Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

Transcription

1 Applied Mathematical Scieces, Vol. 7, 03, o. 6, HIKARI Ltd, Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series Nichaphat Pataarapeelert, ad Vimolyut Varasavag Departmet of Mathematics, Faculty of Applied Sciece Kig Mogkut s Uiversity of Techology North Bagkok 8 Pracharat Sai Road, Wogsawag, Bagsue Bagkok 0800, Thailad ichaphatb@kmutb.ac.th Copyright 03 Nichaphat Pataarapeelert ad Vimolyut Varasavag. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract Chebyshev ad Legedre series have bee widely used i may areas of physics ad egieerig. I this study, we eamie o the efficacy of both seris i terms of series approimatio ad covergece. Some eamples of fuctios that ca be epaded i the form of ifiite series of Chebyshev ad Legedre polyomials are cosidered. Usig the same umber of polyomials coefficiets, we fid that Chebyshev series give a better approimatio tha Legedre series. I additio, we fid that the rate of covergece of ifiite series geerated from the same fuctio i terms of Chebyshev polyomial are more rapid tha Legedre polyomial for while for, Chebyshev ad Legedre series give the close rate of covergece. Fially, the covergece of Chebyshev ad Legedre series with the same coefficiets are cosidered. Keywords: Chebyshev polyomial, Legedre polyomial, Series, Covergece

2 36 Nichaphat Pataarapeelert ad Vimolyut Varasavag Itroductio Special fuctios arise i may problems of pure ad applied mathematics, mathematical statistics, physics, ad egieerig. I particular, they play a importat role i umerical aalysis as see i approimatio of itegrals ad solutio of differetial equatios i physic problems e.g. electricity problems [3], [6]. Some special fuctios are ofte employed i approimatig a give fuctio as a appropriate polyomial. Especially, polyomials cosiderig with low order are easy to hadle ad ca be performed with miimal pealty i accuracy for computatioal speed [7]. I additio to use trucated Taylor epasio, Chebyshev ad Legedre polyomials are also helpful tools to approimate cotiuous fuctios [3]. However, the compariso of which series gives better approimatio may ot be straightforward. I this research, we fill this gap by rearragig Chebyshev ad Legedre series i the form of power series so that we are able to compare the accuracy of approimatio. We will give some eamples of epadig fuctios i term of ifiite series of Chebyshev Polyomials ad Legedre Polyomials. Usig these epasios, we will approimate the fuctios uder the same umber of polyomials coefficiets. Moreover, we compare the rate of covergece of ifiite series i terms of Chebyshev ad Legedre polyomials that are epaded from the same fuctio. Fially, we aalyze ad determie the coditio for covergece of ifiite series represeted i terms of Chebyshev or Legedre polyomials. Usig such coditios, we compare the rate of covergece of series of Chebyshev ad Legedre polyomials where the coefficiets of such series are the same. Approimatios of Fuctios Usig Chebyshev ad Legedre Series Epasios Cosider a fuctio f ( ) where that ca be writte i the form of Chebyshev series epasio as [6] f( ) ct( ) () 0 where T ( ) is Chebyshev polyomial of the first kid of degree. The coefficiets ad c determied from the orthogoal property are give by f( T ) ( ) c 0 d ()

3 Compariso study of series approimatio 37 f( T ) ( ) c ;,, 3,... d (3) For Legedre series epasio of f, ( ) the series is i the followig form where ( ) coefficiet a is give by f ( ) ap ( ) (4) 0 P is Legedre polyomial of the first kid of degree ad the a f ( ) P ( ) d ; 0,,, 3,.... () As kow that the fiite Chebyshev ad Legedre series epasios are very useful i the approimatio of various fuctios, here we preset some fuctios with their Chebyshev ad Legedre series epasio forms ad the approimate their partial sums as follows.. Trigoometric Fuctio We first epad a fuctio f ( ) si ;, (6) as Chebyshev series ct ( ) 0. To determie the coefficiets c, we substitute cos i equatios (3). We the have 4 c ;, 3,,.... (7) Sice si is odd fuctio, we fid that c 0 for 0,, 4, 6,.... Therefore, Chebyshev series epasio of si is i the form 4 si ( ) m m 0 (m ) T. (8) To approimate this series, we will cosider its partial sum for five terms as 4 si T ( ) T ( ) T ( ) T ( ) T ( ). (9) we substitute T ( ), T 3 () 4 3, T 3 ( ) 6 0, T ( ) , ad T ( ) i equatio (9). Therefore, the iverse sie fuctio ca be writte i term of degree of with a five terms trucated series as si (0)

4 38 Nichaphat Pataarapeelert ad Vimolyut Varasavag Further, we will fid Legedre series epasio ap ( ) of (6). 0 Employig orthogoal relatio, we fid that that a 0for, 4, 6,... while.3...( ) a ad a ( ( ) ( )) ; 3,, 7,... P P ( ) () We hece obtai Legedre series epasio as.3...(m ) f ( ) P ( ) ( ( ) ( )) P P m m m ( m). () Similarly, usig the same umber of polyomials coefficiets as Chebyshev series 3 epasio, ad substitutig P( ), P ( ) ( 3 ), P ( ) (63 70 ), P 7( ) ( ), ad P9 ( ) ( ), we have si (3) The comparisos of usig these two approimatios are show i followig Table ad Table. Table. Approimatios of f ( ) si obtaied from equatios (0) ad (3) by usig Chebyshev ad Legedre polyomials, respectively. si Chebyshev series Legedre series epasio epasio Table. Errors compariso from calculatig five terms of series epasios betwee Chebyshev ad Legedre of f ( ) si. Error from usig Error from usig Chebyshev series epasio Legedre series epasio

5 Compariso study of series approimatio 39 From Table, we fid that the values from Chebyshev series epasio for five terms are more accurate tha Legedre series epasio whe the umber of polyomials coefficiets is the same. However, we fid that whe, the error of usig Chebyshev is while it is zero from Legedre. Moreover, whe 0, the error from usig partial sums of both of Chebyshev ad Legedre have is zero.. Epoetial Fuctio For epoetial fuctio, f ( ) e ;, (4) it ca be approimated by usig Chebyshev ad Legedre series epasios as the previous fuctio. To compare the efficiecy of usig these series, we proceed as follows. We begi with determiig the coefficiets of Chebyshev series epasio as give i ()-(3). Substitutig cos, we fid that cos c0 e d () 0 ad cos c e cos d ;,, 3,... (6) 0 We calculate the itegrals umerically i equatios () ad (6) by employig Simpso s 3 rule with the equal space h. We the obtai 0 h cosh cos h cos9h c0 e 4e e... 4e e.66 3, (7) h cosh cos h cos9h c e 4e cosh e cos h... 4e cos9h e (8) We ivestigate c, c3, c 4 by usig similar process ad the obtai c 0.7, c , adc Therefore, the approimatio of epoetial fuctio i term of partial sum of Chebyshev polyomial is give by e.66 T ( ).303 T ( ) 0.7 T ( ) T ( )+0.00 T ( ) (9) After substitutig T ( ) for 0,,, 3, 4 i terms of, we obtai 3 4 e (0) For the coefficiets of Legedre series epasio give i equatio (), we fid that a e P( ) d ; 0,,, 3,.... ()

6 330 Nichaphat Pataarapeelert ad Vimolyut Varasavag We ca determie a for 0,,, 3, 4, by substitutig P ( ) i term of ad usig by part itegratio. Fially, we obtai the approimatio of epoetial fuctio writte i the form of partial sum of Legedre polyomials 4 0 ap ( ) as e.7 P0( ).036 P( ) P( ) P3( )+0.0 P4( ). () We ote that this is the same umber of polyomials coefficiets as Chebyshev series i (9). Agai, we rewrite () i term of as 3 4 e (3) The comparisos of usig these two approimatios are show as followig Table 3 ad Table 4. Table 3. Approimatios of f ( ) e obtaied from equatios (0) ad (3) by usig Chebyshev ad Legedre polyomials, respectively. e Chebyshev series Legedre series epasio epasio Table 4. Errors compariso from calculatig five terms of series epasios betwee Chebyshev ad Legedre of f ( ) e. Error from usig Chebyshev series epasio Error from usig Legedre series epasio From Table 4, we observe that the errors from usig Chebyshev series epasio of f ( ) e are smaller tha Legedre series epasio whe we calculate the partial sum for five terms. For this five terms calculatio, however, we ote that at 0 the error obtaied from usig Chebyshev is while it is zero whe we use Legedre series epasio.

7 Compariso study of series approimatio 33 3 The Rate of Covergece of Chebyshev ad Legedre Series Epasios 3. The series epaded from the same fuctio It is kow that both of T ( ) ad P ( ) are orthogoal polyomials for. This meas that if we defie some aalytic fuctios for, we may write those fuctios i term of T ( ) ad P ( ) give by () ad (4), respectively. Although, both series coverge, the rate of covergece may be differet. I this sectio, we study the rate of covergece of series obtaied from Chebyshev ad Legedre epasios. For, we cosider the step fuctio give by 0 ; 0 f ( ) (4) ; 0. This give fuctio ca be writte i the form of Chebyshev ad Legedre series epasios as m f ( ) ( ) T ( 0 ) ( ) ( ) m T m 4m T, () ad m ( ) (m )!(4m ) f ( ) P( ) P ( ) ( ) 0 P, (6) m 4 m m ( m )!( m )! respectively. To compare the rate of covergece betwee these two series, we determie the value of ide m used for umerical calculatio for each series uder the tolerace of order 0. The results are show i Table as follows. Table. Value of ide m for covergece of () ad (6) subject to the error tolerace of 0. m for Chebyshev m for Legedre We observe from Table that the ifiite series of Chebyshev i () requires less term for approimatio tha ifiite series of Legedre i (6) uder the tolerace of order 0. Therefore, we ca coclude that Chebyshev series epasio are quite coverge rapidly with respect to Legedre series epadig from the step fuctio (4).

8 33 Nichaphat Pataarapeelert ad Vimolyut Varasavag I a iterestig case, we focus o the covergece of ifiite series i term of Chebyshev ad Legedre polyomials for. I the followig eample, we cosider the Chebyshev ad Legedre series epasio of the fuctio l. Let 8 8 w 7 8 (7) 8 where w, we fid that. Sice w 87w l l 8 l 8 7w 8 7w (8) ad this fuctio is odd, we ca fid the series epasio of l as 8 7w 8 l c T ( w) m m 8 7w (9) m0 where cos c l cos(m ) d m cos (30) ad 8 7w 8 l a P ( w) m m 8 7w (3) 0 where 4m 3 87w a 8 l P ( w) dw m m 8 7w (3) which are represeted i terms of Chebyshev ad Legedre polyomials, respectively. We et determie the value of ide m used for umerical calculatio for each series for l uder the tolerace of order 0. The results are show i Table 6. Table 6. Value of ide m for covergece of (9) ad (3) subject to the error tolerace of 0. m for Chebyshev m for Legedre We observe from Table 6 that the ifiite series of Chebyshev i (9) requires more or equal terms i approimatio tha Legedre i (3). Therefore, we might

9 Compariso study of series approimatio 333 coclude that the rate of covergece of Legedre series epaded from l is close to Chebyshev series epasio i this case. 3. The series with the same coefficiet I this sectio, we fid the coditios for covergece of Chebyshev series ct ( ) ad Legedre series 0 ap ( ). For, the ifiite 0 series i term of Chebyshev polyomial is absolutely coverget if ct ( ) 0 coverges. Sice T ( ) cos, the ecessary coditio for absolute covergece is that c must coverge. While, the ifiite series i term of 0 Legedre polyomial is absolutely coverget if ap coverges. 0 Sice P ( ), this implies that the ecessary coditio is that coverge. ad I additio, we determie the coditios for covergece of 0 a must 0 0 ct ( ) ap ( ) whe. For Chebyshev series, we let cosh t ; t 0. By usig the geeratig fuctio for ( ) Therefore, the Chebyshev series T, we fid that T (cosh t) cosh t. ct ( ) for is i the form f ( ) c T (cosh t) c cosh t. (33) Usig the ratio test of covergece, we get c cosh( ) t lim c cosh t Lettig L lim Moreover, sice c e e lim t t c e e c t lim e. c c c t e ( ) t ( ) t, the Chebyshev series is absolutely coverget if e t e t (34). (3) L, the coditio (3) becomes

10 334 Nichaphat Pataarapeelert ad Vimolyut Varasavag L L. (36) We et cosider the ifiite series preseted i term of Legedre polyomials for. Similarly, we defie cosh t ; t 0. The closed form approimatio of Legedre polyomial P ( ) for large [8] is i the form ( ) t e P (cosh t). (37) siht Therefore, the ifiite series i terms of Legedre polyomial the form ap ( ) is i 0 ( ) t N e ap(cosh t) (cosh ) ap t a. (38) 0 0 N siht whe N is sufficietly large. This series coverges whe the secod term o the right side of (38) coverges. Usig the same procedure as Chebyshev series, the ecessary coditio for absolutely covergece of ifiite series i terms of Legedre series is give by [8] M M (39) a where lim M. We observe that the covergece of such series deped a o the behavior of their coefficiets. We ow compare the rate of covergece of ifiite series i terms of Chebyshev ad Legedre polyomials where the coefficiets for both series are the same. For, employig the give coefficiet c a, (40) ( ) we fid that c a coverges which is satisfied the 0 0 0( ) above coditio. Therefore, the ifiite series of Chebyshev ad Legedre polyomials are absolutely coverget. To compare the rate of covergece, we determie the umber of terms for each series uder the tolerace of order 0. After calculatig, we fid that the ifiite series i terms of Chebyshev polyomials usig the same coefficiet employ much more term to coverge tha Legedre polyomials as show i Table 7.

11 Compariso study of series approimatio 33 Table 7. The umber of term for covergece of T ( ) ad 0 ( ) P ( ) subject to the error tolerace of 0. ( ) 0 Number of term for Number of term for Chebyshev Legedre For, we cosider the give coefficiet c a 0.. (4) We fid from (4) that L M 0.. Therefore, the rage of satisfied with the coditios (36) ad (39) that is.0. Here, the umbers of terms used for covergece of these series are show i Table 8. Table 8. The umber of term for covergece of subject to the error tolerace of 0. Number of term for Chebyshev 0 0. T ( ) ad 0. P ( ) 0 Number of term for Legedre We observe that ifiite series of Chebyshev polyomials requires more terms, uder the same coefficiet, for covergece tha Legedre polyomials for both case ad. 4 Coclusio I summary, the elemetary fuctios were eemplified i order to compare the accuracy of approimatio i which both series are rearraged i the form of trucated power series of five terms. Although, it is observed that Chebyshev series gives a better approimatio, the test for more geeral case which eeds theoretical support is a key issue for further study. Nevertheless, our results may be useful i practical ad ca be used as a guide for deeper ivestigatio.

12 336 Nichaphat Pataarapeelert ad Vimolyut Varasavag I additio, we compared the rate of coverget of Chebyshev ad Legedre series epasio epaded from the same fuctio. Whe, the result showed that Chebyshev series epasio of the give fuctio is quite covergig rapidly with respect to Legedre series epasio. While whe, the result showed that the rate of covergece of Legedre series is similar to Chebyshev series. Fially, we provide some ifiite series i terms of Chebyshev ad Legedre polyomials uder the same coefficiet. We foud that the series i term of Chebyshev requires more terms for covergece tha Legedre polyomials for both case ad. Although, these eperimets might ot be claimed for compariso of the rate of covergece from usig Chebyshev ad Legedre polyomials because they both coverge to the differet fuctios, oe may study further about the characteristic of those fuctios ad ivestigate why Legedre polyomials require less umber of terms tha Chebyshev polyomials uder the same coefficiet. Ackowledgemet This research is supported by Sciece ad Techology Research Istitute, Kig Mogkut s Uiversity of Techology North Bagkok. Refereces [] A. Gil, J. Segura, ad N. M. Temme, Numerical Methods for Special Fuctios, Society for Idustrial ad Applied Mathematics, 007. [] G. E. Adrews, R. Askey ad R. Roy, Special Fuctios, Cambridge Uiversity Press, 000. [3] H. Wag ad S. Xiag, O the covergece rates of Legedre approimatio, Mathematics of Computatio, 8(0), [4] J. Maso, Chebyshev Polyomials: Theory ad Applicatios. Kluwer Academic, 996. [] J. C. Maso ad D. C. Hadscomb, Chebyshev Polyomials, CRC Press, New York, 003. [6] M. A. Abutheraa ad D. Lester, Computable fuctio represetatios usig effective Chebyshev polyomial, World Academy of Sciece, Egieerig ad Techology, (3)007,

13 Compariso study of series approimatio 337 [7] M. A. Cohe ad C. O. Ta, A polyomial approimatio for arbitrary fuctios, Applied Mathematics Letters, 0. [8] N. Bookorkuea, V. Varasavag, ad S. Rataapu, A Effective Method for Calculatig the Sum of a Ifiite Series of Legedre Polyomails, 8(00), -3. [9] R. E. Attar, Legedre Polyomials ad Fuctios, Createspace, 009. [0] Z. C. She ad J. Jiamig, Computatio of Special Fuctios, Wiley-Itersciece, New York, 996. Received: April, 03