References

Mie scattering

• C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley (1983), Chapter 4. The conventions, the series and the recurrences of \(\pi_n\) and \(\tau_n\).

• W. J. Wiscombe, Mie Scattering Calculations: Advances in Technique and Fast, Vector-Speed Computer Codes, NCAR Technical Note NCAR/TN-140+STR (1979, revised 1996), and “Improved Mie scattering algorithms”, Applied Optics 19 (1980), 1505–1509. The MIEV0 baseline, its truncation rule and its published test cases.

• W. Gautschi, “Computational aspects of three-term recurrence relations”, SIAM Review 9 (1967), 24–82. Minimal solutions and Pincherle’s theorem, behind the enclosure of the logarithmic derivatives.

• W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions”, Applied Optics 15 (1976), 668–671. The continued fraction for \(D_n\), which scatrs replaces by a contracting downward recurrence, and its guard against a zero denominator, which scatrs keeps.

• R. L. Smith, “Algorithm 116: Complex division”, Communications of the ACM 5 (1962), 435. The complex reciprocal.

• I. Gargantini and P. Henrici, “Circular arithmetic and the determination of polynomial zeros”, Numerische Mathematik 18 (1972), 305–320. Disk arithmetic, used by the reference generator.

Layered spheres

• A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres”, Journal of Applied Physics 22 (1951), 1242–1246. The coated sphere, by direct matching at the interfaces, as the reference generator and the textbook method of the comparison do.

• C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley (1983), Chapter 5 and Section 8.1. The electrostatic polarizability of a coated sphere (the Rayleigh limit checked in the external validation) and the coated sphere’s coefficients.

• W. Yang, “Improved recursive algorithm for light scattering by a multilayered sphere”, Applied Optics 42 (2003), 1710–1720. The recursion for the effective logarithmic derivatives.

• O. Peña and U. Pal, “Scattering of electromagnetic radiation by a multilayered sphere”, Computer Physics Communications 180 (2009), 2348–2354. Yang’s recursion in the form of scattnlay, the code of the comparison.

• O. B. Toon and T. P. Ackerman, “Algorithms for the calculation of scattering by stratified spheres”, Applied Optics 20 (1981), 3657–3660. The coated sphere in atmospheric optics, and the stability of its recurrences.

• T. C. Bond and R. W. Bergstrom, “Light absorption by carbonaceous particles: an investigative review”, Aerosol Science and Technology 40 (2006), 27–67. The refractive index of black carbon in Example: coated black carbon.

Size distributions and refractive indices

• J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres”, Space Science Reviews 16 (1974), 527–610. The effective radius and effective variance, and the gamma distribution of Gamma.

• D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions, Elsevier (1969). The modified gamma distribution of ModifiedGamma.

• I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica”, Journal of the Optical Society of America 55 (1965), 1205–1209. The Sellmeier coefficients of fused silica in Refractive-index models and Example: size distributions and refractive-index models.

• C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley (1983), Chapter 9. The Lorentz oscillator model of an absorption band in Example: size distributions and refractive-index models.

Quadrature

• F. G. Tricomi, “Sugli zeri dei polinomi sferici ed ultrasferici”, Annali di Matematica Pura ed Applicata (4) 31 (1950), as cited in N. Hale and A. Townsend, SIAM Journal on Scientific Computing 35 (2013). The starting approximation of the Gauss–Legendre nodes.

• G. Szegő, Orthogonal Polynomials, 4th ed., American Mathematical Society Colloquium Publications 23 (1975), Theorem 6.21.2. Bruns’s inequality for the zeros of \(P_n\), used by the tests.

• I. Bogaert, “Iteration-free computation of Gauss–Legendre quadrature nodes and weights”, SIAM Journal on Scientific Computing 36(3) (2014), A1008–A1026. The \(O(n)\) method of the f64 rules above 100 points.

• L. N. Trefethen, Approximation Theory and Approximation Practice, SIAM (2013), Theorem 19.3. The error bound for Gauss quadrature of analytic functions, used in Legendre moments of a phase function and for the nodes of spheroids (Spheroids). The theorem is stated for rules with \(n + 1\) points.

Spheroids and the T-matrix method

• P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering”, Physical Review D 3 (1971), 825–839. The extended boundary condition method.

• M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles”, Journal of the Optical Society of America A 8 (1991), 871–882. The orientation averages from the T-matrix, by its rank decomposition.

• M. I. Mishchenko and L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles”, Optics Communications 109 (1994), 16–21. The loss of precision of the EBCM for spheroids, and the extended-precision code tmatd of the comparison.

• M. I. Mishchenko, L. D. Travis and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, Cambridge University Press (2002), Chapter 5. The T-matrix of an axisymmetric particle (§5.8), its conventions and the convergence procedure.

• W. R. C. Somerville, B. Auguié and E. C. Le Ru, “Severe loss of precision in calculations of T-matrix integrals”, Journal of Quantitative Spectroscopy and Radiative Transfer 113 (2012), 524–535; “A new numerically stable implementation of the T-matrix method for electromagnetic scattering by spheroidal particles”, 123 (2013), 153–168; and “Accurate and convergent T-matrix calculations of light scattering by spheroids”, 160 (2015), 29–35. The cancellations in the integrals for spheroids, an algorithm that avoids them (with the arbitrary-precision values External validation compares), and its convergence.

• F. Kuik, J. F. de Haan and J. W. Hovenier, “Benchmark results for single scattering by spheroids”, Journal of Quantitative Spectroscopy and Radiative Transfer 47 (1992), 477–489. The benchmark spheroid whose arbitrary-precision values External validation compares.

• W. R. C. Somerville, B. Auguié and E. C. Le Ru, “SMARTIES: User-friendly codes for fast and accurate calculations of light scattering by spheroids”, Journal of Quantitative Spectroscopy and Radiative Transfer 174 (2016), 39–55. The code, and the range it reaches, compared in External validation.

• M. R. A. Majić, L. Pratley, D. Schebarchov, W. R. C. Somerville, B. Auguié and E. C. Le Ru, “Approximate T matrix and optical properties of spheroidal particles to third order with respect to size parameter”, Physical Review A 99 (2019), 013853. The long-wavelength expansion of the T-matrix that the tests check.

• M. R. A. Majić, F. Gray, B. Auguié and E. C. Le Ru, “Electrostatic limit of the T-matrix for electromagnetic scattering: Exact results for spheroidal particles”, Journal of Quantitative Spectroscopy and Radiative Transfer 200 (2017), 50–58. The electrostatic limit of the EBCM matrices.

• O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon, M. Sorokin and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust”, Journal of Geophysical Research 111 (2006), D11208. Spheroids as the model of mineral dust in Example: mineral dust as spheroids.

• N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd edition, SIAM (2002), Chapters 12 and 15. Iterative refinement in mixed precision, and condition estimation.

Wigner d-functions and polarized radiative transfer

• M. I. Mishchenko, L. D. Travis and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, Cambridge University Press (2002), Appendix B. The convention of the Wigner d-functions, the recurrence of Eq. B.22, and the expansion of the phase matrix in Greek coefficients (Eqs. 4.75–4.80) that scatrs follows.

• J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 3rd ed., Springer (2002), the section on the algorithms of Goertzel and Reinsch. Reinsch’s difference form, the idea behind the difference forms that scatrs uses next to the poles.

• J. W. Hovenier, C. van der Mee and H. Domke, Transfer of Polarized Light in Planetary Atmospheres: Basic Concepts and Practical Methods, Kluwer Academic Publishers (2004). The expansion of the phase matrix in generalized spherical functions, behind the Greek coefficients.

• L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the Galaxy”, Astrophysical Journal 93 (1941), 70–83. The Henyey–Greenstein phase function.

Arithmetic

• M. Joldes, J.-M. Muller and V. Popescu, “Tight and rigorous error bounds for basic building blocks of double-word arithmetic”, ACM Transactions on Mathematical Software 44(2) (2017). The double-double algorithms and their error bounds.

• J. R. Shewchuk, “Adaptive precision floating-point arithmetic and fast robust geometric predicates”, Discrete & Computational Geometry 18 (1997), 305–363. The floating-point expansions of the double-double remainder.

• T. Ogita, S. M. Rump and S. Oishi, “Accurate sum and dot product”, SIAM Journal on Scientific Computing 26(6) (2005), 1955–1988. The compensated sum that the tests use to measure errors against the reference values.

• J. Revels, M. Lubin and T. Papamarkou, “Forward-mode automatic differentiation in Julia”, arXiv:1607.07892 (2016). ForwardDiff.jl, whose dual numbers made the same choice of strict zero tests (in version 1.0).

Reference values

• F. Johansson, “Arb: efficient arbitrary-precision midpoint-radius interval arithmetic”, IEEE Transactions on Computers 66(8) (2017), 1281–1292. The ball arithmetic behind the reference suite, used through python-flint.