ON THE PRECISION OF THE SUCCESSIVE APPROXIMATIONS METHOD TESTED ON KEPLER'S EQUATION FROM ASTRONOMY

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1 NATURA RESOURCES AND SUSTAINABE DEVEOPMENT, 03 ON THE PRECISION OF THE SUCCESSIVE APPROXIMATIONS METHOD TESTED ON KEPER'S EQUATION FROM ASTRONOMY 39 Curilă Mircea, Curilă Sori Uiversity of Oradea, Faculty of Evirometal Protectio, Deartmet of Evirometal Egieerig, 6, Ge. Magheru street, Oradea, Româia, Uiversity of Oradea, Faculty of Electrotechics ad Iformatic, Deartmet of Electroics ad Telecommuicatios, 5 Armatei Româe street, zi code 40087, Oradea, Româia, scurila@uoradea.ro Abstract I this aer we oit out a method to esure the same recisio for the eact solutio as the recisio obtaied by the last two iteratios i the method of successive aroimatios. The method of double ad multile iteratio could be a imrovemet of the method of successive aroimatios i the aim to eted the iterval for the ichitz costat, such that the maimum umber of eact digits to be esured i the aroimatio of the Keler's equatio solutio. Key words: Keler's equatio, successive aroimatios, trascedetal equatio INTRODUCTION The method of successive aroimatios is oe of the owerful tools to solve oliear equatios umerically. A remarkable trascedetal equatio to which oe ca aly the method of successive aroimatios is Keler's equatio arisig i Astroomy. Keler's equatio gives the relatio betwee the olar coordiates of a celestial body, such as a laet or a comet, ad the time elased from a give iitial oit. Keler's equatio is of fudametal imortace i Astroomy, but caot be directly iverted i terms of simle fuctios i order to determie where the laet will be at a give time. Keler s equatio ad other oliear equatios were early solved umerically ad itesively studied usig the method of successive aroimatios, as ca be see i (Aaboe A., 0), (Colwell P., 993), (Coma G. et al., 976), (Daby J.M.A., 988), (Demidowitch B. et al., 979), (Swerdlow N.M., 000), but the umber of eact digits i these aroimatios could ot be established with certitude whe the ischitz costat of the ivolved fuctio is greater tha 0.5. I order to imrove this status we roose a refiemet of successive aroimatios method by icludig double or multile iteratios such that the umber of eact digits i the aroimatio to be determied with assuredess. This is imortat from alicatios oit of view sice Keler s equatio ad similar other equatios aear i several cotets i alied scieces ad egieerig. Moo Margaret of laet Uraus was discovered back i 00 by S.S. Sheard ad D.C. Jewitt. More recet observatios showed it to be a distict celestial body ad o October 9, 003, its recogitio was officially acceted ad icluded i the list as the 7 th moo of Uraus. The discovery was made usig the 8.3 meter Subaru telescoe at Maua Kea (Hawaii). I 008, Margaret's curret eccetricity was This temorarily gives Margaret the most eccetric orbit of ay moo i the solar system, though Nereid's mea eccetricity is greater.

2 MATERIA AND METHOD I the case that the equatio: f ( ) 0, R ca be trasformed ito the equivalet form, ( ), I R with I a iterval, the method of successive aroimatios geerates a recurret sequece: ( ), N which i certai coditios coverges to oe of the solutio of the equatio (). Whe aly the method of successive aroimatios, each real solutio of the equatio () is isolated i a secified iterval by usig the method of roots searatio geerated by the techique of the Rolle's sequece, see (Demidowitch B. et al., 979). After these, each such iterval is shorted as ca as ossible such that the first derivative ' to satisfy the iequality '() o the whole iterval. This iequality leads to the covergece of the sequece of successive aroimatios, give by the recurrece ( ), N, to the solutio, startig from ay first iterative ste 0. et q be the mea aomaly (a arameterizatio of time) ad the eccetric aomaly (a arameterizatio of olar agle) of a celestial body (laet or comet) orbitig o a ellise with eccetricity E, the Keler's equatio is give by: E si( ) q () where E 0, ad q > 0, see (Aaboe A., 0) ad (Swerdlow N.M., 000). I (Aaboe A., 0), the method of successive aroimatios is alied to equatio () for Mars ad Mercur with q ad , while i (Motebruck O. et al., 009) the Newto's method is alied to equatio (). If I is the eact solutio of the equatio ( ) ad '(), I, the lim ad the followig a riori error estimate ad a osteriori error estimate ' hold, where ma ( ) : I, is the ischitz costat of, see (Coma 0, N (), N (3) G. et al., 976) ad (Demidowitch B. et al., 979). The estimate (3) offers a ractical stoig criterio of the sequece ( ), eouced as follows: for give 0, fid the first atural umber for which ad sto to this. The the recisio i the aroimatio of by the term is. 30

3 The urose of this aer is to secify the recisio of the aroimatio by idicatig which digits are eact i the decimal reresetatio of. How ca hel the usual method of successive aroimatios i this aim? For istace, if =0.8, =0.000=0-4 ad =3.658, the we have , that is So, 3.654,3.65 ad we caot say that the third digit is eact i = But, if 0. 5, the 0 K ad all K- of the first digits i the reresetatio of are eact. More recisely, if =0.5, =3.658, ad the ad 3.653, ,. Thus, the third digit is eact. From these, we see that if the ischitz costat is such that 0. 5, the the iequality offers the isurace that the first K- digits after dot are eact. But if 0.5<<, the the iequality ot esure the eactess of the first K- digits. Ca be ameliorated this situatio? How? I order to resod to this questio we ass to the idea of double iteratio ad more geeral, to multile iteratio. The method of double iteratio is described by the sequece ( ( )) N (4), K Theorem: If the the a osteriori error estimate 0 leads to K- eact first digits after dot, usig the method of double iteratio (4). Proof: I the double iteratio method, the ischitz roerty ( ) ( y) y,, y I leads to ( ) ( ) ( ) ( ) N, Thus,... ad assig to limit for, we get: ( )...,, N which is the a osteriori error estimate corresodig to the double iteratio method. If K, the ad 0, N, that is the recisio of, N (5) ad are the same. So, the first K- digits are eact i the term. Sice the case 0.5 eteds for the iterval [0, 0.5] which esure the 3

4 eactess of the digits, we ifer that the method of double iteratio ameliorates the roblem of determiig the recisio whe >0.5, but. Cosequetly, by the method of double iteratio, the iterval (0, 0.5] is eteded to the iterval 0, for the ischitz costat, i order to esure the maimum umber of eact digits after dot. Remark: The method of multile iteratio based o the iterative sequece (... )( ), N times eted the iterval (0, 0.5] to 0, for the ischitz costat, because i this case, the a osteriori error estimate becomes, N NUMERICA EXPERIMENT The eerimet tests the method of successive aroimatios alied to umerical solvig of Keler's equatio arisig i Astroomy. Cosider Keler's equatio with E= ad q=0.5 (corresodig to a comet): si( ) 0.5,, 4 Here ( ) si( ) 0. 5 ad ma '( ) :, ma cos( ) :, cos Thus, the method double iteratio is eough to esure the maimum umber of 4 8 eact digits. Takig 0 ad 0,0,0 we obtai with double iteratios 4 the results reseted i table : Table The method of double iteratio eact digits With simle iteratio the results are reseted i table : Table The method of successive aroimatios obtaiig the same values for. 3

5 which The stoig criterio was: for give 0 fid the first atural umber for ad sto to this iteratio, retaiig the value as aroimatio of the solutio. I the case of moo Margaret of laet Uraus, the eerimetal results obtaied for the aroimatio of Keler's equatio solutio with the method of successive aroimatios ad of double iteratio are reseted i table 3. As ca be viewed i these two eamles, we ca secify that after 6 double iteratios, the aroimate of the solutio of Keler s equatio alied to arabolic orbits has eact first four digits after dot, ad after 0 double iteratios, the aroimate solutio of Keler s equatio alied to the orbit of the moo Margaret has first 0 eact digits after dot. By usig the method of successive aroimatios we eed to reeat oce the iteratios whe the sequece becomes statioary, i order to be sure about the umber of iteratios. So, our roosed method offers the certitude for the umber of eact digits. Table 3 Aroimatio of Keler's equatio solutio for Margaret moo of Uraus laet successive aroimatios double iteratio =e-0 =e-0 =9 = The method eemlified here for Keler s equatio could be alied to other oliear equatio arisig i egieerig, hysics, ad biology, with the same beefit. CONCUSIONS I order to solve trascedetal Keler's equatio arisig i Astroomy, oe ca aly the method of successive aroimatios which is oe of the owerful tools to solve oliear equatios umerically. 33

6 The method of double ad multile iteratio could be a imrovemet of the method of successive aroimatios i the aim to eted the iterval (0,0.5] to 0, for the ichitz costat, such that the maimum umber of eact digits to be esured i the aroimatio of the solutio. This method is a real imrovemet of the method of successive aroimatios from umerical oit of view. The alicability of the method cosists i the followig asects: - requires the same coditios as the method of successive aroimatios; - i comariso with the method of successive aroimatios offers the certitude about the umber of eact digits of the aroimatio; - it is a imrovemet of the method of successive aroimatios secifyig the ecessary umber of iterative stes i order to esure a secified tolerace i the aroimatio of the solutio; - ca be alied to the same eamles from the area of scieces ad egieerig as the method of successive aroimatios. REFERENCES. Aaboe A., 0, Eisodes from the early history of astroomy, Sriger, Berli, Bica A.M., 004, Mathematical models i biology govered by diferetial equatios, PhD Thesis, "Babeş-Bolyai" Uiversity Cluj-Naoca Bica A.M., Curilă M., Curilă S., 0, About a umerical method of successive iterolatios for fuctioal Hammerstei itegral equatios, J. of Comutatioal & Alied Mathematics, 36, Bica A.M., Curilă M., Curilă S., 03, The method of successive iterolatios solvig iitial value roblems for secod order fuctioal diferetial equatios, Fied Poit Theory 4, o., Bica A.M., Curilă M., Curilă S., 0, The method of successive iterolatios for fuctioal itegral equatios, Proc. of the It. Cof. o Noliear Oerators, Dif. Eq. ad Al. ICNODEA, Cluj-Naoca. 6. Bica A.M., Curilă M., Curilă S., 03, Etedig the method of successive aroimatios for itegral equatios, i: Fied Poit Theory ad its Alicatios (eds. R. Esiola, A. Petruşel, S. Prus), Proc. of the 0th It. Cof. FPTA, House of the Book of Sciece, Cluj-Naoca, Colwell P., 993, Solvig Keler's Equatio over Three Ceturies, Publisher: VA: Willma- Bell, ISBN-3: Coma G., Pavel G., Rus I., Rus A., 976, Itroducere î teoria ecuaţiilor oeraţioale, Ed. Dacia, Cluj-Naoca. 9. Daby J.M.A., 988, Fudametals of Celestial Mechaics, Publisher: VA: Willma-Bell, ISBN-3: Demidowitch B., Maro I., 979, Elemets de calcul umerique, Ed. Mir, Moscow.. Iacu C., 98, O the cubic slie of iterolatio, Sem. of Fuct. Aal. ad Num. Meth. Cluj- Naoca 4, Iacu C., Mustăţa C., 987, Error estimatio i the aroimatio of fuctios by iterolatio cubic slies, Mathematica, Tome9 (5), No, Ioakimids N.I., Paadakis K.E., 985, A New Simle Method for the Aalytical Solutio of Keler's Equatio, Celestial Mechaics ad Dyamical Astroomy, Volume 35, Issue 4, , ISSN: Motebruck O., Pfleger T., 009, Astroomy o the ersoal comuter, Sriger-Verlag, Berli Heidelberg, Swerdlow N.M., 000, Keler's iterative solutio to Keler's equatio, J. for History of Astroomy, 3,

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