New methods for numerical evaluation of ultra-high degree and order associated Legendre functions

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F49777513%3A23520%2F22%3A43966135" target="_blank" >RIV/49777513:23520/22:43966135 - isvavai.cz</a>

Výsledek na webu

<a href="https://link.springer.com/article/10.1007/s11200-022-0830-9" target="_blank" >https://link.springer.com/article/10.1007/s11200-022-0830-9</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s11200-022-0830-9" target="_blank" >10.1007/s11200-022-0830-9</a>

Jazyk výsledku

angličtina

Název v původním jazyce

New methods for numerical evaluation of ultra-high degree and order associated Legendre functions

Popis výsledku v původním jazyce

We improve the precision and computation speed of the fully-normalized associated Legendre functions (fnALFs) for ultra-high degrees and orders of spherical harmonic transforms. We take advantage of their numerical behaviour of and propose two new methods for solving an underflow/overflow problem in their calculation. We specifically discuss the application of the two methods in the fixed-order increasing-degree recursion computation technique. The first method uses successive ratios of fnALFs and the second method, called the Midway method, starts iteration from tiny initial values, which are still in the range of the IEEE double-precision environment, rather than from sectorial fnALFs. The underflow/overflow problem in the successive ratio method is handled by using a logarithm-based method and the extended range arithmetic. We validate both methods using numerical tests and compare their results with the X-number method in terms of precision, stability, and speed. The results show that the relative precision of the proposed methods is better than 10-9 for the maximum degree of 100000, compared to results derived by the high precision Wolfram’s Mathematica software. Average CPU times required for evaluation of fnALFs over different latitudes demonstrate that the two proposed methods are faster by about 10-30% and 20-90% with respect to the X-number method for the maximum degree in the range of 50-65000.

Název v anglickém jazyce

New methods for numerical evaluation of ultra-high degree and order associated Legendre functions

Popis výsledku anglicky

We improve the precision and computation speed of the fully-normalized associated Legendre functions (fnALFs) for ultra-high degrees and orders of spherical harmonic transforms. We take advantage of their numerical behaviour of and propose two new methods for solving an underflow/overflow problem in their calculation. We specifically discuss the application of the two methods in the fixed-order increasing-degree recursion computation technique. The first method uses successive ratios of fnALFs and the second method, called the Midway method, starts iteration from tiny initial values, which are still in the range of the IEEE double-precision environment, rather than from sectorial fnALFs. The underflow/overflow problem in the successive ratio method is handled by using a logarithm-based method and the extended range arithmetic. We validate both methods using numerical tests and compare their results with the X-number method in terms of precision, stability, and speed. The results show that the relative precision of the proposed methods is better than 10-9 for the maximum degree of 100000, compared to results derived by the high precision Wolfram’s Mathematica software. Average CPU times required for evaluation of fnALFs over different latitudes demonstrate that the two proposed methods are faster by about 10-30% and 20-90% with respect to the X-number method for the maximum degree in the range of 50-65000.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10508 - Physical geography

Projekt

<a href="/cs/project/GA21-13713S" target="_blank" >GA21-13713S: Odhady nejistot pro integrální transformace v geodézii</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2022

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Studia Geophysica et Geodaetica

ISSN

0039-3169

e-ISSN

1573-1626

Svazek periodika

66

Číslo periodika v rámci svazku

3-4

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

17

Strana od-do

81-97

Kód UT WoS článku

000883430500001

EID výsledku v databázi Scopus

2-s2.0-85141979721