On computation and use of Fourier coefficients for associated Legendre functions

Abstract

The computation of spherical harmonic series in very high resolution is known to be delicate in terms of performance and numerical stability. A major problem is to keep results inside a numerical range of the used data type during calculations as under-/overflow arises. Extended data types are currently not desirable since the arithmetic complexity will grow exponentially with higher resolution levels. If the associated Legendre functions are computed in the spectral domain, then regular grid transformations can be applied to be highly efficient and convenient for derived quantities as well. In this article, we compare three recursive computations of the associated Legendre functions as trigonometric series, thereby ensuring a defined numerical range for each constituent wave number, separately. The results to a high degree and order show the numerical strength of the proposed method. First, the evaluation of Fourier coefficients of the associated Legendre functions has been done with respect to the floating-point precision requirements. Secondly, the numerical accuracy in the cases of standard double and long double precision arithmetic is demonstrated. Following Bessel’s inequality the obtained accuracy estimates of the Fourier coefficients are directly transferable to the associated Legendre functions themselves and to derived functionals as well. Therefore, they can provide an essential insight to modern geodetic applications that depend on efficient spherical harmonic analysis and synthesis beyond [$5~\times ~5$] arcmin resolution.

Appendices

Appendix 1

To demonstrate how the Fourier coefficient round-off errors will infiltrate the functional domain, we consider Bessel’s inequality. Following Sansone (1991), we can use the theorem:

“Let {$f_k$} be a sequence of orthonormal functions in $\sigma$, F square integrable in $\sigma$  and {$a_{k}$} the sequence of Fourier coefficients with respect to {$f_k$},

$$\begin{aligned} a_k = \int _\sigma F f_k~\mathrm{d}\sigma , \quad (k = 1,2,\ldots ), \end{aligned}$$

(33)

then the series

$$\begin{aligned} \sum \limits _{k=0}^\infty a_k^2 \end{aligned}$$

(34)

is convergent, and, moreover, the $a_k$ will satisfy Bessel’s inequality:

$$\begin{aligned} \sum \limits _{k=0}^\infty a_k^2 \ \leqq \int _\sigma F^2~\mathrm{d}\sigma .\text {''} \end{aligned}$$

(35)

Concerning the round-off errors $\epsilon$ of our coefficients, we put:

$$\begin{aligned} b_k= & {} a_k +~\epsilon _k, \end{aligned}$$

(36)

and for the square residuals of the finite series

$$\begin{aligned} \phi = \int _\sigma \! \left[ P_{lm} -\! \sum \limits _{k=0}^N b_k~f_k \right] ^2\!\!\!\!\mathrm{d}\sigma . \end{aligned}$$

(37)

Note that $\sigma \in [0;2\pi ],$ although the Legendre functions $P_{lm}$ are known to be square integrable (i.e., they form another orthonormal system) only in $[0; \pi ]$. However, we need only a well-defined subset of the sequence of orthonormal functions $f_k' \subset f_k$, where the Fourier integrals may be used to obtain the appropriate coefficients $a_{lmk}$. These are (Eq. 1):

$$\begin{aligned} f_k' = \left\{ \begin{array}{llll} \cos k\, \vartheta &{} \quad (m- \text {even},&{}~l,k - &{}\text {even}),\\ \sin k\, \vartheta &{} \quad (m- \text {odd},&{}~l,k - &{}\text {even}),\\ \cos k\, \vartheta &{} \quad (m- \text {even},&{}~l,k - &{}\text {odd}),\\ \sin k\, \vartheta &{} \quad (m- \text {odd},&{}~l,k - &{}\text {odd}). \end{array} \right. \end{aligned}$$

(38)

In all other cases, $\int \limits _0^{2\pi } P_{lm}~f_k~\mathrm{d}\sigma$ are not guaranteed to be solved correctly. We then continue for the subset $f_k'$ and obtain

$$\begin{aligned} \phi&= \int _\sigma P_{lm}^2~\mathrm{d}\sigma + \sum _{k=0}^N b_k^2 \int _\sigma f^{'2}_k\nonumber \\&\quad + 2\sum _{k\ne n} b_k~b_n \int _\sigma f'_k~f'_n~\mathrm{d}\sigma - 2\sum _{k=0}^N b_k \int _\sigma P_{lm}~f'_k~\mathrm{d}\sigma \nonumber \\&= \int _\sigma P_{lm}^2~\mathrm{d}\sigma + \sum \limits _{k=0}^N b^2_k -2\sum _{k=0}^N b_k~ a_k \nonumber \\&= \int _\sigma P_{lm}^2~\mathrm{d}\sigma + \sum \limits _{k=0}^N (a^2_k+ 2 a_k~\epsilon _k + \epsilon ^2_k) \nonumber \\&\quad - 2\sum \limits _{k=0}^N (a^2_k+ a_k~\epsilon _k) \nonumber \\&= \int _\sigma P_{lm}^2~\mathrm{d}\sigma - \sum \limits _{k=0}^N a^2_k + \sum \limits _{k=0}^N\epsilon ^2_k, \end{aligned}$$

(39)

where

$$\begin{aligned} \int _\sigma f_k f_n~\mathrm{d}\sigma = 0,\quad k \ne n; \quad \int _\sigma f^2_k~\mathrm{d}\sigma = 1. \end{aligned}$$

(40)

Because the error-free solution is defined by

$$\begin{aligned} \int _\sigma \left[ P_{lm} - \sum \limits _{k=0}^N a_k~f_k' \right] ^2 \mathrm{d}\sigma \equiv 0, \end{aligned}$$

(41)

we obtain for the square residuals

$$\begin{aligned} \phi = \sum \limits _{k=0}^N\epsilon ^2_k, \end{aligned}$$

(42)

which corresponds to what we have been investigating throughout Eqs. (24–27) and also in Eq. (28) and can thus be considered as corresponding deviation in the functional domain.

Appendix 2

The dynamical scaling works as follows: the sectorial Legendre function $(l=m)$ serves as starting value and is subject to numerical underflow such that extended exponent values have to be stored. In the course of the recursive computations of the respective degrees, the external exponents are then partially re-applied before overflow values occur until the scales have fully dissolved.

$$\begin{aligned}&\mathrm {\% { S}_c\ vector\ of\ external\ scales\ with \ dimension:}\ \theta \\&\mathrm {\% \ { p}\ \ vector\ of\ Legendre \ Functions} \nonumber \end{aligned}$$

$$\begin{aligned} \begin{array}{llll} &{} p &{} = \ \mathrm {init\ } &{} \mathrm {\% factorials} \\ \mathrm {for} &{} i &{}= \ 1\ :\ m \\ &{} p &{} = \ p \cdot \sin \theta \\ &{} ip &{} = \ \mathrm {find}(\log | p | < -700) &{} \mathrm { \%underflow?}\\ &{} \mathrm {if} &{} \mathrm {not\ empty (ip)} &{} \\ &{} \quad s_{c} &{} = \ \mathrm {round}( \log |p(ip)| ) &{} \mathrm {\% extract\ exponent} \\ &{} \quad p(ip) &{} = \ p(ip) \cdot 2^{-s_{c}} &{} \mathrm {\% and\ apply\ scaling} \\ &{} \quad S_c(ip) &{} = \ S_c(ip) + s_{c} &{} \mathrm {\% update \ scale} \\ &{} \mathrm {end\ if} &{}&{}\\ \mathrm {end} &{}&{}&{} \end{array} \end{aligned}$$

During l-recursion, the Legendre functions are then re-scaled in the following way:

$$\begin{aligned} \begin{array}{llll} \mathrm {for} &{} l &{}= m+1: \mathrm {lmax} &{} \nonumber \\ &{} p &{} = func(p_{-1}, p_{-2}) &{} \%l-\mathrm {recursion}\\ &{} p_{-2} &{}= p_{-1} &{}\\ &{} p_{-1} &{}= p &{}\\ &{} ip &{}= \mathrm {find}(\log |p| > 700) &{} \mathrm {\% overflow?} \\ &{} \mathrm {if} &{} \mathrm {not \ empty (ip)} &{}\\ &{} \quad s_c &{}= \max ( -700, S_c(ip) ) &{} \mathrm {\%partial \ scale}\\ &{} \quad p_{-1}(ip) &{} = p_{-1}(ip) \cdot 2^{s_c} &{} \mathrm {\% dissolve \ scaling}\\ &{} \quad p_{-2}(ip) &{} = p_{-2}(ip) \cdot 2^{s_c} &{} \\ &{} \quad S_c(ip)&{} = S_c(ip) - s_c &{} \mathrm {\% update \ scale}\\ &{} \mathrm {end\ if} &{}&{}\\ &{}p &{}= p_{-1} \cdot 2^{S_c} &{} \mathrm {\%final\ re-scale}\\ \mathrm {endif}. \end{array} \end{aligned}$$