Numerical computation of spherical harmonics of arbitrary degree and order by extending exponent of floating point numbers: III integral (original) (raw)
The coordinate-free approach to spherical harmonics
2008
We present in a unified and self-contained manner the coordinate-free approach to spherical harmonics initiated in the mid 19th century by James Clerk Maxwell, William Thomson and Peter Guthrie Tait. We stress the pedagogical advantages of this approach which leads in a natural way to many physically relevant results that students find often difficult to work out using spherical coordinates and associated Legendre functions. It is shown how most physically relevant results of the theory of spherical harmonics - such as recursion relations, Legendre's addition theorem,surface harmonics expansions, the method of images, multipolar charge distributions, partial wave expansions, Hobson's integral theorem, rotation matrix and Gaunt's integrals - can be efficiently derived in a coordinate free fashion from a few basic elements of the theory of solid and surface harmonics discussed in the paper.
Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order
Constructive Approximation, 2017
Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions, belong to types classified by Schwarz, with dihedral, tetrahedral, or octahedral monodromy. The dihedral Legendre functions are expressed in terms of Jacobi polynomials. For the last two monodromy types, an underlying 'octahedral' polynomial, indexed by the degree and order and having a nonclassical kind of orthogonality, is identified, and recurrences for it are worked out. It is a (generalized) Heun polynomial, not a hypergeometric one. For each of these families of algebraic associated Legendre functions, a representation of the rank-2 Lie algebra so(5, C) is generated by the ladder operators that shift the degree and order of the corresponding solid harmonics. All such representations of so(5, C) are shown to have a common value for each of its two Casimir invariants. The Dirac singleton representations of so(3, 2) are included.
In order to accelerate the spherical harmonic synthesis and/or analysis of arbitrary function on the unit sphere, we developed a pair of procedures to transform between a truncated spherical harmonic expansion and the corresponding two-dimensional Fourier series. First, we obtained an analytic expression of the sine/cosine series coefficient of the 4pi4 \pi4pi fully normalized associated Legendre function in terms of the rectangle values of the Wigner ddd function. Then, we elaborated the existing method to transform the coefficients of the surface spherical harmonic expansion to those of the double Fourier series so as to be capable with arbitrary high degree and order. Next, we created a new method to transform inversely a given double Fourier series to the corresponding surface spherical harmonic expansion. The key of the new method is a couple of new recurrence formulas to compute the inverse transformation coefficients: a decreasing-order, fixed-degree, and fixed-wavenumber three-term formula for general terms, and an increasing-degree-and-order and fixed-wavenumber two-term formula for diagonal terms. Meanwhile, the two seed values are analytically prepared. Both of the forward and inverse transformation procedures are confirmed to be sufficiently accurate and applicable to an extremely high degree/order/wavenumber as 230approx1092^{30} \approx 10^9230approx109. The developed procedures will be useful not only in the synthesis and analysis of the spherical harmonic expansion of arbitrary high degree and order but also the evaluation of the derivatives and integrals of the spherical harmonic expansion.
Rectangular rotation of spherical harmonic expansion of arbitrary high degree and order
In order to move the polar singularity of arbitrary spherical harmonic expansion to a point on the equator, we rotate the expansion around the y-axis by 90 degree such that the x-axis becomes a new pole. The expansion coefficients are transformed by multiplying a special value of Wigner D-matrix and a normalization factor. The transformation matrix is unchanged whether the coefficients are 4π fully normalized or Schmidt quasi-normalized. The matrix is recursively computed by the so-called X-number formulation (Fukushima in J Geodesy 86: 271–285, 2012a). As an example, we obtained 2190×2190 coefficients of the rectangular rotated spherical harmonic expansion of EGM2008. A proper combination of the original and the rotated expansions will be useful in (i) integrating the polar orbits of artificial satellites precisely and (ii) synthesizing/analyzing the gravitational/geomagnetic potentials and their derivatives accurately in the high latitude regions including the arctic and antarctic area.
New recurrence relations for spherical harmonic functions and their derivatives
Physics of the Earth and Planetary Interiors, 1968
A single pair of recurrence relations are presented for the harmonic analyses of the Earth's main magnetic field. The same rapid numerical evaluation of spherical harmonic functions pair ofrecurrence relations may also be applied to the evaluation P~m (cos O)~mçb and their first derivatives X~m(cos 6)~mçb, of second derivatives. Y~'"(cos O)~/mçb, Z~m(cos 6)~mci which occur in spherical
Recursive Computation of Spherical Harmonic Rotation Coefficients of Large Degree
Applied and Numerical Harmonic Analysis, 2015
Computation of the spherical harmonic rotation coefficients or elements of Wigner's dmatrix is important in a number of quantum mechanics and mathematical physics applications. Particularly, this is important for the Fast Multipole Methods in three dimensions for the Helmholtz, Laplace and related equations, if rotation-based decomposition of translation operators are used. In these and related problems related to representation of functions on a sphere via spherical harmonic expansions, computation of the rotation coefficients of large degree n (of the order of thousands and more) may be necessary. Existing algorithms for their computation, based on recursions, are usually unstable, and do not extend to n. We develop a new recursion and study its behavior for large degrees, via computational and asymptotic analyses. Stability of this recursion was studied based on a novel application of the Courant-Friedrichs-Lewy condition and the von Neumann method for stability of finite-difference schemes for solution of PDEs. A recursive algorithm of minimal complexity O n 2 for degree n and FFTbased algorithms of complexity O n 2 log n suitable for computation of rotation coefficients of large degrees are proposed, studied numerically, and cross-validated. It is shown that the latter algorithm can be used for n 10 3 in double precision, while the former algorithm was tested for large n (up to 10 4 in our experiments) and demonstrated better performance and accuracy compared to the FFT-based algorithm.
Fourier-series representation and projection of spherical harmonic functions
Journal of Geodesy, 2012
Computations of Fourier coefficients and related integrals of the associated Legendre functions with a new method along with their application to spherical harmonics analysis and synthesis are presented. The method incorporates a stable three-step recursion equation that can be processed separately for each colatitudinal Fourier wavenumber. Recursion equations for the zonal and sectorial modes are derived in explicit single-term formulas to provide accurate initial condition. Stable computations of the Fourier coefficients as well as the integrals needed for the projection of Legendre functions are demonstrated for the ultra-high degree of 10,800 corresponding to the resolution of one arcmin. Fourier coefficients, computed in double precision, are found to be accurate to 15 significant digits, indicating that the normalized error is close to the machine round-off error. The orthonormality, evaluated with Fourier coefficients and related integrals, is shown to be accurate to O(10 −15) for degrees and orders up to 10,800. The Legendre function of degree 10,800 and order 5,000, synthesized from Fourier coefficients, is accurate to the machine round-off error. Further extension of the method to even higher degrees seems to be realizable without significant deterioration of accuracy. The Fourier series is applied to the projection of Legendre functions to the high-resolution global relief data of the National Geophysical Data Center of the National Oceanic and Atmospheric Administration, and the spherical harmonic degree variance (power spectrum) of global relief data is discussed.