Journal of Geodesy

, Volume 86, Issue 4, pp 271–285 | Cite as

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

Original Article


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 232 =  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.


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


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Supplementary material

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.National Astronomical ObservatoryTokyoJapan

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