The Experts below are selected from a list of 141 Experts worldwide ranked by ideXlab platform
Toshio Fukushima - One of the best experts on this subject based on the ideXlab platform.
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Fast computation of sine/cosine series coefficients of Associated Legendre Function of arbitrary high degree and order
Journal of Geodetic Science, 2018Co-Authors: Toshio FukushimaAbstract:Abstract In order to accelerate the spherical/spheroidal harmonic synthesis of any Function, we developed a new recursive method to compute the sine/cosine series coefficient of the 4π fully- and Schmidt quasi-normalized Associated Legendre Functions. The key of the method is a set of increasing-degree/order mixed-wavenumber two to four-term recurrence formulas to compute the diagonal terms. They are used in preparing the seed values of the decreasing-order fixed-degree, and fixed-wavenumber two- and three-term recurrence formulas, which are obtained by modifying the classic relations. The new method is accurate and capable to deal with an arbitrary high degree/ order/wavenumber. Also, it runs significantly faster than the previous method of ours utilizing the Wigner d Function, say around 20 times more when the maximum degree exceeds 1,000.
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Transformation between surface spherical harmonic expansion of arbitrary high degree and order and double Fourier series on sphere
Journal of Geodesy, 2018Co-Authors: Toshio FukushimaAbstract: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 $$4 \pi $$ 4 π fully normalized Associated Legendre Function in terms of the rectangle values of the Wigner d 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 $$2^{30}\,{\approx }\,10^9$$ 2 30 ≈ 10 9 . 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 in the evaluation of the derivatives and integrals of the spherical harmonic expansion.
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Numerical Computation of Point Values, Derivatives, and Integrals of Associated Legendre Function of the First Kind and Point Values and Derivatives of Oblate Spheroidal Harmonics of the Second Kind of High Degree and Order
International Association of Geodesy Symposia, 2015Co-Authors: Toshio FukushimaAbstract:This article reviews the recent works of the author on the numerical computation of the point values, the derivatives, and the integrals of the Associated Legendre Function (ALF) of the first kind as well as the point values and the derivatives of the oblate spheroidal harmonics of the second kind (Fukushima T, 2012a, J. Geodesy, 86, 271; ibid., 2012b, J. Geodesy, 86, 745; ibid., 2012c, J. Geodesy, 86, 1019; ibid., 2012d, Comp. Geosci., 49, 1; ibid., 2013, J. Geodesy, 87, 303; ibid., 2014, Comp. Geosci., 63,17. First, a sort of exponent extension of the floating point numbers, named the X-number formulation, resolved the underflow problem in the computation of the point values of the fully-normalized ALF of the first kind of high degree and order such as 216 000 or more. Similarly, the formulation precisely computes their derivatives and integrals. Second, a dynamic switch from the X-number to the ordinary floating point number during the fixed-order increasing-degree recursions significantly reduces the increase in the CPU time caused by the exponent extension. Third, the sectorial integrals obtained by the forward recursion cause no troubles in the subsequent non-sectorial recursions. Fourth, the fixed-order increasing-degree recursions can be accelerated on PCs with multiple or many cores by the folded parallel computation, namely by the parallel computation the load balance of which is equalized by pairing the recursion of orders m and M − m, where M is the maximum order to be computed. Finally, a recursive formulation is developed to compute the point values and the derivatives of the oblate spheroidal harmonics of the second kind, i.e. the unnormalized ALF of the second kind with a pure imaginary argument. The relating Fortran programs as well as the output examples are available at the author’s WEB page in ResearchGate: https://www.researchgate.net/profile/Toshio_Fukushima/
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Prolate Spheroidal Harmonic Expansion of Gravitational Field
The Astronomical Journal, 2014Co-Authors: Toshio FukushimaAbstract:As a modification of the oblate spheroidal case, a recursive method is developed to compute the point value and a few low-order derivatives of the prolate spheroidal harmonics of the second kind, Qnm (y), namely the unnormalized Associated Legendre Function (ALF) of the second kind with its argument in the domain, 1 < y < ∞. They are required in evaluating the prolate spheroidal harmonic expansion of the gravitational field in addition to the point value and the low-order derivatives of , the 4π fully normalized ALF of the first kind with its argument in the domain, |t| ≤ 1. The new method will be useful in the gravitational field computation of elongated celestial objects.
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numerical computation of spherical harmonics of arbitrary degree and order by extending exponent of floating point numbers ii first second and third order derivatives
Journal of Geodesy, 2012Co-Authors: Toshio FukushimaAbstract:We confirm that the first-, second-, and third-order derivatives of fully-normalized Legendre polynomial (LP) and Associated Legendre Function (ALF) of arbitrary degree and order can be correctly evaluated by means of non-singular fixed-degree formulas (Bosch in Phys Chem Earth 25:655–659, 2000) in the ordinary IEEE754 arithmetic when the values of fully-normalized LP and ALF are obtained without underflow problems, for e.g., using the extended range arithmetic we recently developed (Fukushima in J Geod 86:271–285, 2012). Also, we notice the same correctness for the popular but singular fixed-order formulas unless (1) the order of differentiation is greater than the order of harmonics and (2) the point of evaluation is close to the poles. The new formulation using the fixed-order formulas runs at a negligible extra computational time, i.e., 3–5 % increase in computational time per single ALF when compared with the standard algorithm without the exponent extension. This enables a practical computation of low-order derivatives of spherical harmonics of arbitrary degree and order.
J S Gardner - One of the best experts on this subject based on the ideXlab platform.
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asymptotic expansion of the Associated Legendre Function over the interval 0 leq theta leq pi
IEEE Transactions on Antennas and Propagation, 2011Co-Authors: J S GardnerAbstract:The Associated Legendre Function is generally defined by an integral or differential equation. However, a closed-form representation of the Associated Legendre Function of order 0 or 1 could be beneficial to model the propagation and scattering of waves in spherical coordinates. To this end, an Associated Legendre approximation with accuracy over the entire defining interval is presented that is particularly conducive to integral evaluation using stationary phase or steepest descents.
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Asymptotic Expansion of the Associated Legendre Function Over the Interval $0 \leq \theta \leq \pi$
IEEE Transactions on Antennas and Propagation, 2011Co-Authors: J S GardnerAbstract:The Associated Legendre Function is generally defined by an integral or differential equation. However, a closed-form representation of the Associated Legendre Function of order 0 or 1 could be beneficial to model the propagation and scattering of waves in spherical coordinates. To this end, an Associated Legendre approximation with accuracy over the entire defining interval is presented that is particularly conducive to integral evaluation using stationary phase or steepest descents.
Demni Nizar - One of the best experts on this subject based on the ideXlab platform.
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Explicit expressions of the Hua-Pickrell semi-group
HAL CCSD, 2021Co-Authors: Arista Jonas, Demni NizarAbstract:A paraitre dans Theory of Probability and its ApplicationsIn this paper, we study the one-dimensional Hua-Pickrell diffusion. We start by revisiting the stationary case considered by E. Wong for which we supply omitted details and write down a unified expression of its semi-group density through the Associated Legendre Function in the cut. Next, we focus on the general (not necessarily stationary) case for which we prove an intertwining relation between Hua-Pickrell diffusions corresponding to different sets of parameters. Using Cauchy Beta integral on the one hand and Girsanov's Theorem on the other hand, we discuss the connection between the stationary and general cases. Afterwards, we prove our main result providing novel integral representations of the Hua-Pickrell semi-group density, answering a question raised by Alili, Matsumoto and Shiraishi (S\'eminaire de Probabilit\'es, 35, 2001). To this end, we appeal to the semi-group density of the Maass Laplacian and extend it to purely-imaginary values of the magnetic field. In the last section, we use the Karlin-McGregor formula to derive an expression of the semi-group density of the multi-dimensional Hua-Pickrell particle system introduced by T. Assiotis
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Explicit expressions of the Hua-Pickrell semi-group
2020Co-Authors: Arista Jonas, Demni NizarAbstract:In this paper, we study the one-dimensional Hua-Pickrell diffusion. We start by revisiting the stationary case considered by E. Wong for which we supply omitted details and write down a unified expression of its semi-group density through the Associated Legendre Function in the cut. Next, we focus on the general (not necessarily stationary) case for which we prove an intertwining relation between Hua-Pickrell diffusions corresponding to different sets of parameters. Using Cauchy Beta integral on the one hand and Girsanov's Theorem on the other hand, we discuss the connection between the stationary and general cases. Afterwards, we prove our main result providing novel integral representations of the Hua-Pickrell semi-group density, answering a question raised by Alili, Matsumoto and Shiraishi (S\'eminaire de Probabilit\'es, 35, 2001). To this end, we appeal to the semi-group density of the Maass Laplacian and extend it to purely-imaginary values of the magnetic field. In the last section, we use the Karlin-McGregor formula to derive an expression of the semi-group density of the multi-dimensional Hua-Pickrell particle system introduced by T. Assiotis.Comment: 19 pages, the final expression of the semi-group density is obtaine
Arista Jonas - One of the best experts on this subject based on the ideXlab platform.
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Explicit expressions of the Hua-Pickrell semi-group
HAL CCSD, 2021Co-Authors: Arista Jonas, Demni NizarAbstract:A paraitre dans Theory of Probability and its ApplicationsIn this paper, we study the one-dimensional Hua-Pickrell diffusion. We start by revisiting the stationary case considered by E. Wong for which we supply omitted details and write down a unified expression of its semi-group density through the Associated Legendre Function in the cut. Next, we focus on the general (not necessarily stationary) case for which we prove an intertwining relation between Hua-Pickrell diffusions corresponding to different sets of parameters. Using Cauchy Beta integral on the one hand and Girsanov's Theorem on the other hand, we discuss the connection between the stationary and general cases. Afterwards, we prove our main result providing novel integral representations of the Hua-Pickrell semi-group density, answering a question raised by Alili, Matsumoto and Shiraishi (S\'eminaire de Probabilit\'es, 35, 2001). To this end, we appeal to the semi-group density of the Maass Laplacian and extend it to purely-imaginary values of the magnetic field. In the last section, we use the Karlin-McGregor formula to derive an expression of the semi-group density of the multi-dimensional Hua-Pickrell particle system introduced by T. Assiotis
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Explicit expressions of the Hua-Pickrell semi-group
2020Co-Authors: Arista Jonas, Demni NizarAbstract:In this paper, we study the one-dimensional Hua-Pickrell diffusion. We start by revisiting the stationary case considered by E. Wong for which we supply omitted details and write down a unified expression of its semi-group density through the Associated Legendre Function in the cut. Next, we focus on the general (not necessarily stationary) case for which we prove an intertwining relation between Hua-Pickrell diffusions corresponding to different sets of parameters. Using Cauchy Beta integral on the one hand and Girsanov's Theorem on the other hand, we discuss the connection between the stationary and general cases. Afterwards, we prove our main result providing novel integral representations of the Hua-Pickrell semi-group density, answering a question raised by Alili, Matsumoto and Shiraishi (S\'eminaire de Probabilit\'es, 35, 2001). To this end, we appeal to the semi-group density of the Maass Laplacian and extend it to purely-imaginary values of the magnetic field. In the last section, we use the Karlin-McGregor formula to derive an expression of the semi-group density of the multi-dimensional Hua-Pickrell particle system introduced by T. Assiotis.Comment: 19 pages, the final expression of the semi-group density is obtaine
Judd S. Gardner - One of the best experts on this subject based on the ideXlab platform.
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Uniform Asymptotic Expansion of the Associated Legendre Function to Leading Term for Complex Degree and Integral Order
IEEE Transactions on Antennas and Propagation, 2007Co-Authors: Judd S. GardnerAbstract:The Associated Legendre Function arises naturally in the study of spherical waves. Since in practical applications it is most often symbolically represented by Pn m(xi) for m les n and Pn m(xi) equiv 0 for m > n where m is the integer order and n is the integer degree, this form will be employed to develop the uniform asymptotic expansion. The considerable extent to which this Function appears in literature substantiates its importance in engineering and science, and particularly to spherical harmonics. In his book, "Partial Differential Equations in Physics" Sommerfeld covers a variety of subjects including spherical harmonics, and gives a detailed account of obtaining an expansion of the Associated Legendre Function, Pn m(cos(thetas)), by the method of steepest descents over the interval 0 les thetas les pi. The results he obtains are quite accurate for n Gt m except as thetas approaches the critical points, thetas rarr 0 or thetas rarr pi. Beginning with the same integral representation of the Associated Legendre Function with integer order and degree that Sommerfeld employed, a uniform asymptotic expansion is found that is applicable to the neighborhoods of thetas = 0 and thetas = pi and that becomes increasingly more accurate as n increases beyond m. Furthermore, the accuracy of the resulting uniform asymptotic expansion remains for real degree and complex degree as well. The results are plotted in order to assess the accuracy and the domain of validity of the uniform asymptotic expansion. The results of the uniform asymptotic expansion are also compared to the available approximation of the Associated Legendre Function given in terms of Bessel Functions for small values of thetas.
