Numerical computation of spherical harmonics of arbitrary degree and order by extending exponent of floating point numbers

作者：Fukushima Toshio^*

来源：Journal of Geodesy, 2012, 86(4): 271-285.

DOI：10.1007/s00190-011-0519-2

• Associated Legendre functions
• Exponent extension
• Floating point number
• Spherical harmonics
• Underflow problem

By extending the exponent of floating point numbers with an additional integer as the power index of a large radix, we compute fully normalized associated Legendre functions (ALF) by recursion without underflow problem. The new method enables us to evaluate ALFs of extremely high degree as 2(32) = 4,294,967,296, which corresponds to around 1 cm resolution on the Earth's surface. By limiting the application of exponent extension to a few working variables in the recursion, choosing a suitable large power of 2 as the radix, and embedding the contents of the basic arithmetic procedure of floating point numbers with the exponent extension directly in the program computing the recurrence formulas, we achieve the evaluation of ALFs in the double-precision environment at the cost of around 10% increase in computational time per single ALF. This formulation realizes meaningful execution of the spherical harmonic synthesis and/or analysis of arbitrary degree and order.

• 出版日期2012-4
• 单位中国科学院国家天文台