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

Abstract

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/.