, T_m(z) can be expressed as the contour integral,

$$T_m(z)=\frac{1}{2\pi i}\underset{\text{Γ}}{\oint }\frac{T_m(v)}{v-z}dv,$$ (4.1)

where Γ is any simple closed contour, and z is a point in the interior of Γ. Similarly,

$$\frac{1}{2\pi i}\underset{\text{Γ}}{\oint }\frac{T_m(v)}{v-z}dv=0,$$ (4.2)

for all z outside Γ. Recall from 3.2.2 that for any mth order Chebyshev polynomial, if ρ = M^1/m, then, within the Bernstein ellipse E_ρ, T_m(z) is bounded by the constant M. Furthermore, within the interior of E_ρ, the Chebyshev polynomial oscillates exactly once along any possible Gustafsson contour (see Section 3.6). We also note that the Bernstein ellipse E_ρ² has minor axis twice the length of the minor axis of E_ρ, and major axis four times the length of the major axis of E_ρ (see Section 3.6 and Figure 5).

Fig. 5: Contours of interest with respect to the function R_m(z) in the z = cos ϕ-plane.

Gustafsson’s contours are labeled as γ₁ and γ₂. The inner Bernstein ellipse is denoted by E_ρ. The outer ellipse is denoted by $E_{{\rho }^2}$. The intersection of γ₁ with E_ρ is denoted by p₁, and the intersection of γ₂ with E_ρ is denoted by p₂. The arc of E_ρ between p₂ and p₁ is denoted by C_ρ. The contours highlighted in red and region shaded in red correspond to the values of z on which the quadrature in (4.4) must be accurate, in the sense of (4.6)-(4.9).

Let C_ρ denote the part of the Bernstein ellipse E_ρ between the contours γ₁ and γ₂, which corresponds to the portion of E_ρ between p₁ and p₂, where $p_1\in ℂ$ and $p_2\in ℂ$ are the intersection points of γ₁ and γ₂ with E_ρ, respectively (see Figure 5). We split the Cauchy integral into two parts,

$$T_m(z)=\frac{1}{2\pi i}\underset{C_{\rho }}{\int }\frac{T_m(v)}{v-z}dv+\frac{1}{2\pi i}\underset{E_{\rho }\backslash C_{\rho }}{\int }\frac{T_m(v)}{v-z}dv.$$ (4.3)

Now, suppose that θ₁, … , θ_n, w₁, … , w_n are the nodes and weights of a quadrature formula such that

$$\frac{1}{2\pi i}\underset{C_{\rho }}{\int }\frac{T_m(v)}{v-z}dv\approx \frac{1}{2\pi i}\sum _{i=1}^n\frac{T_m(v_i)}{v_i-z}d{v}_i{w}_i,$$ (4.4)

where $v_i=E_{\rho }({\theta }_i)$, $d{v}_i=E_{\rho }^{\prime }({\theta }_i)$ and the quadrature is accurate to precision ϵ > 0 for all z ∈ [−1, 1], $z\in {\gamma }_1\cap E_{{\rho }^2}^o$, $z\in {\gamma }_2\cap E_{{\rho }^2}^o$, $z\in ℂ\backslash E_{{\rho }^2}^o$, where $E_{{\rho }^2}^o$ is the interior of $E_{{\rho }^2}$ (see Figure 5). Now, let R_m(z) be defined by

$$R_m(z)=\frac{1}{2\pi i}\underset{E_{\rho }\backslash C_{\rho }}{\int }\frac{T_m(v)}{v-z}dv+\frac{1}{2\pi i}\sum _{i=1}^N\frac{T_m(v_i)}{v_i-z}d{v}_i{w}_i.$$ (4.5)

We observe that, due to formula (4.1), we have that

$$|T_m(z)-R_m(z)|<ϵ,$$ (4.6)

for z ∈ [−1, 1]. We also observe that, due to formula (4.1), we have that

$$|T_m(z)-R_m(z)|<ϵ,$$ (4.7)

for $z\in {\gamma }_1\cap E_{\rho }^o$ and $z\in {\gamma }_2\cap E_{\rho }^o$. Likewise, due to formula (4.2),

$$|R_m(z)|<ϵ,$$ (4.8)

for $z\in {\gamma }_1\backslash E_{\rho }^o$ and $z\in {\gamma }_2\backslash E_{\rho }^o$. We also observe that, due to formula (4.2),

$$|R_m(z)|<ϵ,$$ (4.9)

for $z\in ℂ\backslash E_{{\rho }^2}^o$.

4.1.1. Deformation of the Contour

Recall from Section 3.7 that, after the variable substitution of z = cos ϕ, dz = − sin ϕ dϕ, the formula for the modal Green’s function, G_m, is

$$G_m=\underset{[-1,1]}{\int }\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{T}_m(z)dz.$$ (4.10)

Our approximation, R_m(z), by formula (4.1), is approximately equal to T_m(z) on the interval [−1, 1]. Therefore, substituting R_m(z) for T_m(z), we arrive at a formula for G_m,

$$G_m\approx \underset{[-1,1]}{\int }\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{R}_m(z)dz.$$ (4.11)

The integrand of (4.11) is analytic everywhere in the complex plane except for a finite number of poles, so the integral can be deformed. Recall that, for any closed contour $\text{Γ}\subset ℂ$, by Cauchy’s residue theorem,

$${\oint }_{\text{Γ}}\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{R}_m(z)dz=2\pi i\sum _{k=1}^n\underset{z=z_k}{\text{Res}}\left(\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{R}_m(z)\right),$$ (4.12)

where z₁, …, z_n are the poles inside Γ. For brevity, let the portion of the integrand in (4.12) corresponding to the product of the spherical-wave term and the $1/\sqrt{1-z^2}$ term be represented by the function H(z), given by

$$H(z)=\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}.$$ (4.13)

If Γ is a closed contour which includes the interval [−1, 1], we have that

$$G_m\approx -\underset{\text{Γ}\backslash \left[-1,1\right]}{\int }H(z)R_m(z)dz+2\pi i\sum _{k=1}^n\underset{z=z_k}{\text{Res}}(H(z)R_m(z)),$$ (4.14)

where we have substituted formula (4.13) to abbreviate the integral. We select Γ \ [−1, 1] to be Gustafsson’s contours within the outer ellipse, E_ρ², with both segments connected by a short segment ${\gamma }_c\subset ℂ\backslash E_{{\rho }^2}^o$ (see Figure 6). Substituting this choice of Γ \ [−1, 1] into (4.14), we have

$$G_m\approx -\underset{{\gamma }_1\cap E_{{\rho }^2}^o}{\int }H(z)R_m(z)dz-\underset{{\gamma }_2\cap E_{{\rho }^2}^o}{\int }H(z)R_m(z)dz-\underset{{\gamma }_c}{\int }H(z)R_m(z)dz+2\pi i\sum _{k=1}^n\underset{z=z_k}{\text{Res}}(H(z)R_m(z)),$$ (4.15)

where γ₁ and γ₂ are Gustafsson’s contours as described in Section 3.1.1, and $E_{{\rho }^2}^o$ is the interior of the Bernstein ellipse introduced earlier (see Figure 6). We split the integral corresponding to the γ₁ contour into

$$\underset{{\gamma }_1\cap E_{{\rho }^2}^o}{\int }H(z)R_m(z)dz=\underset{{\gamma }_1\cap E_{\rho }^o}{\int }H(z)R_m(z)dz+\underset{({\gamma }_1\cap E_{{\rho }^2}^o)\backslash E_{\rho }^o}{\int }H(z)R_m(z)dz.$$ (4.16)

Fig. 6: Contours used in formula (4.15) in the z = cos ϕ-plane.

The interior Bernstein ellipse is denoted by E_ρ and is drawn in blue. The exterior ellipse is denoted by $E_{{\rho }^2}$. Gustafsson’s contours within the exterior ellipse are denoted by ${\gamma }_2\cap E_{{\rho }^2}^o$ and ${\gamma }_2\cap E_{{\rho }^2}^o$ and are drawn in red. The contour ${\gamma }_c\subset ℂ\backslash E_{{\rho }^2}^o$, connecting the γ₁ and γ₂ segments, is drawn in green. The intersection of γ₁ with E_ρ is denoted by p₁, and the intersection of γ₂ with E_ρ is denoted by p₂.

Recall that by formula (4.7), R_m(z) ≈ T_m(z) for $z\in {\gamma }_1\cap E_{\rho }^o$ and for $z\in {\gamma }_2\cap E_{\rho }^o$. Also, recall that by formula (4.8), R_m(z) ≈ ϵ for $z\in {\gamma }_1\backslash E_{\rho }^o$ and for $z\in {\gamma }_2\backslash E_{\rho }^o$. Combining (4.7) and (4.8) with (4.16), we arrive at a formula for the integral over the γ₁ contour within the interior of E_ρ², given by

$$\underset{{\gamma }_1\cap E_{{\rho }^2}^o}{\int }H(z)R_m(z)dz\approx \underset{{\gamma }_1\cap E_{\rho }^o}{\int }H(z)T_m(z)dz.$$ (4.17)

Likewise, the formula for the integral over the γ₂ contour within the interior of E_ρ² is

$$\underset{{\gamma }_2\cap E_{{\rho }^2}^o}{\int }H(z)R_m(z)dz\approx \underset{{\gamma }_2\cap E_{\rho }^o}{\int }H(z)T_m(z)dz.$$ (4.18)

We also observe that, due to formula (4.9), the integral corresponding to γ_c evaluates to zero. We now substitute our formulae for the ${\gamma }_1\cap E_{{\rho }^2}^o$, ${\gamma }_2\cap E_{{\rho }^2}^o$, and γ_c contour integrals into (4.15) to arrive at

$$\begin{array}{l}{G}_m\approx -\underset{{\gamma }_1\cap E_{\rho }^o}{\int }\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{T}_m(z)dz-\underset{{\gamma }_2\cap E_{\rho }^o}{\int }\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{T}_m(z)dz\\ +2\pi \text{i}\sum _{k=1}^n\underset{z=z_k}{\text{Res}}\left(\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{R}_m(z)\right).\end{array}$$ (4.19)

4.1.2. Interpretation of the Residues in Formula (4.19) as a Quadrature Formula over the Contour C_ρ

By Cauchy’s integral theorem,

$$G_m\approx -\underset{{\gamma }_1\cap E_{\rho }^o}{\int }H(z)T_m(z)dz-\underset{{\gamma }_2\cap E_{\rho }^o}{\int }H(z)T_m(z)dz-\underset{C_{\rho }}{\int }H(z)T_m(z)dz,$$ (4.20)

where γ₁, γ₂, $E_{\rho }^o$, and C_ρ are described in Section 4.1.

Subtracting (4.19) from (4.20), and rearranging, we arrive at a formula for the integral over the C_ρ contour,

$$-\underset{C_{\rho }}{\int }\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{T}_m(z)dz\approx 2\pi i\sum _{k=1}^n\underset{z=z_k}{\text{Res}}\left(\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{R}_m(z)\right).$$ (4.21)

Recall from Section 4.1 that

$$R_m(z)=\frac{1}{2\pi i}\underset{E\rho \backslash C_{\rho }}{\int }\frac{T_m(v)}{v-z}dv+\frac{1}{2\pi i}\sum _{i=1}^n\frac{T_m(v_i)}{v_i-z}d{v}_i{w}_i,$$ (4.22)

where $v_i=E_{\rho }({\theta }_i)$, $d{v}_i=E_{\rho }^{\prime }({\theta }_i)$, and θ₁, … , θ_n, w₁, … , w_n are nodes and weights of the quadrature formula constructed in (4.4). Thus, the residues in (4.21) at the points z₁, … , z_n correspond to residues at the points v₁, … , v_n and

$$2\pi i\sum _{i=1}^n\underset{z=z_i}{\text{Res}}\left(\frac{e^{-i\kappa }\sqrt{1-\alpha z}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{R}_m(z)\right)=-\sum _{i=1}^n\frac{e^{-i\kappa \sqrt{1-\alpha v_i}}}{\sqrt{1-\alpha v_i}\sqrt{1-v_i^2}}{T}_m(v_i)d{v}_i{w}_i.$$ (4.23)

Substituting (4.23) into (4.21), we have that

$$\sum _{i=1}^n\frac{e^{-i\kappa \sqrt{1-\alpha v_i}}}{\sqrt{1-\alpha v_i}\sqrt{1-v_i^2}}{T}_m(v_i)d{v}_i{w}_i\approx \underset{C_{\rho }}{\int }\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{T}_m(z)dz,$$ (4.24)

which resembles a quadrature formula for the contour integral on C_ρ. Substituting formula (4.24) into formula (4.19), we arrive at

$$\begin{array}{l}{G}_m\approx \underset{{\gamma }_1\cap E_{\rho }^o}{\int }\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{T}_m(z)dz-\underset{{\gamma }_2\cap E_{\rho }^o}{\int }\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{T}_m(z)dz\\ -\sum _{i=1}^n\frac{e^{-i\kappa \sqrt{1-\alpha v_i}}}{\sqrt{1-\alpha v_i}\sqrt{1-v_i^2}}{T}_m(v_i)d{v}_i{w}_i,\end{array}$$ (4.25)

where $v_i=E_{\rho }({\theta }_i)$, $d{v}_i=E_{\rho }^{\prime }({\theta }_i)$, and θ₁, … , θ_n, w₁, … , w_n are the nodes and weights of the quadrature formula constructed in (4.4).

4.2. Evaluation of the Integral on Gustafsson’s Contours when α ≈ 1

Recall from Section 4.1.2 that the formula for the mth modal Green’s function is

$$\begin{array}{r}\hfill G_m\approx -\underset{{\gamma }_1\cap E_{\rho }^o}{\int }\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{T}_m(z)dz-\underset{{\gamma }_2\cap E_{\rho }^o}{\int }\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}{T}_m(z)dz\\ \hfill -\sum _{i=1}^n\frac{e^{-i\kappa \sqrt{1-\alpha v_i}}}{\sqrt{1-\alpha v_i}\sqrt{1-v_i^2}}{T}_m(v_i)d{v}_i{w}_i,\end{array}$$ (4.26)

where $v_i=E_{\rho }({\theta }_i)$, $d{v}_i=E_{\rho }^{\prime }({\theta }_i)$, and θ₁, … , θ_n, w₁, … , w_n are the nodes and weights of the quadrature formula constructed in (4.4). Recall also from Section 3.8 that the integrals in (4.26) can be written as

$$\begin{array}{r}\hfill G_m\approx \frac{4}{\sqrt{\alpha }}{\int }_0^{{\tau }_1}\frac{F_1(\tau )}{\sqrt{{\tau }^2+2i{\beta }_{-}}}d\tau -\frac{4i}{\sqrt{\alpha }}{\int }_0^{{\tau }_2}\frac{F_2(\tau )}{\sqrt{{\tau }^2+2i{\beta }_{+}}}d\tau \\ \hfill -\sum _{i=1}^n\frac{e^{-i\kappa \sqrt{1-\alpha v_i}}}{\sqrt{1-\alpha v_i}\sqrt{1-v_i^2}}{T}_m(v_i)d{v}_i{w}_i,\end{array}$$ (4.27)

where F₁(τ) and F₂(τ) are smooth functions corresponding to the γ₁ and γ₂ contours, respectively (see Section 3.8, formulae (3.92) and (3.96)), τ₁ and τ₂ are positive parameters such that γ₁(τ₁) and γ₂(τ₂) are the intersections of γ₁ and γ₂ with E_ρ, respectively, and

$${\beta }_{-}=\sqrt{1/\alpha -1},{\beta }_{+}=\sqrt{1/\alpha +1}.$$ (4.28)

When $\alpha \approx 1$, the parameter ${\beta }_{+}\approx 2$, meaning that the integrand in (4.27) corresponding to γ₂ remains a smooth function of τ for all values of 0 < α < 1, and can be evaluated efficiently with a Gauss-Legendre quadrature. In contrast, when $\alpha \approx 1$, the parameter ${\beta }_{-}\approx 0$. Consequently, for $\alpha \approx 1$, the integrand in (4.27) corresponding to the γ₁ contour has a singularity resembling 1/τ at τ = 0.

4.2.1. Evaluation of the Integral on the Contour γ₁ when $\alpha \approx 1$

We integrate along the contour γ₁ using the following procedure. Observe that for τ sufficiently large, the integrand is smooth. Thus we split the integral into two parts,

$$\frac{-4}{\sqrt{\alpha }}{\int }_0^{{\tau }_1}\frac{F_1(\tau )}{\sqrt{{\tau }^2+i{\beta }_{-}}}d\tau =\frac{-4}{\sqrt{\alpha }}{\int }_0^{{\tau }_0}\frac{F_1(\tau )}{\sqrt{{\tau }^2+i{\beta }_{-}}}d\tau +\frac{-4}{\sqrt{\alpha }}{\int }_{{\tau }_0}^{{\tau }_1}\frac{F_1(\tau )}{\sqrt{{\tau }^2+i{\beta }_{-}}}d\tau .$$ (4.29)

The integral corresponding to the interval [τ₀, τ₁] can be efficiently computed using a Gauss-Legendre quadrature. The integral corresponding to the interval [0, τ₀] is evaluated with a specialized recurrence based on the technique used by Gustafsson (see [9], Section 4.2), described below. Recall that F₁(τ) is smooth, given by the formula

$$F_1(\tau )=\frac{e^{-i\kappa }\sqrt{1-\alpha {\tilde{\gamma }}_1(\tau )}}{\sqrt{1+{\tilde{\gamma }}_1(\tau )}}{T}_m({\tilde{\gamma }}_1(\tau )),$$ (4.30)

where ${\tilde{\gamma }}_1(\tau )$ is

$${\tilde{\gamma }}_1(\tau )={\tau }^4+i{\beta }_{-}{\tau }^2+1.$$ (4.31)

We expand F₁(τ) in k terms of its Taylor series about the point τ = 0, given by the formula

$$F_1(\tau )\approx \sum _{n=0}^k{a}_n{\tau }^n.$$ (4.32)

We compute the coefficients a_n by first forming a Legendre expansion of F₁(τ) on the interval [0, τ₀], and then repeatedly spectrally differentiating this expression as described in Section 2.8.

We substitute (4.32) into the integral corresponding to the interval [0, τ₀] in formula (4.29), resulting in

$$\begin{array}{l}\frac{-4}{\sqrt{\alpha }}{\int }_0^{{\tau }_0}\frac{F_1(\tau )}{\sqrt{{\tau }^2+2i{\beta }_{-}}}d\tau \approx \frac{-4}{\sqrt{\alpha }}{\int }_0^{{\tau }_0}\frac{{\sum }_{n=0}^k{a}_n{\tau }^n}{\sqrt{{\tau }^2+i{\beta }_{-}}}d\tau \\ =\frac{-4}{\sqrt{\alpha }}\sum _{n=0}^k{a}_n{\int }_0^{{\tau }_0}\frac{{\tau }^n}{\sqrt{{\tau }^2+2i{\beta }_{-}}}d\tau .\end{array}$$ (4.33)

Recall from Section 2.6 that the integral of τⁿ divided by $\sqrt{a{\tau }^2+b}$ has the recurrence relation

$$\int \frac{{\tau }^n}{\sqrt{a{\tau }^2+b}}d\tau =\frac{{\tau }^{n-1}\sqrt{a{\tau }^2+b}}{na}-\frac{{(n-1)}^b}{na}\int \frac{{\tau }^{n-2}}{\sqrt{a{\tau }^2+b}}d\tau ,$$ (4.34)

for $n\ge 2$, where the bases cases have the formulae

$$\int \frac{1}{\sqrt{a{\tau }^2+b}}d\tau =\frac{1}{\sqrt{a}}\ln \left(\tau \sqrt{a}+\sqrt{a{\tau }^2+b}\right),\int \frac{\tau }{\sqrt{a{\tau }^2+b}}d\tau =\frac{\sqrt{a{\tau }^2+b}-\sqrt{b}}{a}.$$ (4.35)

This recurrence is known to be stable when $|b|<|a|$. We observe that ${\beta }_{-}\ll 1$ when $\alpha \approx 1$, meaning that the recurrence given by (4.34) is stable when applied to (4.33).

Remark 4.1.

The same treatment is used to evaluate G_s,m, with one minor change. After making the variable substitution z = cos ϕ, the singular term $\sqrt{{\tau }^2+i{\beta }_{-}}$ appears in the numerator rather than the denominator, which is handled by the recurrence described in Section 2.7.

4.3. Construction of the Quadratures to Evaluate the Integral over the Contour C_ρ

Recall from Section 4.1 that our approximation R_m(z) of the mth order Chebyshev polynomial, T_m(z), has the formula

$$R_m(z)=\frac{1}{2\pi i}\underset{E_{\rho }\backslash C_{\rho }}{\int }\frac{T_m(v)}{v-z}dv+\frac{1}{2\pi i}\sum _{i=1}^n\frac{T_m(v_i)}{v_i-z}d{v}_i{w}_i,$$ (4.36)

where $v_i=E_{\rho }({\theta }_i)$, $d{v}_i=E_{\rho }^{\prime }({\theta }_i)$, C_ρ is the region of the Bernstein ellipse E_ρ between the contours γ₁ and γ₂, and θ₁, … , θ_n and w₁, … , w_n are the nodes and weights of a quadrature formula such that

$$\frac{1}{2\pi i}\underset{C_{\rho }}{\int }\frac{T_m(v)}{v-z}dv\approx \frac{1}{2\pi i}\sum _{i=1}^n\frac{T_m(v_i)}{v_i-z}d{v}_i{w}_i,$$ (4.37)

for z ∈ [−1, 1], $z\in {\gamma }_1\cap E_{{\rho }^2}^o$, $z\in {\gamma }_2\cap E_{{\rho }^2}^o$ and $z\in ℂ\backslash E_{{\rho }^2}^o$ where $E_{{\rho }^2}^o$ is the interior of $E_{{\rho }^2}$ (see Figure 5). Recall also from Section 4.1 that, by Cauchy’s integral formula,

$$T_m(z)={\oint }_{E_{\rho }}\frac{T_m(v)}{v-z}dv,$$ (4.38)

for $z\in E_{\rho }$, where E_ρ is the Bernstein ellipse described in Section 4.1. Finally, recall from Section 3.2.2 that, if $\rho =M^{1/m}$, then $|T_m(z)|<M$ for all $z\in E_{\rho }^o$. For the sake of simplicity, we first assume that M = e, and then consider the case for general M in Sections 4.3.3 and 4.3.4.

We summarize the contours on which R_m(z) approximates T_m(z) pointwise by stating that

$$\left|{\int }_{C_{\rho }}\frac{T_m(v)}{v-z}dv-\sum _{i=1}^n\frac{T_m(v_i)d{v}_i{w}_i}{v_i-z}\right|<ϵ,$$ (4.39)

for all z ∈ [−1, 1], $z\in {\gamma }_1\cap E_{{\rho }^2}^o$, $z\in {\gamma }_2\cap E_{{\rho }^2}^o$, and $z\in ℂ\backslash E_{{\rho }^2}^o$, where $E_{{\rho }^2}^o$ is the interior of $E_{{\rho }^2}$. Let $p_1\in ℂ$ and $p_2\in ℂ$ denote the intersections of γ₁ and γ₂ with E_ρ, respectively (see Figure 7). Let ${\tilde{C}}_{\rho }\subset C_{\rho }$ denote the portion of C_ρ no closer than 1/m² from the points p₁ and p₂, defined by

$${\tilde{C}}_{\rho }=\left\{z:z\in C_{\rho },|p_1-z|>1/m^2,|p_2-z|>1/m^2\right\}.$$ (4.40)

Fig. 7: Splitting of the Bernstein ellipse into ${\tilde{C}}_{\rho }$ and $C_{\rho }\backslash {\tilde{C}}_{\rho }$ based on proximity to Gustafsson’s contours in the cos ϕ-plane.

Gustafsson’s contours are denoted as γ₁ and γ₂, and drawn in green. The points where γ₁ and γ₂ intersect E_ρ are denoted as p₁ and p₂, respectively. The region of the ellipse bounded by the intersections p₁ and p₂ defines the segment C_ρ. The segment of C_ρ not close to the points p₁ and p₂ is denoted as ${\tilde{C}}_{\rho }$ and drawn in red. The segments of C_ρ which are close to the points p₁ and p₂ are denoted as $C_{\rho }\backslash {\tilde{C}}_{\rho }$ and drawn in blue. The remainder of the ellipse is denoted as $E_{\rho }\backslash C_{\rho }$ and drawn in black.

We split the integral in (4.37) into integrals over ${\tilde{C}}_{\rho }$ and $C_{\rho }\backslash {\tilde{C}}_{\rho }$, arriving at

$${\int }_{C_{\rho }}\frac{T_m(v)}{v-z}dv={\int }_{{\tilde{C}}_{\rho }}\frac{T_m(v)}{v-z}dv+{\int }_{C_{\rho }\backslash {\tilde{C}}_{\rho }}\frac{T_m(v)}{v-z}dv.$$ (4.41)

The domain of integration ${\tilde{C}}_{\rho }$ is relatively well-separated from all values of z on which the quadrature rule in (4.39) must hold. In contrast, the domain of integration $C_{\rho }\backslash {\tilde{C}}_{\rho }$ is not well-separated.

4.3.1. Quadratures for the Portion of C_ρ Away From Gustafsson’s Contours

Recall from formula (4.40) that, by construction, ${\tilde{C}}_{\rho }$ is separated from γ₁ and γ₂ by 1/m², and that the Bernstein ellipse E_ρ is separated from the interval [−1, 1] by $\approx 1/m^2$ near ±1. If we use the smooth change of variables ϕ described in Section 3.3 to transform the contour ${\tilde{C}}_{\rho }$ into a subset [a, b] of the interval [−1, 1], and then perform an affine transformation A to map [a, b] to [−1, 1], it is easy to see that any point z on [−1, 1], ${\gamma }_1\cap E_{{\rho }^2}^o$, ${\gamma }_2\cap E_{{\rho }^2}^o$, or $ℂ\backslash E_{{\rho }^2}^o$, will be mapped to a point $A\circ \varphi (z)$ on or outside of the Bernstein ellipse E_ρ. Note that the mapping ϕ is smooth, and that both ϕ and A approach the identity map for large m. It follows that

$${\int }_{{\tilde{C}}_{\rho }}\frac{T_m(v)}{v-z}dv\approx {\int }_{-1}^1\frac{p(u)}{u-A\circ \varphi (z)}du,$$ (4.42)

for some polynomial p(u) of order ≈ m, where $A\circ \varphi (z)$ is on or outside of the Bernstein ellipse E_ρ. Thus, Corollary 3.3 tells us that the right-hand side of (4.42) is well-approximated by an O(m) point Gauss-Legendre quadrature, since the integrand of the right-hand side is well-approximated by a Chebyshev expansion with O(m) terms. From the fact that the change-of-variables mapping $u=A\circ \varphi (v)$ is smooth and nearly equal to the identity for large m, it follows that

$${\int }_{{\tilde{C}}_{\rho }}\frac{T_m(v)}{v-z}dv$$ (4.43)

is also well-approximated by an O(m) point Gauss-Legendre quadrature, for all z ∈ [−1, 1], $z\in {\gamma }_1\cap E_{{\rho }^2}^o$, $z\in {\gamma }_2\cap E_{{\rho }^2}^o$, and $z\in ℂ\backslash E_{{\rho }^2}^o$. Specifically, Corollary 3.3 indicates (after a mapping by $A\circ \varphi$, recalling that $A\circ \varphi (p_1)=1$ and $A\circ \varphi (p_2)=-1$) that there exists an O(m) point quadrature such that

$$\left|{\int }_{{\tilde{C}}_{\rho }}\frac{T_m(v)}{v-z}dv-\sum _{i=1}^n\frac{T_m(v_i)d{v}_i{w}_i}{v_i-z}\right|\lesssim ϵ,$$ (4.44)

when z is well-separated from p₁ and p₂, and that

$$\left|{\int }_{{\tilde{C}}_{\rho }}\frac{T_m(v)}{v-z}dv-\sum _{i=1}^n\frac{T_m(v_i)d{v}_i{w}_i}{v_i-z}\right|\lesssim \frac{ϵ}{|\sqrt{(z-p_1)(z-p_2)}|},$$ (4.45)

when $z\approx p_1$ or $z\approx p_2$ , for all z ∈ [−1, 1], $z\in {\gamma }_1\cap E_{{\rho }^2}^o$, $z\in {\gamma }_2\cap E_{{\rho }^2}^o$, and $z\in ℂ\backslash E_{{\rho }^2}^o$.

Recall that, in the integrands of (4.11) and (4.14), R_m(z) is multiplied by the term H(z), which is a bounded function over the interval [−1, 1] and all of the relevant contours. Thus, for formula (4.25) to be an accurate approximation to G_m, the integral (4.43) only needs to be approximated to within an error ϵ in the L¹-sense, over z ∈ [−1, 1], $z\in {\gamma }_1\cap E_{{\rho }^2}^o$, $z\in {\gamma }_2\cap E_{{\rho }^2}^o$, and $z\in ℂ\backslash E_{{\rho }^2}^o$. From formulae (4.44) and (4.45), we see that this is indeed the case. Thus, an O(m) point quadrature for (4.43) is sufficient to compute G_m with no loss of accuracy.

4.3.2. Quadratures for the Portions of C_ρ Near Gustafsson’s Contours

In this section, we present the construction of a quadrature rule which approximates the contour integral

$${\int }_{C_{\rho }\backslash {\tilde{C}}_{\rho }}\frac{T_m(v)}{v-z}dv,$$ (4.46)

for z ∈ [−1, 1], $z\in {\gamma }_1\cap E_{{\rho }^2}^o$, $z\in {\gamma }_2\cap E_{{\rho }^2}^o$, and $z\in ℂ\backslash E_{{\rho }^2}^o$, where $E_{{\rho }^2}^o$ is the interior of $E_{{\rho }^2}$. Since [−1, 1] is well-separated from $C_{\rho }\backslash {\tilde{C}}_{\rho }$, we focus only on $z\in ({\gamma }_1\cap E_{{\rho }^2})\cup ({\gamma }_2\cap E_{{\rho }^2})$. Observe that $C_{\rho }\backslash {\tilde{C}}_{\rho }$ consists of two disjoint segments (see Figure 7). One segment of $C_{\rho }\backslash {\tilde{C}}_{\rho }$ contains the point p₁, which denotes the intersection of γ₁ (associated with z = 1) with E_ρ, the other segment of $C_{\rho }\backslash {\tilde{C}}_{\rho }$ contains the point p₂, which denotes the intersection of γ₂ (associated with z = −1) with E_ρ. We denote the points where $C_{\rho }\backslash {\tilde{C}}_{\rho }$ ends and ${\tilde{C}}_{\rho }$ begins by ${\tilde{p}}_1$ and ${\tilde{p}}_2$, where ${\tilde{p}}_1$ is the point closer to p₁, and ${\tilde{p}}_2$ is the point closer to p₂. We analyze the segment of $C_{\rho }\backslash {\tilde{C}}_{\rho }$ near p₂, with the understanding that the Bernstein ellipse is symmetric and an identical argument applies to the segment of $C_{\rho }\backslash {\tilde{C}}_{\rho }$ near p₁. We define B_δ as

$$B_{\delta }=\left\{z:|\text{Arg}(z)|\ge \frac{\pi }{6},|z|\le \delta \right\}.$$ (4.47)

Recall from Section 3.6 that γ₂, in the vicinity of p₂, always lies in $p_2+{\widehat{B}}_{1/m^2}$, where ${\widehat{B}}_{1/m^2}$ is a rotated version of B_1/m², such that the opening in B_1/m² is bisected by C_ρ (see Figure 8). We note that, when z is in one of the regions of interest but outside of B_1/m², it is sufficiently well-separated from the domain of integration $C_{\rho }\backslash {\tilde{C}}_{\rho }$, so that a Gauss-Legendre quadrature with O(1) terms accurately approximates the integral (4.46). Hence, for the remainder of this section we exclusively focus on developing a quadrature rule which approximates (4.46) for $z\in p_2+{\widehat{B}}_{1/m^2}$.

Fig. 8: Region $p_2+{\widehat{B}}_{1/m^2}$ in which the quadrature in formula (4.37) must accurately evaluate the integral over the contour $C_{\rho }\backslash {\tilde{C}}_{\rho }$ for $z\in {\gamma }_2$.

The values of z for which the quadrature must be accurate are the points in the interior of the shaded region denoted by $p_2+{\widehat{B}}_{1/m^2}$, whose boundary is drawn in red. Note that the angle that ${\widehat{B}}_{1/m^2}$ makes with $C_{\rho }\backslash {\tilde{C}}_{\rho }$ is π/6 from above and π/6 from below. Gustafsson’s contour beginning at z = −1 is denoted γ₂, and is drawn in green. The intersection of γ₂ with the Bernstein Ellipse is denoted by p₂. The Bernstein ellipse, E_ρ, is drawn as three contiguous segments. The left segment, colored black, corresponds to the part of the Bernstein ellipse which is not in C_ρ, and is denoted $E_{\rho }\backslash C_{\rho }$. The middle segment, denoted $C_{\rho }\backslash {\tilde{C}}_{\rho }$, is drawn in blue. The right segment, colored black, corresponds to ${\tilde{C}}_{\rho }$. The point where $C_{\rho }\backslash {\tilde{C}}_{\rho }$ ends and where ${\tilde{C}}_{\rho }$ begins is denoted by $\widetilde{p_2}$.

For convenience, we rotate, translate, and rescale $C_{\rho }\backslash {\tilde{C}}_{\rho }$ and $p_2+{\widehat{B}}_{1/m^2}$ (see Figure 9), so that the segment $C_{\rho }\backslash {\tilde{C}}_{\rho }$ is approximated by the interval [0, 1] (i.e., it is translated by p₂, rotated, and scaled by a factor of m²). Likewise, ${\stackrel{⌵}{\gamma }}_2$ represents a similarly translated, rotated, and scaled copy of γ₂. Note that we associate p₂ with the point x = 0 and the point ${\tilde{p}}_2$ with x = 1 (see Figure 9). Consider a quadrature rule x_i, … , x_n and w₁, … , w_n such that

$$\left|{\int }_0^1\frac{\rho (x)}{x-z}dx-\sum _{i=1}^n\frac{\rho (x_i)}{x_i-z}{w}_i\right|<ϵ$$ (4.48)

for all z ∈ B₁, where ρ(x) is smooth. Such a quadrature rule, if used to approximate (4.46), will be accurate to precision ϵ, for all $z\in p_2+{\widehat{B}}_{1/m^2}$. However, recall that, in the integrands of (4.11) and (4.14), R_m(z) is multiplied by the term H(z), which is a bounded function over the interval [−1, 1] and all of the relevant contours. Thus, for formula (4.25) to be an accurate approximation to G_m, the integral (4.46) only needs to be approximated to within an error ϵ in the L¹-sense, over $z\in {\gamma }_1\cap E_{{\rho }^2}^o$ and $z\in {\gamma }_2\cap E_{{\rho }^2}^o$. In the rotated and rescaled coordinates, this means that our quadrature rule for the integral in (4.48) must be accurate to within an error ϵ in the L¹-sense, over $z\in {\stackrel{⌵}{\gamma }}_2\subset B_1$, where ${\stackrel{⌵}{\gamma }}_2$ starts at $z_1\in \partial B_1$ and ends at $z_2\in \partial B_1$, with $|z_1|=|z_2|=1$ (see Figure 9). Thus, the left-hand side of (4.48) has to be bounded by ϵ only in $L^1({\stackrel{⌵}{\gamma }}_2\cap B_1)$, meaning that the integral and the quadrature approximation in (4.48) can disagree on a set of measure ϵ.

Fig. 9: Rescaling and rotation of region of interest depicted in Figure 8.

Region B₁ is a translation, rotation, and rescaling of ${\widehat{B}}_{1/m^2}$ such that B₁ has radius 1. The values of z for which the quadrature must be accurate are the points in the interior of the shaded region B₁.

This allows us to relax (4.48) to the condition

$$\left|{\int }_0^1\frac{\rho (x)}{x-z}dx-\sum _{i=1}^n\frac{\rho (x_i)}{x_i-z}{w}_i\right|<\frac{ϵ}{\left|z\right|},$$ (4.49)

for z ∈ B₁. Thus, for each δ > 0, the quadrature is accurate to within an error $ϵ/\delta$ for all $z\in B_{\delta }$. Since the length of ${\stackrel{⌵}{\gamma }}_2\cap B_{\delta }$ is on the order δ, the corresponding L¹ error in the quadrature rule is $\delta \cdot ϵ/\delta =ϵ$.

We can construct this quadrature by first sampling $z_i\in \partial B_1$, and then computing a generalized Gaussian quadrature (see [4]) on x ∈ [0, 1], where (4.49) is enforced on all the sampled z_i’s. By Cauchy’s theorem, if (4.49) holds on ∂B₁, then it will also hold on B₁. However, this still results in a quadrature rule with several hundred nodes. It turns out that far fewer nodes can be used, due to the following observation.

Recall that, in the integrands of (4.11) and (4.14), R_m(z) is multiplied by the term H(z), given by

$$H(z)=\frac{e^{-i\kappa \sqrt{1-\alpha z}}}{\sqrt{1-\alpha z}\sqrt{1-z^2}}.$$ (4.50)

Because H(z) is smooth near z = p₂, we only need that

$$\left|{\int }_{{\stackrel{⌵}{\gamma }}_2\cap B_1}\sigma (z){\int }_0^1\frac{\rho (x)}{x-z}dxdz-{\int }_{{\stackrel{⌵}{\gamma }}_2\cap B_1}\sigma (z)\sum _{i=1}^n\frac{\rho (x_i)}{x_i-z}{w}_{i}dz\right|<ϵ,$$ (4.51)

for all sufficiently smooth functions σ(z). Since σ(z) is smooth, it can be represented by a Taylor series of a small order k, so that

$$\sigma (z)\approx \sum _{j=0}^k{a}_j{z}^j.$$ (4.52)

Thus, inequality (4.51) becomes

$$\left|{\int }_{{\stackrel{⌵}{\gamma }}_2\cap B_1}{z}^j{\int }_0^1\frac{\rho (x)}{x-z}dxdz-{\int }_{{\stackrel{⌵}{\gamma }}_2\cap B_1}{z}^j\sum _{i=1}^n\frac{\rho (x_i)}{x_i-z}{w}_{i}dz\right|<ϵ,$$ (4.53)

for each j = 0, 1, … , k. Exchanging the order of integration,

$$\left|{\int }_0^1\rho (x){\int }_{{\stackrel{⌵}{\gamma }}_2\cap B_1}\frac{z^j}{x-z}dzdx-\sum _{i=1}^n\rho (x_i){\int }_{{\stackrel{⌵}{\gamma }}_2\cap B_1}\frac{z^j}{x_i-z}dz,w_i\right|<ϵ,$$ (4.54)

Recall from Section 2.9 that

$${\int }_{{\stackrel{⌵}{\gamma }}_2\cap B_1}\frac{z^j}{x-z}dz=\varphi (x)+\psi (x)\log \left(\frac{x-z_1}{x-z_2}\right),$$ (4.55)

where ϕ and ψ are polynomials of order j, and z₁ and z₂ are the endpoints of ${\stackrel{⌵}{\gamma }}_2\cap B_1$. Due to the geometry of B₁, we have that

$$|z_1-x|\ge \frac{1}{2},|z_2-x|\ge \frac{1}{2},$$ (4.56)

for all x ∈ [0, 1]. We also observe that the branch cut of

$$\log \left(\frac{x-z_1}{x-z_2}\right)$$ (4.57)

does not intersect the interior of [0, 1], so (4.57) is smooth on [0, 1]. Since ρ(x) is smooth and (4.57) is smooth, we observe that the integrand in (4.54), given by

$$\rho (x){\int }_{{\stackrel{⌵}{\gamma }}_2\cap B_1}\frac{z^j}{x-z}dz,$$ (4.58)

is a smooth function of x for x ∈ [0, 1]. Hence, a Gauss-Legendre quadrature with O(1) points will satisfy (4.54).

Because (4.54) is satisfied, (4.51) is satisfied, and so the contour deformation argument presented in Section 4.1.1 can be carried out without change, using a Gauss-Legendre quadrature with O(1) points on $C_{\rho }\backslash {\tilde{C}}_{\rho }$ and O(m) points on ${\tilde{C}}_{\rho }$ (see Section 4.3.1).

4.3.3. The Error in the Approximation R_m(z)

In order to derive the approximation (4.25) to the Green’s function G_m, we approximated the Chebyshev polynomial T_m(z) by the function R_m(z), defined by (4.22) (see also (4.11) and (4.14)). Recall from Section 3.2.2 that, when $\rho =M^{1/m}$, we have that $|T_m(z)|\approx M$ for all z ∈ E_ρ. If the formula for R_m(z) is evaluated numerically, then the integrand and summand in that formula will both have size approximately M, while the sum, R_m(z), will have size approximately one for z ∈ [−1, 1]. Thus, due to cancellation error, $|R_m(z)-T_m(z)|\approx Mϵ$ for all z ∈ [−1, 1], where ϵ is equal to machine precision. This means that, for $\rho =M^{1/m}$, the approximation for G_m given by formula (4.25) has an error of M ϵ.

4.3.4. The Number of Quadrature Nodes on ${\tilde{C}}_{\rho }$

In Section 4.3.1, we showed that O(m) nodes are required on ${\tilde{C}}_{\rho }$ by pointing out that the distance from ${\tilde{C}}_{\rho }$ to the nearest pole in the integrand of (4.43) is $\approx 1/m^2$ at its endpoints and ≈ 1/m in the middle. We then used Corollary 3.3 to state that the number of terms required to expand the integrand of (4.43) in Chebyshev polynomials is O(m), which means that O(m) nodes are needed in the corresponding quadrature formula.

In fact, Corollary 3.3 provides a quantitative bound on how many terms are required. We observe that $|T_m(z)|<M$ for all $z\in E_{\rho }^o$, and that the minimum attainable error in the evaluation of (4.43) is M ϵ. Replacing ϵ with M ϵ and L with M in Corollary 3.3, we find that the number of Chebyshev expansion terms required to approximate the integrand of (4.43) to precision M ϵ is

$$k_0\approx m(-\log (ϵ/2))/\log (M).$$ (4.59)

Thus, $O(k_0/2)=O(m)$ Gauss-Legendre nodes are required on ${\tilde{C}}_{\rho }$. The parameter M allows for a tradeoff between the number of quadrature nodes and the error in the approximation. This is illustrated for various values of M in Tables 1 and 2, both for $ϵ\approx {10}^{-16}$ and $ϵ\approx {10}^{-34}$, respectively.

Table 1: The required number of Gauss-Legendre nodes on ${\tilde{C}}_{\rho }$ to approximate (4.43), in double precision.

In this table, k₀/2 is the required number of nodes, $ϵ={10}^{-16}$, and $\rho =M^{1/m}$.

M	Mϵ	k 0	k0/2
10	10−15	16.3m	8.15m
100	10−14	8.15m	4.08m
103	10−13	5.43m	2.72m
106	10−10	2.72m	1.36m
109	10−7	1.81m	0.91m
1012	10−4	1.34m	0.68m

Table 2: The required number of Gauss-Legendre nodes on ${\tilde{C}}_{\rho }$ to approximate (4.43), in quadruple precision.

In this table, k₀/2 is the required number of nodes, $ϵ={10}^{-34}$, and $\rho =M^{1/m}$.

M	Mϵ	k 0	k0/2
10	10−33	34.3m	17.15m
100	10−32	17.15m	8.58m
103	10−31	11.43m	5.72m
106	10−28	5.72m	2.86m
109	10−25	3.81m	1.91m
1012	10−22	2.86m	1.43m
1015	10−19	2.29m	1.14m

Remark 4.2.

In Section 4.3.2, we demonstrate that only O(1) nodes are required on $C_{\rho }\backslash {\tilde{C}}_{\rho }$. We observe that, in practice, we can place a single O(m) Gauss-Legendre quadrature with k₀/2 nodes on the entire contour C_ρ, rather than placing two O(1) quadratures on each part of $C_{\rho }\backslash {\tilde{C}}_{\rho }$ and one O(m) quadrature on ${\tilde{C}}_{\rho }$. We also note that, in practice, the minimum number of quadrature nodes required to achieve the accuracy M ϵ matches the estimates in Tables 1 and 2 very closely.

Remark 4.3.

We note that integrating (4.43) with an O(m) point Gauss-Legendre quadrature with respect to arc length requires an arc length parameterization of the contour C_ρ, which is a section of the Bernstein ellipse E_ρ. Such arc length parameterizations are given by incomplete elliptic integrals, and are not available analytically, although accurate and efficient algorithms are available. Since the precise locations of the quadrature nodes on C_ρ depend on the intersection points p₁ and p₂ of Gustafsson’s contours with the Bernstein ellipse E_ρ, and since these points are constantly changing for each choice of parameters κ, m, and α (see Section 1), the evaluation of such elliptic integrals becomes computationally expensive even with an efficient algorithm. It turns out that it is possible to use a Gauss-Legendre quadrature with respect to the ellipse parameter θ (see formula (2.8)), instead of arc length. Applying a quadrature rule with respect to the ellipse parameter turns out to be only slightly suboptimal in terms of the required number of nodes. For M = 100, an error of 10⁻¹⁴ is attained by 5m quadrature nodes with respect to θ, instead of the expected 4m nodes with respect to arc length, and an error of 10⁻³² is attained by 11m quadrature nodes with respect to θ, instead of the expected 8.6m nodes with respect to arc length (See Tables 1 and 2).

4.4. Extension to Complex Wavenumber

In Section 4.1.2, we presented an algorithm which evaluates G_m for real-valued wavenumber by summing the contributions from three components: the two steepest descent contours γ₁ and γ₂ (i.e., Gustafssons’ contours), and the connecting segment, denoted by C_ρ (see formula (4.25)). Our approach ultimately sums these three contributions using O(m) quadrature nodes. To evaluate G_m for complex-valued wavenumber, we replace the steepest descent contours for real wavenumber with the steepest descent contours for complex wavenumber, constructed in Section 3.1.1 (see formulae (3.13) and (3.14)). The geometry of these contours also allows us to evaluate G_m with O(m) nodes. However, unlike the real case, the lengths of the three segments, γ₁, γ₂, and C_ρ, vary with the wavenumber’s complex argument, and, therefore, so does the allocation of the O(m) quadrature nodes among the three segments. In this section, we characterize how γ₁, γ₂, and C_ρ change with complex-valued κ. We briefly comment on the intersection of these contours with the Bernstein ellipse. Lastly, we demonstrate that one of the two contours approaches a singularity when the wavenumber is almost purely imaginary. For this case, we replace the three contours with a different set of contours which are well-separated from the singularity and is also evaluated with O(m) nodes.

4.4.1. Behavior of the Steepest Descent Contours and Connecting Contour for Complex Wavenumber

Recall from Section 3.1.1 (see formulae (3.13) and (3.14)) that the steepest descent contours for the numerator of the spherical-wave term with complex wavenumber can be parameterized as

$${\gamma }_1(s)=e^{-2i\varphi }{s}^2+2i{\beta }_{-}{e}^{-i\varphi }s+1,$$ (4.60)

$${\gamma }_2(s)=e^{-2i\varphi }{s}^2+2i{\beta }_{+}{e}^{-i\varphi }s-1.$$ (4.61)

Recall also from Section 1.2 that, with the negative time-harmonic convention, the wavenumber for the retarded modal Green’s function is in quadrant IV of the complex plane (i.e., $-\pi /2\le \varphi \le 0$). It is easy to see from formulae (4.60) and (4.61) that, as $\varphi \to -\pi /2$, both contours are rotated counterclockwise compared to the contours associated with real wavenumber (compare Figure 1 and Figure 10). Unlike the steepest descent contours for real wavenumber, for complex wavenumber, γ₁’s intersection with the Bernstein ellipse may occur anywhere on the Bernstein ellipse in the upper half of the complex plane. Consequently, the arclength of γ₁ and the connecting contour, C_ρ, varies with ϕ, with the arclength of C_ρ approaching zero in the limit $\varphi \to -\pi /2$.

Fig. 1: Phase plot of the numerator of the spherical-wave term in the z = cos ϕ-plane.

A phase plot of the function $\exp (-i\kappa \sqrt{1-\alpha z})$ with the parameters β₋ = 0.95 and κ = 45 is shown. The interval [−1, 1] and the steepest descent contours γ₁ and γ₂ are superimposed on the plot. Note that the branch cut in the term $\sqrt{1-\alpha z}$ is visible on the right of the figure. The distance from the point z = 1 to the branch cut is equal to ${\beta }_{-}^2$.

Fig. 10: Phase plot of the numerator of the spherical-wave term in the z = cos ϕ-plane for complex wavenumber.

A phase plot of the function $\exp (-i\kappa \sqrt{1-\alpha z})$ with the parameters β₋ = 0.95 and κ = 45 exp(−i0.44π) is shown. The interval [−1, 1] and the steepest descent contours γ₁ and γ₂ are superimposed on the plot. Note that the branch cut in the term $\sqrt{1-\alpha z}$ is visible on the right of the figure. The distance from the point z = 1 to the branch cut is equal to ${\beta }_{-}^2$. The angle that the branch cut makes with the real axis is equal to −2ϕ.

4.4.2. Intersection of the Steepest Descent Contours for Complex Wavenumber with the Bernstein Ellipse

For real wavenumber, we are able to solve for the steepest descent contours’ intersections with the Bernstein ellipse by using the quadratic formula. However, for general complex wavenumber, the steepest descent contours’ intersections with the Bernstein ellipse can only be found by solving a quartic equation. Rather than implement the quartic formula, we apply Newton’s algorithm to find the intersection points. An appropriate initialization can be found by approximating the steepest descent contour by a line, and then finding the intersection of this linear approximation with the Bernstein ellipse by using the quadratic formula. A convenient linearization is as follows.

When $2{\beta }_{-}<1$ we approximate the contours (4.60) and (4.61) as

$${\gamma }_1(s)\approx e^{-2i\varphi }{s}^2+1,$$ (4.62)

$${\gamma }_2(s)\approx e^{-2i\varphi }{s}^2-1.$$ (4.63)

When $2{\beta }_{-}\ge 1$, we approximate the contours as

$${\gamma }_1(s)\approx 2i{\beta }_{-}{e}^{-i\varphi }s+1,$$ (4.64)

$${\gamma }_2(s)\approx 2i{\beta }_{+}{e}^{-i\varphi }s-1.$$ (4.65)

With this initialization, Newton’s method convergences in under 15 iterations for all possible choices of complex wavenumber and source-to-target distance. To solve for the intersection in terms of the Bernstein ellipse parameter, we use the inverse Joukowski transformation (see Section 2.2, formula 2.3).

4.4.3. The Number of Quadrature Nodes Needed for Complex Wavenumber

When the wavenumber is complex, the lengths of the contours within the Bernstein ellipse change with the argument of the complex wavenumber (see formulae (4.60) and (4.61)), meaning that the number of times the integrand oscillates on each contour varies with the argument of the complex wavenumber. Thus, the number of nodes in their respective quadrature rules must also vary. We now demonstrate that the total number of quadrature nodes remains O(m) when the wavenumber is complex, and estimate the number of nodes required to resolve each contour integral in terms of the intersections of the contours with the Bernstein ellipse. Observe that, along the steepest descent contour with respect to the spherical-wave term, all oscillations in the integrand of (4.25) arise from the mth-order Chebyshev polynomial term, T_m(z). Let $E_{\rho }({\theta }_1)$ be the intersection of γ₁ with the Bernstein ellipse, and $E_{\rho }({\theta }_2)$ be the intersection of γ₂ with the Bernstein ellipse. Consider an extension to γ₁ which includes a short segment connecting ${\gamma }_1(0)$ to the point $E_{\rho }(0)$, and an extension to γ₂ which includes a segment connecting γ₂(0) to the point $E_{\rho }(\pi )$, which we denote as ${\gamma }_1^e$ and ${\gamma }_2^e$, respectively (see Figure 11). From the Taylor expansion of the major axis of the Bernstein ellipse in terms of the Fourier mode m, the point $E_{\rho }(0)$ can be shown to be O(1/m²) from z = 1, and by symmetry, $E_{\rho }(\pi )$ is O(1/m²) from z = −1. Consequently, on the segment from z = 1 to $E_{\rho }(0)$, T_m(z) oscillates at most once, and likewise, T_m(z) oscillates at most once from z = −1 to $E_{\rho }(\pi )$. Observe that, ${\gamma }_1^e$ intersects the Bernstein ellipse at $E_{\rho }({\theta }_1)$ and $E_{\rho }(0)$, and ${\gamma }_2^e$ intersects the Bernstein ellipse at $E_{\rho }({\theta }_2)$ and $E_{\rho }(\pi )$. It can be shown that the number of times T_m(z) oscillates on the extended contour ${\gamma }_1^e$ within the Bernstein ellipse is at most twice the number of oscillations of T_m(z) along the Bernstein ellipse beginning and ending at the intersection points of ${\gamma }_1^e$ with the Bernstein ellipse (i.e., $E_{\rho }(\theta )$ from 0 to θ₁). Thus, because ${\gamma }_1\subset {\gamma }_1^e$, the number of oscillations on γ₁ is bounded by $2m{\theta }_1/\pi$; when κ is in quadrant IV, this bound can be further improved to $m{\theta }_1/\pi$. A similar argument shows that the number of oscillations on γ₂ is bounded by $2m(\pi -{\theta }_2)$, and when κ is in quadrant IV, this can be further improved to $m(\pi -{\theta }_2)$. Lastly, T_m(z) on the connecting contour oscillates exactly $m({\theta }_2-{\theta }_1)/\pi$ times. Hence, the three contours γ₁, γ₂, and C_ρ are resolved with a total O(m) nodes, with nodes allocated in proportion to their respective numbers of oscillations. It turns out that when the total number of nodes over these three contours matches the required number of nodes estimated in Section 4.3.4 and Remarks 4.2–4.3, our method achieves full accuracy for complex wavenumber.

Fig. 11: The intersections of the extended contours ${\gamma }_1^e$ and ${\gamma }_2^e$ with the Bernstein ellipse.

The extended steepest descent contours ${\gamma }_1^e$ and ${\gamma }_2^e$, for a complex wavenumber, are plotted in green, together with their intersection points with the Bernstein ellipse, E_ρ. The connecting contour, C_ρ, is plotted in blue. A circle of radius a_ρ − 1 centered at z = 1 is plotted with a dotted circumference.

Recall from Section 4.2 that the integrand associated with γ₁ has a 1/τ singularity in the limit as β₋ approaches zero (see formula 4.29). For real wavenumber, we address this by splitting the integral over γ₁ into a singular part and a smooth part. We then evaluate the smooth part with a Gauss-Legendre quadrature rule and the singular part with a specialized recurrence used by Gustafsson [9], which altogether is accomplished with O(1) nodes. For complex wavenumber, a similar approach can be taken, except that special care is necessary to ensure that the domain of integration of the singular part has O(1) oscillations, and thus, can be evaluated with O(1) nodes.

Let [0, τ₀] be the domain of integration for the singular integral and [τ₀, τ₁] be the domain of integration for the smooth integral, where τ₁ is the value of the contour parameter such that γ₁(τ₁) intersects the Bernstein ellipse. The following heuristic provides a robust method to select τ₀ such that T_m(z) oscillates at most once on the contour γ(τ) from zero to τ₀. Recall that T_m(z) oscillates at most once in the disc of radius $a_{\rho }-1\approx 1/m^2$ centered at z = 1 (see Figure 11). Then, we choose τ₀ such that $|{\gamma }_2({\tau }_0)-1|=a_{\rho }(0)-1$. This solution is easily approximated by using formulae (4.62) and (4.63) when $2{\beta }_{-}<1$, and by using formulae (4.64) and (4.65) when $2{\beta }_{-}\ge 1$. With this choice of τ₀, T_m(z) oscillates at most once, meaning it is resolved with a O(1) nodes.

4.4.4. Avoiding the Singularity Associated with γ₁ for Nearly-Imaginary Wavenumber

Recall from Section 3.8 that, after reparameterizing Gustafsson’s contours and splitting the integral into a γ₁ term and a γ₂ term, the integral associated with γ₁ has a singularity at z = −1 (see formula 3.83). When the wavenumber is real, γ₁ is well-separated from the singularity. In the limit when the wavenumber approaches a purely imaginary value (i.e., $\varphi \approx -\pi /2$), γ₁ approaches the point z = −1 (compare Figure 1 and Figure 10). Therefore, when evaluating G_m for almost purely imaginary wavenumber, integration along γ₁ has loss of accuracy if H^w(z) is not small prior to reaching the singularity. Observe that ${H^w(z)|}_{z=1}\approx e^{-i\kappa \sqrt{1-\alpha }}$, and that ${H^w(z)|}_{z\approx -1}\approx e^{-i\kappa \sqrt{1+\alpha }}$. Hence,

$$\frac{{H^w(z)|}_{z\approx -1}}{{H^w(z)|}_{z=1}}\lesssim e^{-i\kappa (\sqrt{1+\alpha }-\sqrt{1-\alpha })}.$$ (4.66)

When the right-hand-side of formula (4.66) is less than machine epsilon, H^w(z) near z = −1 does not contribute to the integral, so the contour can truncated before reaching the singularity without loss of accuracy. When formula (4.66) is not negligible, integrating on the steepest descent contour for complex wavenumber results in loss of accuracy due to the singularity at z = −1. However, observe that when formula (4.66) is not negligible, $\kappa (\sqrt{1+\alpha }-\sqrt{1+\alpha })$ is small, meaning that the spherical-wave term oscillates slowly on any contour of the form in formula (4.60). Therefore, the integral along any reasonable contour which is well-separated from the singularity may be resolved with O(m) nodes. Hence, when the right-hand-side of formula (4.66) is not negligible, rather than integrate on the steepest descent contour for complex wavenumber, we simply use a different contour; we choose the steepest descent contour associated with Re κ, which is well-separated from the singularity for all possible κ and β₋.

4.4.5. Bounding the Angle of the Intersection of the Steepest Descent Contours with the Bernstein Ellipse

The proof in Section 4.3.2, showing that only O(m) nodes are required to compute G_m, relied on the fact that the angle of intersection between the steepest descent contour and the Bernstein ellipse is bounded from below. For complex wavenumber in quadrant IV, it is also possible to show that the angle of intersection of the contours with the ellipse is bounded from below. Interestingly, for the case of complex wavenumber in quadrant I, there exist combinations of wavenumbers and Fourier modes m for which the intersection is oblique.

4.5. Summary of the Algorithm

Recall from Section 1.2 that G_m is a function of κ, m, and α. Recall also that α can be determined from β₋ (see formula 1.19)), and vice versa. We thus consider G_m as a function of κ, m, and β₋. We compute G_m as follows. Recall from Section 4.2 the formula for G_m,

$$G_m\approx \frac{4}{\sqrt{\alpha }}{\int }_0^{{\tau }_1}\frac{F_1(\tau )}{\sqrt{{\tau }^2+2i{\beta }_{-}}}d\tau -\frac{4i}{\sqrt{\alpha }}{\int }_0^{{\tau }_2}\frac{F_2(\tau )}{\sqrt{{\tau }^2+2i{\beta }_{+}}}d\tau -\sum _{i=1}^n\frac{e^{-i\kappa \sqrt{1-\alpha v_i}}}{\sqrt{1-\alpha v_i}\sqrt{1-v_i^2}}{T}_m(v_i)d{v}_i{w}_i,$$ (4.67)

where $F_1(\tau )$ and $F_2(\tau )$ are smooth functions corresponding to the γ₁ and γ₂ contours, respectively defined by (3.91) and (3.95), τ₁ and τ₂ are positive parameters such that ${\gamma }_1({\tau }_1)$ and ${\gamma }_2({\tau }_2)$ are the respective intersections of γ₁ and γ₂ with E_ρ, and T_m is the mth order Chebyshev polynomial. Recall from Section 3.6 that both ${\gamma }_1\cap E_{\rho }$ and ${\gamma }_2\cap E_{\rho }$ have length $\approx 1/m^2$. Hence, T_m(z) oscillates at most once along each contour. By construction, on Gustafsson’s contours (see Section 3.1.1), the numerator of the spherical-wave term in the integrand does not oscillate. Hence, the entire integrand oscillates at most once. By the argument in Section 4.2, the integrand associated with the γ₂ contour is always smooth and hence can be evaluated with an O(1) point Gauss-Legendre quadrature.

The integrand associated with the contour γ₁ has a singularity for ${\beta }_{-}\approx 0$ (i.e., when the source and target are close). For this case, we follow the method in Section 4.2 and evaluate the portion near the singularity by expanding the function $F_1(\tau )$ into its Taylor series, and then use the recurrence described in Section 4.2.1. Due to the smoothness of $F_1(\tau )$, this part of the integral is computed with an O(1) cost. The remainder of the integral is smooth and oscillates at most once, and hence is evaluated with an O(1) Gauss-Legendre quadrature. Hence, both integrals in (4.67) are evaluated in O(1) operations.

The remaining term in (4.67) is a sum of residues evaluated on C_ρ, where C_ρ denotes the portion of a Bernstein ellipse connecting γ₁ and γ₂ (see Section 4.1). We select the residues v₁, … , v_n and weights w₁, … , w_n by constructing a quadrature which approximates

$${\int }_{C_{\rho }}\frac{T_m(v)}{v-z}dv,$$ (4.68)

and which holds for all values of z relevant to the evaluation of G_m (see Section 4.3.2). By the argument in Section 4.3, this is accomplished using O(m) Gauss Legendre nodes on C_ρ.

Therefore, the entire cost of our algorithm for G_m is O(m) and is completely independent of both κ and β₋. Lastly, since the algorithm is entirely quadrature-based, it is embarrassingly parallelizable.

5. Numerical Experiments

In Sections 5.1–5.5 we characterize the performance and accuracy of our method. Importantly, as demonstrated below, we achieve full precision for all possible ranges of β₋ and κ, and our algorithm’s performance is completely independent of β₋ and κ.

We use adaptive integration applied to (1.18) as the gold standard, and measure the error of our algorithm by comparing the two results. We use the change of variables $\varphi =x^3$, $d\varphi =3x^{2}dx$, to ensure that adaptive integration is accurate when $\alpha \approx 1$. We compute the $1-\alpha \cos (\varphi )$ term using the double angle formula to avoid cancellation error. The error in evaluating the modal Green’s function for very large κ is not measured, as adaptive integration is too expensive and no prior method can compute the modal Green’s function for large κ.

An implementation of the previously described algorithm was written in Fortran 77 and wrapped in MATLAB as a MEX file. Our code is available on https://doi.org/10.5281/zenodo.7040462. In our implementation, we chose M = 100, and used 5m quadrature nodes with respect to the ellipse parameter on C_ρ in double precision, and 11m quadrature nodes with respect to the ellipse parameter on C_ρ in extended precision (see Section 4.3.4 and Remarks 4.2 and 4.3). The timing and performance experiments in Sections 5.1–5.4 were performed using a consumer laptop with a four-core 2.6 GHz Intel i7 processor running a timing script in MATLAB 2018b with two threads. The parallel computing experiment in Section 5.5 was run on a server with a 16-core Intel Xeon 2.9 GHz processor.

5.0.1. The Interpretation of β₋ and κ

Recall from Section 1.2 that the modal Green’s function can be thought of as a function of four parameters: m, k, α, and R₀. After the introduction of the parameters κ and β₋ (see formula (1.16)), the R₀ term exclusively appears as a 1/R₀ scaling outside the integral. Hence, with this parameterization, R₀ is of no independent consequence to the performance of our algorithm, so we only characterize our algorithm’s performance as a function of κ, β₋, and m. Recall also that β₋ is defined as

$${\beta }_{-}=\frac{\text{Δ}}{{\rho }_0},$$ (5.1)

where Δ is the minimum source-to-target distance and ${\rho }_0=2r{r}^{\prime }$, with r and $r^{\prime }$ being the radial distances of the source and target in cylindrical coordinates. Recall finally from Section 1.2 that κ is defined as

$$\kappa =k{R}_0.$$ (5.2)

5.1. Performance of the Algorithm with Varying Source-to-Target Distance

We examined the performance of our algorithm over a wide range of source-to-target distances. As shown in Table 3 and Table 4, our algorithm’s performance is independent of β₋.

Table 3: The evaluation of the modal Green’s function in double precision for varying β₋ with a large wavenumber (κ = 10, 000).

The error is evaluated by using adaptive Gaussian quadrature as the gold standard. For brevity, ${\beta }_{-}={10}^{-18}$ is omitted.

	κ = 10, 000, m = 10		κ = 10, 000, m = 1000
β −	Evaluation Time	Absolute Error	Evaluation Time	Absolute Error
100	3.76×10−5 secs	3.34×10−14	1.44×10−3 secs	4.71×10−13
10−3	3.57×10−5 secs	3.43×10−14	1.44×10−3 secs	1.79×10−12
10−6	3.51×10−5 secs	3.92×10−14	1.44×10−3 secs	3.43×10−13
10−9	3.59×10−5 secs	5.50×10−14	1.44×10−3 secs	5.27×10−13
10−12	3.52×10−5 secs	3.33×10−14	1.44×10−3 secs	5.28×10−13
10−15	3.46×10−5 secs	1.69×10−14	1.44×10−3 secs	4.81×10−13
10−21	3.44×10−5 secs	6.63×10−14	1.44×10−3 secs	5.11×10−13

Table 4: The evaluation of the modal Green’s function in quadruple precision for varying β₋ with a large wavenumber (κ = 10, 000).

The error is evaluated by using adaptive Gaussian quadrature as the gold standard. For brevity, ${\beta }_{-}={10}^{-18}$ is omitted.

	κ = 10, 000, m = 10		κ = 10, 000, m = 1000
β −	Evaluation Time	Absolute Error	Evaluation Time	Absolute Error
100	5.82×10−3 secs	2.89×10−32	2.11×10−1 secs	6.59×10−31
10−3	5.84×10−3 secs	2.31×10−32	2.11×10−1 secs	1.07×10−30
10−6	5.74×10−3 secs	2.06×10−31	2.11×10−1 secs	1.82×10−30
10−9	6.25×10−3 secs	3.45×10−33	2.12×10−1 secs	2.63×10−31
10−12	6.27×10−3 secs	2.07×10−32	2.12×10−1 secs	1.55×10−31
10−15	6.22×10−3 secs	9.65×10−32	2.12×10−1 secs	5.53×10−31
10−21	5.92×10−3 secs	2.18×10−31	2.11×10−1 secs	6.36×10−31

5.2. Performance of the Algorithm with Varying κ, for Real-Valued κ

We examined the performance of our algorithm over a wide range of real values of κ (for performance with complex-valued κ, see Section 5.3 ). As shown in Tables 5–8, our algorithm’s performance is independent of κ.

Table 5: The evaluation of the modal Green’s function in double precision for varying real-valued κ with large source-to-target distance (β₋ = 1).

The error is evaluated by using adaptive Gaussian quadrature as the gold standard. Note for $\kappa >{10}^6$, the resource requirements of prior methods becomes excessive. For brevity, $\kappa ={10}^{12}$ and $\kappa ={10}^{15}$ are omitted.

	β− = 1 , m = 10		β− = 1 , m = 1000
κ	Evaluation Time	Absolute Error	Evaluation Time	Absolute Error
10−6	1.22×10−4 secs	1.45×10−13	1.84×10−3 secs	2.05×10−12
10−3	6.26×10−5 secs	1.50×10−13	1.64×10−3 secs	2.05×10−12
100	6.08×10−5 secs	1.61×10−13	1.66×10−3 secs	2.02×10−12
101	1.36×10−4 secs	2.71×10−14	2.50×10−3 secs	1.83×10−12
102	1.02×10−4 secs	4.94×10−15	2.43×10−3 secs	2.23×10−12
103	4.56×10−5 secs	1.30×10−14	1.73×10−3 secs	1.51×10−12
104	3.89×10−5 secs	3.34×10−14	1.69×10−3 secs	1.03×10−12
105	3.94×10−5 secs	2.25×10−14	1.70×10−3 secs	5.05×10−13
106	4.15×10−5 secs	2.75×10−13	1.75×10−3 secs	3.32×10−13
107	3.78×10−5 secs	–	8.39×10−4 secs	–
108	3.91×10−5 secs	–	8.33×10−4 secs	–
109	4.46×10−5 secs	–	8.23×10−4 secs	–
1018	3.98×10−5 secs	–	8.33×10−4 secs	–

Table 8: The evaluation of the modal Green’s function in quadruple precision for varying real-valued κ with small source-to-target distance $\left({\beta }_{-}={10}^{-12}\right)$.

The error is evaluated by using adaptive Gaussian quadrature as the gold standard. Note for $\kappa >{10}^6$ , the resource requirements of prior methods becomes excessive.

	β− = 10−12 , m = 10		β− = 10−12 , m = 1000
κ	Evaluation Time	Absolute Error	Evaluation Time	Absolute Error
10−6	7.23×10−3 secs	7.06×10−31	2.08×10−1 secs	5.75×10−29
10−3	7.53×10−3 secs	7.12×10−31	2.08×10−1 secs	5.75×10−29
100	7.16×10−3 secs	6.54×10−31	2.10×10−1 secs	5.70×10−29
101	7.53×10−3 secs	8.03×10−32	2.11×10−1 secs	5.56×10−29
102	7.09×10−3 secs	5.00×10−32	2.11×10−1 secs	4.55×10−29
103	6.25×10−3 secs	6.41×10−32	2.12×10−1 secs	7.93×10−30
104	6.23×10−3 secs	2.07×10−32	2.11×10−1 secs	1.55×10−31
105	6.21×10−3 secs	1.43×10−31	2.12×10−1 secs	2.21×10−31
106	6.21×10−3 secs	2.37×10−31	2.12×10−1 secs	3.88×10−31
107	6.25×10−3 secs	–	1.52×10−1 secs	–
108	6.22×10−3 secs	–	1.53×10−1 secs	–
109	6.19×10−3 secs	–	1.52×10−1 secs	–
1012	5.78×10−3 secs	–	1.52×10−1 secs	–
1015	5.72×10−3 secs	–	1.52×10−1 secs	–
1018	5.69×10−3 secs	–	1.52×10−1 secs	–

5.3. Performance of the Algorithm with Varying κ, for Complex-Valued κ

We examined the performance of our algorithm over a wide range of complex-valued κ, β₋, and Fourier mode m. As shown in Tables 9 – 12, our algorithm’s performance is independent of κ and β₋.

Table 9: The evaluation of the modal Green’s function in double precision for varying complex-valued κ with large source-to-target distance (β₋ = 1).

The error and $\left|G_0\right|$ are evaluated by using adaptive Gaussian quadrature as the gold standard for $\left|\kappa \right|<{10}^6$. For $\left|\kappa \right|\ge {10}^6$, the resource requirements of prior methods becomes excessive, and we instead evaluate $\left|G_0\right|$ using our method.

			β− = 1 , m = 10		β− = 1 , m = 1000
|κ|	Arg(κ)	|G0|	Evaluation Time	Error Scaled by G0	Evaluation Time	Error Scaled by G0
10−3	−π/8	3.31	1.20×10−4 secs	1.52×10−14	4.46×10−4 secs	5.13×10−14
10−3	−π/4	3.31	9.86×10−5 secs	1.52×10−14	4.35×10−4 secs	5.14×10−14
10−3	−3π/8	3.31	1.10×10−4 secs	1.36×10−14	3.75×10−4 secs	5.16×10−14
10−3	−π/2	3.31	7.95×10−5 secs	1.55×10−14	2.84×10−4 secs	5.11×10−14
1	−π/8	2.28	8.04×10−5 secs	2.17×10−14	5.63×10−4 secs	4.74×10−14
1	−π/4	1.69	7.66×10−5 secs	2.24×10−14	2.82×10−4 secs	4.97×10−14
1	−3π/8	1.40	8.02×10−5 secs	2.13×10−14	2.26×10−4 secs	5.19×10−14
1	−π/2	1.31	7.60×10−5 secs	1.79×10−14	2.19×10−4 secs	5.40×10−14
103	−π/8	2.85×10−119	7.56×10−5 secs	3.82×10−14	2.79×10−4 secs	3.36×10−13
103	−π/4	6.72×10−219	7.73×10−5 secs	4.72×10−14	3.24×10−4 secs	1.25×10−13
103	−3π/8	1.81×10−285	1.29×10−4 secs	6.83×10−14	3.76×10−4 secs	6.44×10−15
103	−π/2	7.62×10−309	1.53×10−4 secs	3.76×10−14	7.65×10−4 secs	5.84×10−15
105	−10−3	1.84×10−33	5.93×10−5 secs	4.91×10−12	1.76×10−4 secs	5.90×10−12
106	−10−4	5.82×10−34	6.16×10−5 secs	–	1.83×10−4 secs	–
109	−10−7	1.84×10−35	7.06×10−5 secs	–	1.58×10−4 secs	–
1018	−10−16	5.82×10−40	7.37×10−5 secs	–	1.73×10−4 secs	–

Table 12: The evaluation of the modal Green’s function in quadruple precision for varying complex-valued κ with small source-to-target distance $\left({\beta }_{-}={10}^{-12}\right)$.

The error and $\left|G_0\right|$ are evaluated by using adaptive Gaussian quadrature as the gold standard for $\left|\kappa \right|<{10}^6$. For $\left|\kappa \right|\ge {10}^6$, the resource requirements of prior methods becomes excessive, and we instead evaluate $\left|G_0\right|$ using our method.

			β− = 10−12 , m = 10		β− = 10−12 , m = 1000
|κ|	Arg(κ)	|G0|	Evaluation Time	Error Scaled by G0	Evaluation Time	Error Scaled by G0
10−3	−π/8	4.15×101	1.03×10−2 secs	1.28×10−32	2.92×10−2 secs	1.26×10−31
10−3	−π/4	4.15×101	8.83×10−3 secs	1.19×10−32	3.28×10−2 secs	1.26×10−31
10−3	−3π/8	4.15×101	8.38×10−3 secs	1.62×10−32	2.85×10−2 secs	1.25×10−31
10−3	−π/2	4.15×101	8.99×10−3 secs	1.59×10−32	3.38×10−2 secs	1.26×10−31
1	−π/8	3.98×101	9.18×10−3 secs	1.43×10−32	2.90×10−2 secs	1.30×10−31
1	−π/4	3.96×101	8.42×10−3 secs	1.59×10−32	2.87×10−2 secs	1.30×10−31
1	−3π/8	3.94×101	8.56×10−3 secs	1.52×10−32	3.28×10−2 secs	1.34×10−31
1	−π/2	3.94×101	9.14×10−3 secs	1.81×10−32	3.30×10−2 secs	1.30×10−31
103	−π/8	2.95×101	9.06×10−3 secs	1.88×10−30	2.95×10−2 secs	2.20×10−33
103	−π/4	2.95×101	8.73×10−3 secs	1.78×10−30	2.93×10−2 secs	2.67×10−33
103	−3π/8	2.95×101	9.01×10−3 secs	1.69×10−30	3.62×10−2 secs	2.68×10−33
103	−π/2	2.95×101	7.70×10−3 secs	1.66×10−30	4.00×10−2 secs	1.25×10−33
105	−10−3	2.31×101	7.73×10−3 secs	7.99×10−31	3.63×10−2 secs	7.90×10−31
106	−10−4	1.98×101	7.78×10−3 secs	–	2.27×10−2 secs	–
109	−10−7	1.02×101	7.74×10−3 secs	–	2.27×10−2 secs	–
1018	−10−16	1.77×10−3	7.54×10−3 secs	–	2.24×10−2 secs	–

5.4. Performance of the Algorithm with Varying Fourier Mode (m)

We examined the performance of our algorithm over a wide range of Fourier modes (represented by the parameter m). Because the number of points in the quadrature scales linearly with m, as demonstrated by Table 13, evaluation time scales linearly with the Fourier mode. Recall from the introduction of this section that the evaluation was performed on a four-core processor using two threads.

Table 13:

The evaluation time of the modal Green’s function in double precision for varying m (${\beta }_{-}={10}^{-12}$, κ = 10, 000).

m	Evaluation Time
1	3.88×10−5 secs
10	5.56×10−5 secs
102	1.75×10−4 secs
103	1.46×10−3 secs
104	1.43×10−2 secs
105	1.37×10−1 secs
106	1.36×100 secs
107	1.29×101 secs

5.5. Parallelization of the Algorithm

The cost of our algorithm is O(m) and does not depend on κ or β₋ (see Section 4.5). Because our algorithm is quadrature-based, it is embarrassingly parallelizable.

We measured the algorithm’s performance on a server with a 16-core Intel Xeon 2.9 GHz processor, where each core can run two threads, for a total of 32-threads. We vary the number of threads from 1 to 32, and report the results in Figure 12.

Fig. 12: Evaluation time of the modal Green’s function plotted against m with varying numbers of threads (${\beta }_{-}={10}^{-7}$, κ = 10, 000).

The calculation is performed in double precision. The evaluation times corresponding to 32 threads are not plotted for small m.

Remark 5.1.

We note that, for many applications, the number of source-target interactions is greater than the number of modes, meaning that the practitioner may benefit most by parallelizing over source-target interactions rather than parallelizing over different modes.

6. Conclusions and Generalizations

We have developed an algorithm which evaluates the modal Green’s function for the Helmholtz equation in O(m) time, that is completely independent of both the wavenumber, which is permitted to be complex, and the source-to-target distance. Furthermore, our algorithm is embarrassingly parallelizable. Our algorithm’s method can be readily extended in several directions, described in Sections 6.1–6.4.

6.1. An O(1) Evaluator for Small Wavenumber ($\kappa \ll m$)

Recall that our algorithm is independent of the wavenumber because we integrate along Gustafsson’s contours, which are the steepest descent contours with respect to the numerator of the spherical-wave term (see Section 3.1). When the Fourier mode m is larger than the scaled wavenumber κ, it is more efficient to integrate along a different contour. If instead, we choose the steepest descent contour on which exp (imϕ) does not oscillate, we arrive at an alternative algorithm whose cost is O(κ) and is independent of m . When κ is extremely small, this algorithm is essentially O(1). The case where β₋ is small (i.e., when the source and target are close) is handled in an identical fashion to the method described in Section 4.2. Thus, this alternative algorithm’s cost is completely independent of both m and β₋, and grows as O(κ).

6.2. An O(1) Evaluator for the Modal Green’s Functions for the Laplace Equation

The same method described in Section 6.1 can be applied to the case where κ = 0 to yield an O(1) evaluator of the modal Green’s function for the Laplace equation, whose cost is independent of β₋ (i.e., the cost is independent of the source-to-target distance).

6.3. Extension to an O(m) Evaluator for a Collection of Modal Green’s Functions, with Amortized Cost O(1)

This paper presents an algorithm for the evaluation of a single modal Green’s function for the Helmholtz Equation in O(m) time, independent of β₋ and κ, where β₋ is the scaled minimum source-to-target distance and κ is the scaled wavenumber. It is possible to use this algorithm to compute all of the modal Green’s functions −M, −M + 1, … , M − 1, M in O(M) time using the following method. In [15], Matviyenko presents a five-term recurrence relation for the modal Green’s functions for the Helmholtz equation. He observes that the recurrence relation is stable upwards for one range of Fourier modes and stable downwards for another range of modes. Furthermore, there exists a range of modes for which the recurrence is bi-unstable. Thus, a classical Miller-type algorithm cannot be applied. However, it was recently observed in [16] that if a recurrence relation is represented as a banded matrix, then the inverse power method can be used to find a solution, even when the stability behavior is mixed in the sense just described. We thus apply the inverse power method, as described in [16], to the resulting five-diagonal matrix corresponding to Matviyenko’s recurrence relation. In this fashion, we obtain all the eigenvectors corresponding to the zero eigenvalue; only one vector in this eigenspace corresponds to the vector of modal Green’s functions. We thus use the O(m) evaluator of this paper to select the vector corresponding to the modal Green’s functions. The cost of performing the inverse-power method is O(M), and the cost of the evaluation of the Mth modal Green’s function is O(M), meaning that all M Fourier coefficients are obtained in O(M) time. We note that with this scheme, the M modes are computed simultaneously rather in parallel, however, the computation may still be parallelized over source-target interactions.

6.4. An Evaluator for the Partial Derivatives of the Modal Green’s Function

In Section V of [15], Matviyenko derives an identity expressing the partial derivatives of G_m in terms of G_m, G_m+1, … , G_m+5. Thus, the O(m) evaluator presented in this paper can be used to evaluate the partial derivatives of the modal Green’s function in O(m) time. Furthermore, if the method proposed in Section 6.3 is used to evaluate a collection of modal Green’s functions in amortized O(1) time, then the partial derivatives can be evaluated in amortized O(1) time as well. Finally, we note that the higher order partial derivatives of the modal Green’s function can be expressed in terms of a finite number of functions G_m, and can therefore also be evaluated in O(m) time (or O(1) amortized time) (see Remark 5.1 of [15]).

6.5. An O(1) Evaluator for an Arbitrary Mode of the Modal Green’s Function

It appears that the steepest descent contours for the entire integrand of formula (1.2) do exist, but their relationship with $|\kappa |$, $\text{Arg}(\kappa )$ , m, and β₋ is quite involved. Consider, for example, Figure 13, which is a phase-amplitude plot of the product of the numerator of the spherical-wave term and the Fourier exponential term. Observe that a steepest descent contour can be constructed from −π to π, which passes through the stationary points of the integrand. The steepest descent contour passing through the point ${\varphi }^{\ast }\in \left[-\pi ,\pi \right]$, is the solution to

$$m\gamma (s)+\kappa \sqrt{1-\alpha \cos (\gamma (s))}=-\sqrt{\alpha }is+\kappa \sqrt{1-\alpha \cos ({\varphi }^{\ast })},$$ (6.1)

which is a transcendental equation. We expect that the construction of a completely general-purpose O(1) evaluator will be fairly complicated.

Table 6: The evaluation of the modal Green’s function in quadruple precision for varying real-valued κ with large source-to-target distance (β₋ = 1).

The error is evaluated by using adaptive Gaussian quadrature as the gold standard. Note for $\kappa >{10}^6$, the resource requirements of prior methods becomes excessive. For brevity, $\kappa ={10}^{12}$ and $\kappa ={10}^{15}$ are omitted.

	β− = 1 , m = 10		β− = 1 , m = 1000
κ	Evaluation Time	Absolute Error	Evaluation Time	Absolute Error
10−6	7.17×10−3 secs	4.06×10−31	2.07×10−1 secs	1.12×10−30
10−3	6.43×10−3 secs	3.99×10−31	2.07×10−1 secs	1.12×10−30
100	6.92×10−3 secs	3.94×10−31	2.09×10−1 secs	1.20×10−30
101	7.09×10−3 secs	1.34×10−31	2.11×10−1 secs	9.06×10−31
102	6.93×10−3 secs	7.74×10−34	2.10×10−1 secs	1.32×10−30
103	6.81×10−3 secs	9.49×10−33	2.11×10−1 secs	1.01×10−30
104	5.79×10−3 secs	2.89×10−32	2.12×10−1 secs	6.59×10−31
105	5.78×10−3 secs	1.59×10−31	2.12×10−1 secs	6.05×10−31
106	5.76×10−3 secs	2.23×10−31	2.11×10−1 secs	4.69×10−31
107	5.76×10−3 secs	–	2.11×10−1 secs	–
108	5.82×10−3 secs	–	1.52×10−1 secs	–
109	5.72×10−3 secs	–	1.52×10−1 secs	–
1018	5.70×10−3 secs	–	1.52×10−1 secs	–

Table 7: The evaluation of the modal Green’s function in double precision for varying real-valued κ with small source-to-target distance $\left({\beta }_{-}={10}^{-12}\right)$.

The error is evaluated by using adaptive Gaussian quadrature as the gold standard. Note for $\kappa >{10}^6$, the resource requirements of prior methods becomes excessive.

	β− = 10−12 , m = 10		β− = 10−12 , m = 1000
κ	Evaluation Time	Absolute Error	Evaluation Time	Absolute Error
10−6	4.49×10−5 secs	3.08×10−13	1.34×10−3 secs	2.90×10−11
10−3	4.45×10−5 secs	2.90×10−13	1.37×10−3 secs	2.88×10−11
100	4.77×10−5 secs	1.90×10−13	1.40×10−3 secs	2.84×10−11
101	4.79×10−5 secs	4.35×10−14	1.41×10−3 secs	2.74×10−11
102	4.61×10−5 secs	1.80×10−14	1.43×10−3 secs	2.29×10−11
103	3.57×10−5 secs	1.07×10−14	1.44×10−3 secs	4.19×10−12
104	3.49×10−5 secs	3.33×10−14	1.43×10−3 secs	5.28×10−13
105	3.46×10−5 secs	1.50×10−13	1.44×10−3 secs	6.58×10−13
106	3.41×10−5 secs	5.11×10−13	1.45×10−3 secs	3.04×10−13
107	3.48×10−5 secs	–	7.88×10−4 secs	–
108	3.43×10−5 secs	–	7.87×10−4 secs	–
109	3.38×10−5 secs	–	7.87×10−4 secs	–
1012	3.22×10−5 secs	–	7.85×10—4 secs	–
1015	3.38×10−5 secs	–	7.87×10−4 secs	–
1018	3.33×10−5 secs	–	7.85×10−4 secs	–

Table 10: The evaluation of the modal Green’s function in quadruple precision for varying complex-valued κ with large source-to-target distance (β₋ = 1).

The error and $\left|G_0\right|$ are evaluated by using adaptive Gaussian quadrature as the gold standard for $\left|\kappa \right|<{10}^6$. For $\left|\kappa \right|\ge {10}^6$, the resource requirements of prior methods becomes excessive, and we instead evaluate $\left|G_0\right|$ using our method.

			β− = 1 , m = 10		β− = 1 , m = 1000
|κ|	Arg(κ)	|G0|	Evaluation Time	Error Scaled by G0	Evaluation Time	Error Scaled by G0
10−3	−π/8	3.31	8.44×10−3 secs	9.83×10−32	2.87×10−2 secs	7.32×10−32
10−3	−π/4	3.31	8.07×10−3 secs	9.89×10−32	2.82×10−2 secs	7.36×10−32
10−3	−3π/8	3.31	8.65×10−3 secs	9.98×10−32	3.27×10−2 secs	7.33×10−32
10−3	−π/2	3.31	8.82×10−3 secs	9.88×10−32	3.25×10−2 secs	7.36×10−32
1	−π/8	2.28	8.74×10−3 secs	1.04×10−31	3.32×10−2 secs	8.39×10−32
1	−π/4	1.69	8.82×10−3 secs	1.07×10−31	3.32×10−2 secs	8.69×10−32
1	−3π/8	1.40	8.13×10−3 secs	1.07×10−31	2.83×10−2 secs	8.87×10−32
1	−π/2	1.31	8.22×10−3 secs	1.08×10−31	2.82×10−2 secs	8.87×10−32
103	−π/8	2.85×10−119	9.16×10−3 secs	4.22×10−32	3.30×10−2 secs	8.28×10−31
103	−π/4	6.72×10−219	8.80×10−3 secs	4.77×10−32	3.57×10−2 secs	1.92×10−31
103	−3π/8	1.81×10−285	8.91×10−3 secs	3.97×10−32	3.58×10−2 secs	1.35×10−32
103	−π/2	7.62×10−309	7.66×10−3 secs	3.98×10−32	3.99×10−2 secs	8.62×10−34
105	−10−3	1.84×10−33	7.74×10−3 secs	5.13×10−30	4.72×10−2 secs	5.17×10−30
106	−10−4	5.82×10−34	7.63×10−3 secs	–	2.27×10−2 secs	–
109	−10−7	1.84×10−35	7.55×10−3 secs	–	2.28×10−2 secs	–
1018	−10−16	5.82×10−40	7.51×10−3 secs	–	2.23×10−2 secs	–

Table 11: The evaluation of the modal Green’s function in double precision for varying complex-valued κ with small source-to-target distance $\left({\beta }_{-}={10}^{-12}\right)$.

The error and $\left|G_0\right|$ are evaluated by using adaptive Gaussian quadrature as the gold standard for $\left|\kappa \right|<{10}^6$. For $\left|\kappa \right|\ge {10}^6$, the resource requirements of prior methods becomes excessive, and we instead evaluate $\left|G_0\right|$ using our method.

			β− = 10−12 , m = 10		β− = 10−12 , m = 1000
|κ|	Arg(κ)	|G0|	Evaluation Time	Error Scaled by G0	Evaluation Time	Error Scaled by G0
10−3	−π/8	4.15×101	8.63×10−5 secs	2.42×10−15	3.38×10−4 secs	6.08×10−14
10−3	−π/4	4.15×101	1.18×10−4 secs	2.27×10−15	3.94×10−4 secs	6.13×10−14
10−3	−3π/8	4.15×101	1.16×10−4 secs	2.39×10−15	3.63×10−4 secs	6.13×10−14
10−3	−π/2	4.15×101	1.13×10−4 secs	2.73×10−15	3.44×10−4 secs	6.10×10−14
1	−π/8	3.98×101	1.60×10−4 secs	1.99×10−15	7.86×10−4 secs	6.42×10−14
1	−π/4	3.96×101	8.46×10−5 secs	2.65×10−15	3.21×10−4 secs	6.55×10−14
1	−3π/8	3.94×101	8.61×10−5 secs	2.44×10−15	3.12×10−4 secs	6.66×10−14
1	−π/2	3.94×101	8.61×10−5 secs	2.65×10−15	2.92×10−4 secs	6.70×10−14
103	−π/8	2.95×101	7.49×10−5 secs	4.31×10−13	3.04×10−4 secs	5.83×10−13
103	−π/4	2.95×101	6.54×10−5 secs	4.50×10−13	3.18×10−4 secs	5.32×10−13
103	−3π/8	2.95×101	8.33×10−5 secs	4.01×10−13	3.52×10−4 secs	2.98×10−13
103	−π/2	2.95×101	7.73×10−5 secs	3.99×10−13	1.96×10−4 secs	1.39×10−13
105	−10−3	2.31×101	6.46×10−5 secs	2.55×10−13	1.87×10−4 secs	2.06×10−13
106	−10−4	1.98×101	5.29×10−5 secs	–	1.92×10−4 secs	–
109	−10−7	1.02×101	5.88×10−5 secs	–	1.46×10−4 secs	–
1018	−10−16	1.77×10−3	5.89×10−5 secs	–	1.45×10−4 secs	–