The known standard recursion methods of computing the full normalized associated Legendre functions do not give the necessary precision due to application of IEEE754-2008 standard, that creates a problems of underflow and overflow. The analysis of the problems of the calculation of the Legendre functions shows that the problem underflow is not dangerous by itself. The main problem that generates the gross errors in its calculations is the problem named the effect of “absolute zero”. Once appeared in a forward column recursion, “absolute zero” converts to zero all values which are multiplied by it, regardless of whether a zero result of multiplication is real or not. Three methods of calculating of the Legendre functions, that removed the effect of “absolute zero” from the calculations are discussed here. These methods are also of interest because they almost have no limit for the maximum degree of Legendre functions. It is shown that the numerical accuracy of these three methods is the same. But, the CPU calculation time of the Legendre functions with Fukushima method is minimal. Therefore, the Fukushima method is the best. Its main advantage is computational speed which is an important factor in calculation of such large amount of the Legendre functions as 2 401 336 for EGM2008.
Abramowitz, W. and Stegan, I.A. (1972). Handbook of mathematical functions, 9th edn. Dover Publications, New York (470 pp).
Balmino, G., Vales, N., Bonvalot, S. and Briais, A. (2012). Spherical Harmonic modeling to ultra-high degree of Bouguer and isostatic anomalies. J. geod., 86: 499–520. DOI: 10.1007/s00190-011-0533-4.
Blais, J.A.R. (2008). Discrete spherical harmonic transforms: Numerical preconditioning and optimization. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 5102 LNCS (PART 2), 638–645.
Bucha, B. and Janák, J. (2013). A MATLAB-based graphical user interface program for computing functionals of the geopotential up to ultra-high degrees and orders. Comp. Geosci., 56, 186–196.
Dmitrenko, A. (2012). Modern transformation of determination the geoid [in Russian]. Krivoy Rog: Mineral ISBN 978-966-72-4 (218 pp).
Fantino, E. and Casoto, S. (2005). Comparison among spherical harmonic synthesis methods for functionals of the gravity field. Newton’s Bulletin 3, 32–49.
Fantino, E. and Casoto, S. (2009). Methods of harmonic synthesis for global geopotential models and their first-, second- and third-order gradients. J. Geod., 83, 595–619. DOI: 10.1007/s00190-008-0275-0.
Fukushima, T. (2012a). Numerical computation of spherical harmonics of arbitrary degree and order by extending exponent of floating points numbers. J. Geod., 86, 271–285. DOI: 10.1007/s00190-011-0519-2.
Fukushima, T. (2012b). Recursive computation of finite difference of associate Legendre functions. J. Geod., 86, 745–754. DOI: 10.1007/s00190-012-0553-8.
Fukushima, T. (2012c). Numerical computation of spherical harmonics of arbitrary degree and order by extending exponent of floating point numbers: II first-, second-, and third-order derivatives. J. Geod., 86, 1019–1028. DOI: 10.1007/s00190-012-0561-8.
Fukushima, T. (2013). Recursive computation of oblate spheroidal harmonics of the second kind and their first-, second-, and third-order derivatives. J. Geod., 87, 303–309. DOI: 10.1007/s00190-012-0599-7.
Fukushima, T. (2014a). Numerical computation of spherical harmonics of arbitrary degree and order by extending exponent of floating point numbers: III integral. Comp. Geosci., 63, 17–21. DOI: 10.1016/j.cageo.2013.10.010.
Fukushima, T. (2014b). Prolate spheroidal harmonic expansion of gravitational field. Astron. J., 147, 152–160.
Fukushima, T. (2015). 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 145, 1–5.
Fukushima, T. (2016). Zonal toroidal harmonic expansions of external gravitational fields for ring-like objects, Astron. J., 152, 35 (31pp).
Harris, J.W. and Stocker, H. (1998). Handbook of mathematics and computational science. Springer, Berlin Heidelberg New York (1028 pp).
Heiskanen, W.A. and Moritz, H. (1967). Physical geodesy. Freeman and Co, San Francisco (364 pp).
Hirt, C. (2012). Efficient and accurate high-degree spherical harmonic synthesis of gravity field functionals at the Earth’s surface using the gradient approach. J. Geod., 86, 729–744. DOI:10.1007/s00190-012-0550-y.
Holmes, S.A. and Featherstone, W.E. (2002a). A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalized associated Legendre functions. J. Geod., 76, 279–299. DOI: 10.1007/s00190-002-0216-2.
Holmes, S.A., Featherstone, W.E. (2002b). SHORT NOTE: Extending simplified high-degree synthesis methods to second latitudinal derivatives of geopotential. J. Geod., 76, 447–450. DOI: 10.1007/s00190-002-0268-3.
Holmes, S.A. and Featherstone, W.E. (2002c). A simple and stable approach to high degree and order spherical harmonic synthesis. Vistas for Geodesy in the New Millennium. Of the series International Association of Geodesy Symposia. 125, 259–264.
IEEE Computing Society (2008). 754–2008 – IEEE Standard for Floating-Point Arithmetic. doi:10.1109/IEEESTD.2008.4610935 (70 pp).
Jekeli, C. (2006). Geometric Reference Systems in geodesy: Division of Geodesy and Geospatial Science School of Earth Science, Ohio State University (202 pp). http://www.uacg.bg/filebank/att_1855.pdf.
Jekeli, C., Lee, K.J. and Kwon, J.H. (2007). On the computation and approximation of ultra-high-degree spherical harmonic series. J. Geod., 81, 603–615. DOI: 10.1007/s00190-006-0123-z.
Kwon, J.H. and Lee, J.-K. (2007). Numerical computation of ultra-high-degree legendre function. Journal of the Korean Society of Surveying Geodesy Photogrammetry and Cartography, 25 (1), 63–68.
Liu, Z., Liu, S. and Huang O. (2011). Scale factors in recursion of ultra-high degree and order Legendre functions and Horner’s scheme of summation. Acta Geodaetica et Cartographica Sinica, 40 (4), 454–458.
Lozier, D.W. and Smith, J.M. (1981). Algorithm 567 Extended-range arithmetic and normalized Legendre polynomials. ACM Trans. Math. Softw., 7, 141–146.
Moritz, H. (1980). Geodetic reference system 1980. Bull. Geod., 54(4), 395–405.
NIMA. (2000). Department of Defense World Geodetic System 1984: its definition and relationships with local geodetic systems (3rd ed.). US National Imagery and Mapping Agency Technical Report 8350.2 (175 pp).
Pavlis, N.K., Holmes, S.A., Kenyon, S.C. and Factor, J.K. (2012). The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). J. Geophys. Res., 117, B04406, 1–38.
Peng, F.-Q. and Xia, Z.-R. (2004). Algorithms for computing ultra-high-degree disturbing geopotential elements. Acta Geophysica Sinica, 47 (6), 1023–1028.
Šprlák, M. (2011). On the numerical problems of spherical harmonics: Numerical and algebraic methods avoiding instabilities of the associated legendre’s functions. ZFV – Zeitschrift fur Geodasie, Geoinformation und Landmanagement, 136 (5), 310–320.
Tschering, C.C., Rapp, R.H. and Goal C. (1983). A comparison of methods for computing gravimetric quantities from high degree spherical harmonic expansions. Manuscr. Geod. 8, 249–272.
Wang, Y.M. and Yang, X. (2013). On the spherical and spheroidal harmonic expansion of the gravitational potential of the topographic masses. J. Geod., 87, 909–921. DOI: 10.1007/s00190-013-0654-z.
Wenzel, H.-G. (1998). Ultra-high degree geopotential models GPM98A, B, and C to degree 1800. Submit. Proceedings of the Joint Meeting of the International Gravity Commission and International Geoid Commission, 7–12 September, Trieste, Italy.
Wittwer, T., Klees, R., Seitz, K. and Heck, B. (2008). Ultra-high degree spherical harmonic analysis and synthesis using extended-range arithmetic. J. Geod., 82, 223–229. DOI: 10.1007/s00190-007-0172-y.
Yu, J.-H., Zeng, Y.-Y., Zhu, Y.-C. and Meng, X.-C. (2015). A recursion arithmetic formula for Legendre functions of ultra-high degree and order on every other degree. Chinese J. Geophysics, 58(3), 748–755. DOI: 10.6038/cjg20150305.