Acta Geodaetica et Geophysica

, Volume 53, Issue 2, pp 275–293 | Cite as

Polynomial approximation for fast generation of associated Legendre functions

  • M. R. Seif
  • M. A. Sharifi
  • M. Eshagh
Original Study


Today high-speed computers have simplified many computational problems, but fast techniques and algorithms are still relevant. In this study, the Hermitian polynomial approximation is used for fast evaluation of the associated Legendre functions (ALFs). It has lots of applications in geodesy and geophysics. This method approximates the ALFs instead of computing them by recursive formulae and generate them several times faster. The approximated ALFs by the Newtonian polynomials are compared with Hermitian ones and their differences are discussed. Here, this approach is applied for computing a global geoid model point-wise from EGM08 to degree and order 2160 and in propagating the orbit of a low Earth orbiting satellite. Our numerical results show that the CPU-time decreases at least two times for orbit propagation, and five times for geoid computation comparing to the case where recursive formulae for generation of ALFs are used. The approximation error in the orbit computation is at a sub-millimeter level over two weeks and that the computed geoid 0.01 mm, with a maximum of 1 mm.


Associated Legendre function Polynomial approximation Newton polynomial Hermite polynomial Fast orbit propagation Geoid computation 

Supplementary material


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Copyright information

© Akadémiai Kiadó 2018

Authors and Affiliations

  1. 1.Department of Surveying EngineeringArak University of TechnologyArakIran
  2. 2.School of Surveying and Geospatial Engineering, College of EngineeringUniversity of TehranTehranIran
  3. 3.Research Institute of Geoinformation Technology (RIGT), College of EngineeringUniversity of TehranTehranIran
  4. 4.Department of Engineering ScienceUniversity WestTrollhaättanSweden

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