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

The integrals of fully normalized associated Legendre function (fnALF) of extremely high degree and order such as 2 23 = 8 388 608 can be obtained without underflow problems if the point values of fnALF are properly given by using an exponent extension of the floating point numbers (Fukushima, T., 2012a. J. Geod., 86, 271–285; Fukushima, T., 2012c. J. Geod., 86, 1019–1028). A dynamic termination of the exponent extension during the fixed-order increasing-degree recursions significantly reduces the increase in CPU time caused by the exponent extension. Also, the sectorial integrals are found to be correctly obtained by the forward recursion only even when the backward recursion has been claimed to be necessary (Paul, M.K., 1978, Bull. Geod., 52, 177–190; Gerstl, M., 1980, Manuscr. Geod., 5, 181–199).