UNIVERSITI PUTRA MALAYSIA INTERPOLATION AND APPROXIMATION OF NON DIFFERENTIABLE FUNCTION USING POLYNOMIAL KOO LEE FENG FS

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1 UNIVERSITI PUTRA MALAYSIA INTERPOLATION AND APPROXIMATION OF NON DIFFERENTIABLE FUNCTION USING POLYNOMIAL KOO LEE FENG FS 7 9

2 INTERPOLATION AND APPROXIMATION OF NON DIFFERENTIABLE FUNCTION USING POLYNOMIAL KOO LEE FENG MASTER OF SCIENCE UNIVERSITI PUTRA MALAYSIA 7

3 INTERPOLATION AND APPROXIMATION OF NON DIFFERENTIABLE FUNCTION USING POLYNOMIAL By KOO LEE FENG Thesis Submitted to the School of Graduate Studies, Uiversiti Putra Malaysia, i Fulfilmet of the Requiremet for the Degree of Master of Sciece February 7

4 DEDICATION To my God Jehovah for his grace, kidess ad love. All glories, praises ad thaksgivigs to Him i the highest. For the LORD gives wisdom; From His mouth come kowledge ad uderstadig. Proverbs : 6 Scripture Psalm 4: - I waited patietly for the Lord; Ad He iclied to me, Ad heard my cry. He also brought me up out of a horrible pit, Out of the miry clay, Ad set my feet upo a rock, Ad established my steps. ii

5 Abstract of thesis preseted to the Seate of Uiversiti Putra Malaysia i fulfilmet of the requiremet for the degree of Master of Sciece INTERPOLATION AND APPROXIMATION OF NON DIFFERENTIABLE FUNCTION USING POLYNOMIAL By KOO LEE FENG February 7 Chairma: Faculty: Associate Professor Adem Kilicma, PhD Sciece I ay approimatio problem, we cocered with a measure of closeess of polyomial p ( ) to a fuctio f ( ). Approimatios are ofte obtaied by power series epasios i which the higher order terms are dropped. Oe of the fudametal idea i the differetial calculus is that a fuctio ca be locally approimated by its taget lie. If f be a fuctio defied o a ope iterval I ad let c I ad N. Suppose that the fuctio has -th derivative at all I. The, the polyomial T ( f, c)( c) is called as Taylor polyomial of order of fuctio at the poit c. If the fuctio is ifiitely differetiable o I, so the series is called Taylor Series of f at poit c. Taylor Series is oe of the fudametal idea i differetial calculus. However, Taylor Series oly ca apply if ad oly if the fuctio f be differetiated o its iterval stated. A differetiable fuctio is a cotiuous fuctio. But, this is ot always true that a cotiuous fuctio is a differetiable fuctio. Weierstass Approimatio Theorem iii

6 states that every cotiuous fuctio defied o a iterval [ a, b] ca be uiformly approimated as closely as the decided fuctio. Thus, this theorem esure that every fuctio ca be approimate by a polyomial. Cosequetly, i the research, we develop a ew approimatio method to approimate the o differetiable fuctio which has sigularity at oe poit, two poits ad three poits by usig Fourier series, Lagrage Iterpolatio ad covolutio method. We will also discover the asymptotic of Lagrage Iterpolatio for fuctio f ( t) = t where > with equidistat odes ad discover the covergece of f ( t) = t, > at poit t =. Lastly, we compare the effective of Fourier series ad Lagrage method i approimate the o-differetiable fuctio. iv

7 Abstrak tesis yag dikemukaka kepada Seat Uiversiti Putra Malaysia sebagai memeuhi keperlua utuk ijazah Master Sais INTERPOLASI DAN PENGHAMPIRAN FUNGSI TAK TERBEZAKAN DENGAN MENGGUNAKAN POLYNOMIAL Oleh KOO LEE FENG Februari 7 Pegerusi: Falkuti: Profesor Madya Adem Kilicma, PhD Sais Dalam sebarag masalah peyelesaia hampira, merujuk kepada pegukura yag terdekat poliomial p ( ) terhadap fugsi asal f ( ). Biasaya, suatu peghampira f ( ) diperoleh apabila perluasa kuasa siri kuasa yag lebih tiggi dihapuska. Salah satu idea asa dalam kalkulus pembezaa ialah suatu fugsi dapat dihampiri secara setempat dega megguaka garisa tage fugsi tersebut. Sekiraya, sesuatu fugsi, f ditakrifka dalam selag terbuka I sedemikia higga c I da N. Jika fugsi tersebut mempuyai terbita ke- pada semua lai I, maka polyomial ( f c)( c) T, dikeali sebagai fugsi poliomial hampira Taylor berdarjah pada titik c. Jikalau fugsi tesebut dapat dibezaka ifiiti pada selag I, maka siri tersebut dikeali sebagai siri Taylor fugsi f pada titik c. walaupu,siri Taylor merupaka idea asas dalam terbita kalkulus,amu begitu, peghampira dega kaedah siri Taylor haya dapat diaplikasika jika dega haya jika fugsi f tersebut dapat diterbitka dalam selag yag diyataka. v

8 Dikataka bahawa sesuatu fugsi terbita adalah fugsi selajar.namu begitu, keyataa jika fugsi selajar megimplikasika bahawa fugsi tersebut juga mempuyai fugsi terbita adalah tidak bear. Theorem Peghampira Weierstrass meyataka bahawa setiap fugsi selajar yag tertakrif dalam selag tertutup [ a, b] dapat dihampirka secara seragam, sehampir mugki dega fugsi yag diputuska. Dega itu, theorem ii memastika bahawa setiap fugsi dapat dihampiri dega megguaka poliomial. Justeru itu, dalam kajia ii telah membaguka satu cara peghampira yag baru utuk meghampiri fugsi yag tidak dapat dibezaka pada satu, dua da tiga titik dega megguaka siri Fourier, iterpolasi Lagrage da kovolasi.seterusya,kita juga aka membicagka asimptot Iterpolasi Lagrage pada fugsi f ( t) = t di maa > berdasarka titik sama jarak da seterusya membicagka peumpua fugsi f ( t) = t di maa > pada titk t=. Akhirya, perbadiga keberkesaa tetag cara pegguaa siri Fourier da iterpolasi Lagrage dalam proses peghampira fugsi tapa terbita dibuat. vi

9 ACKNOWLEDGEMENTS It s my pleasure to ackowledge the cooperatio of those who ivolved directly or idirectly i the jourey of completig thesis. My deepest appreciatio ad gratitude goes to all mighty God, Jesus Christ for His woderful guidace ad blessig. I am etremely grateful to my supervisor, Assoc Prof Dr Adem Kilicma for his ecellet supervisio, ivaluable guidace, patiece, helpful discussios ad cotiuous ecouragemet. My thaks also goes to the members of my supervisor committee Assoc Prof Dr Fudziah Ismail ad Dr Zaiidi K. Eshkuvatov for their ivaluable discussio, commet ad help. They had give their precious time to share their epertise i preparig, researchig ad writig up this thesis. Also, I would like to eted my thaks to all i Mathematics Departmet, Uiversity Putra Malaysia for their kid assistace durig my studies. Also, my appreciatio goes to my best buddies who edured with patiece ad uderstadig i the ups ad dows throughout the preparatio of this thesis. Last but ot least, I would like to thak my beloved family especially my parets for their ucoditioal love, support ad uderstadig throughout the course of my studies. vii

10 DECLARATION I hereby declare that the thesis is based o my origial work ecept for quotatios ad citatios which have bee duly ackowledged. I also declare that it has ot bee previously or cocurretly submitted for ay other degree at UPM or other istitutios. KOO LEE FENG Date : 6APRIL7 viii

11 This thesis submitted to the Seate of Uiversiti Putra Malaysia ad has bee accepted ad fulfillmet of the requiremet for the degree of Master of Sciece. The members of the Supervisory Committee as follows : Adem Kilicma, PhD Associate Professor Faculty of Sciece Uiversiti Putra Malaysia ( Chairma ) Fudziah Ismail, PhD Associate Professor Faculty of Sciece Uiversiti Putra Malaysia (Member ) Zaiidi K. Eshkuvatov, PhD Faculty of Sciece Uiversiti Putra Malaysia (Member ) AINI IDERIS, PhD Professor / Dea School of Graduate Studies Uiversity of Putra Malaysia Date : MAY7 i

12 I certify that a Eamiatio Committee has met o d February7 to coduct the fial eamiatio of Koo Lee Feg o her master thesis etitled Iterpolatio ad Approimatio of No Differetiable Fuctio usig Polyomial i accordace with Uiversiti Pertaia Malaysia (Higher Degree) Act 98 ad Uiversiti Pertaia Malaysia (Higher Degree) Regulatios 98.The Committee recommeds that the cadidate be awarded the relevat degree. Members of the Eamiatio Committee are as follows: Nik Asri Nik Log, PhD Faculty of Sciece Uiversiti Putra Malaysia (Chairma) Azmi Jaafar, PhD Associate Professor Faculty of Computer Sciece ad Iformatio Techology Uiversiti Putra Malaysia (Iteral Eamier) Leog Wah Jue, PhD Faculty of Sciece Uiversiti Putra Malaysia (Iteral Eamier) Maslia Darus, PhD Professor Faculty of Sciece ad Techology Uiversiti Kebagsaa Malaysia (Eteral Eamier) HASANAH MOHD. GHAZALI, PhD Professor/Deputy Dea School of Graduate Studies Uiversiti Putra Malaysia Date :

13 LIST OF TABLES Table Page. Lagrage Iterpolatio 8 5. Lagrage Iterpolatio With Fuctio f ( t) = t Lagrage Iterpolatio With Fuctio f ( t) = t o [-,] with Equidistat Nodes Lagrage Iterpolatio With Fuctio ( t) = t Lagrage Iterpolatio With Fuctio ( t) = t 4 f 65 f Lagrage Iterpolatio With Fuctio ( t) = ( t 4) ( t 3) f 68 i

14 LIST OF FIGURES Figure Page 3. Approimatio By Fourier Polyomial f ( t) = t 3 3. Error Estimatio of f ( t) = t With Fourier Polyomial Approimatio By Fourier Polyomial ( t) = t 3.4 Error Estimatio of ( t) = t f 6 f With Fourier Polyomial Approimatio By Fourier Polyomial ( ) 3.6 Error Estimatio of ( ) f t f t = 4 t 3 = 4 t With Fourier Polyomial Approimatio By Fourier Polyomial ( ) ( )( f t t 3 4 t ) 3.8 Error Estimatio of f ( t) = ( t 3)( 4 t ) With Fourier Polyomial 4. Covolutio of ( f g)( t) where ), g( ) = = 4 f ( = 4. Covolutio of ( f g)( t) where ), g( ) = 4 f ( = 5. Lagrage Iterpolatio Polyomial at Several Degree of f t = Fuctio () t 5.3 Error Estimate of f ( t) = t With Lagrage Iterpolatio Polyomial 5.5 Lagrage Iterpolatio Polyomial at Several Degree of f t = Fuctio () t 5.6 Error Estimatio of f ( t) = t With Lagrage Iterpolatio Polyomial 5.7 Iterpolatio of ( t) t Degree Error Estimate of ( t) t Degree Iterpolatio of ( t) t Degree 4 5. Error Estimate of ( t) t Degree 4 5. Iterpolatio of ( t) t Degree f = by usig Lagrage ad Fourier at f = by usig Lagrage ad Fourier at f = by usig Lagrage ad Fourier at f = by usig Lagrage ad Fourier at f = by usig Lagrage ad Fourier at ii

15 5. Error Estimate of f ( t) = t by usig Lagrage ad Fourier at Degree 5.4 Lagrage Iterpolatio Polyomial at Several Degree of Fuctio f () t = t Error Estimate of f ( t) = t 4 With Lagrage Iterpolatio Polyomial 5.7 Lagrage Iterpolatio Polyomial at Several Degree of Fuctio f () t = t Error Estimate of f ( t) = t 4 With Lagrage Iterpolatio Polyomial 5. Lagrage Iterpolatio Polyomial at Several Degree of f t = t 4 t 3 Fuctio () ( ) ( ) 5. Error Estimate of f ( t) = ( t 4) ( t 3) Iterpolatio Polyomial 5. D ± ε = { z = ± iε : a M ε } With Lagrage 5.3 Graph y=a+b ad y=cos iii

16 TABLE OF CONTENTS DEDICATION ABSTRACT ABSTRAK ACKNOWLEDGEMENTS DECLARATION APPROVAL LIST OF TABLES LIST OF FIGURES Page ii iii v vii viii i i ii CHAPTER INTRODUCTION MATHEMATICS BACKGROUND AND 5 LITERATURE REVIEW. Taylor Approimatio 5. Lagrage Iterpolatio 7.3 Fourier Series 9.4 Covolutio.5 Weierstrass Approimatio Theorem 3 APPROXIMATION TO THE NON 5 DIFFERENTIABLE FUNCTION 3. Orthogoal Polyomial 6 3. Approimatio Of The Fuctio No Differetiable 7 At Oe Poit By Fourier Polyomial 3.3 Approimatio Of The Fuctio No Differetiable 8 At Two Poits By Fourier Polyomial 3.4 Approimatio Of The Fuctio No Differetiable 3 At Three Poits By Fourier Polyomial 3.5 Coclusio 37 4 APPROXIMATION AND CONVOLUTION 38 METHOD 4. Covolutio Of No-Differetiable Fuctio Covolutio By Usig Approimate Idetity Cotiuous Property Of Fuctio Limit Of No-Differetiable Fuctio 53 5 APPROXIMATION BY LAGRANGE METHOD The Fuctio No-Differetiable At Oe Poit The Fuctio No-Differetiable At Two Poits The Fuctio No-Differetiable At Three Poit s 68 iv

17 5.4 Strog Asymptotics I Lagrage Iterpolatio With 69 Equidistat Nodes 5.5 Coclusio 4 6 CONCLUSION AND FURTHER RESEARCH 4 REFERENCES 44 APPENDICES 46 BIODATA OF THE AUTHOR 56 v

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19 CHAPTER ONE INTRODUCTION Numerical method is a procedure which is ofte useful, either to approimate a mathematical problem with a umerical problem or to solve a umerical problem or at least to reduce the comple umerical problem to a simple problem [9]. Approimatio ad iterpolatio are two importat studies i Numerical aalysis. Approimatio is a method to say aythig sigificat about how well f approimates f requires qualitative iformatio about f uder appropriate coditios such as its cotiuity ad differetiability. I approimatio, we prefer umerical measure of closeess. Differet approimatio schemes correspod to differet otios of covergece [9, 8]. Besides, it sometimes may be a very hard task for us to solve a problem by usig eact formula, thus we ofte solve it by approimate it. Hece, approimatio is a alterative techique for this problem. I egieerig ad sciece, oe ofte has a umber of data poits, obtaied by samplig or some eperimet, ad tried to costruct a fuctio which closely fits those data poits. We called this process as curve fittig. Iterpolatio is specific case of curve fittig.

20 Thus, we ca say that iterpolatio is a method of costructig ew data poits from discrete set of kow data poits. Iterpolatio cosists fudametally i fidig a simple fuctio which kow as a iterpolatig fuctio which takes the give fuctio values at the give values of ad represets y = f ( ) adequately i a rage icludig the o-tabular value of. The value of the iterpolatig fuctio at the latter value of is take to the value ( ) of y = f. The differece betwee fuctio approimatio ad iterpolatio is the iterpolatig fuctio f is used to replace or simply the origial fuctio g with certai smooth property preserved by the discrete iterpolatio odes ad their eighborhood. The behavior of Lagrage iterpolatio polyomial of fuctio () t = t, > f where t (, ) have attracted much attetio of several geeratios of mathematicias [8]. It bega with Berstei i 96 proved that the sequece of Lagrage iterpolatio polyomial to t based o the equidistat odes diverges at ay of the iterval of[, ]. The ivestigatios of this problem cotiue by a few mathematicias ad cocluded that Lagrage iterpolatio polyomials based o equidistat odes may have very poor approimatio properties [,,,].

21 The fuctio of t has two taget lies. Thus, it is a type of o-differetiable fuctio. Hece, i this thesis, our mai objective is to develop a ew approimatio method i approimatig the possible of o-differetiable for the fuctio () t f where =. Based o the problem, we will review the mathematical backgroud which will assists us to have a clear picture i our research. This icludes the review of a few types of approimatio ad iterpolatio ad the Weierstrass Approimatio Theorem, which provide the basic idea approaches our target that each cotiuous fuctio is always ca be approimated by the polyomial. We obtai our mai result through a few eamples i order to show that the odifferetiable fuctio ca be approimated by usig Fourier polyomial. First of all, we geerate the orthogoal polyomial, ad the we approimate the o-differetiable fuctio by usig the orthogoality ad the approimatio of Fourier polyomial. For each of the eample give, we will start with the fuctio has sigularity at oe poit, the two poits ad followig by three poits. For each eample give, we provide two graphs. The first graph is the approimatio by usig Fourier polyomial of the odifferetiable fuctio at differet orders ad followig by the error estimatio graph of the fuctio with Fourier polyomial at several orders. Net, we cotiue to ivestigate a few ideas i the process of approimate the o-differetiable fuctio by usig covolutio ad discuss the cotiuous property of o-differetiable fuctio. 3

22 We approimated the o-differetiable fuctio by usig the Lagrage iterpolatio polyomial method. We plot the graph of Lagrage iterpolatio fuctio at differet order ad followig by the graph of error estimatio of the fuctio with Lagrage iterpolatio at several orders for each eample that we used. After that, we will discuss the result regardig the error estimatio of Lagrage iterpolatio for fuctio () t = t, > f with Equidistat Nodes, describe the rate of divergece of the Lagrage iterpolatio ( f t) L N, for < t <, ad discuss their covergece at t =. Lastly, we compare the effective of usig Lagrage Iterpolatio method ad Fourier Polyomial method i approimatio of the o-differetiable fuctio. 4

23 CHAPTER TWO MATHEMATICS BACKGROUND AND LITERATURE REVIEW Approimatio to o-differetiable fuctio have a very close relatioship to the differetiable fuctio, Fourier Series[5,8], Covolutio[,4,5], Weierstrass Approimatio Theorem[6,8,7,8,9], Taylor Approimatio Polyomial[6,7,6] ad Lagrage Iterpolatio[,7]. Thus, i this chapter, we will review topics which we will use i our ivestigatio.. Taylor Approimatio Taylor approimatio is the most basic approimatio techique compared to other methods. We use Taylor approimatio method to approimate a fuctio with aother fuctio. If a fuctio f () is give, we try to approimate fuctio f by fuctio g by fidig the derivative of fuctio f i the order that we wish to fid. 5

24 Defiitio. [6] If fuctio f has derivatives at poit [ a, b] at is writte as, the Taylor polyomial of order for f P ( ) = f ( ) + f ( )( ) = k = f ( k ) ( ) k! k + f ( ( ) ) ( ) ( ) f +... ( )!! Theorem. Taylor Theorem With Cauchy Remaider [6] ( ) Let fuctio f has + cotiuous derivatives o iterval [ a, b] ad let [ a, b],. The f k = P + R where P ( ) = f is Taylor epasio at k! ( ) ( ) ( ) k = k ( ) ( ) ( ) degree while R R ( ) + is the remaider with the formula ( ( ) = ( t) f ) ()dt t. (.)! This itegral form of remaider does require the additioal hypothesis that the ( +) st derivative is Riema itegrabled. This theorem esures that the series coverges to the fuctio. Sice Taylor approimatio requires may derivatives, ad it oly uses iformatio at a sigle poit, so it will ot be a ideal method for uiform approimatio over a iterval. I additio, Taylor polyomial is very accurate ear the middle of the iterval ad much 6

25 less accurate ear the eds. Further, Taylor approimatio ca oly be applied whe the fuctio is a differetiable fuctio.. Lagrage Iterpolatio Lagrage approimatio is oe approach for iterpolatig with polyomials. If we wat to approimate a fuctio, [ ] b C a f, with a polyomial b a c p i i i = =, ) (, the most straightforward method is to calculate the value of f at the + distict poits { } i i,...,,,3 ; = of [ ] b a, ad satisfied the equatios.we might have problems with specifyig the coefficiets. Thus, Lagrage iterpolatig polyomial is the best solutio for it. ( ) i f p i i...,,, ) ( = = { i c i,...,, : = } Defiitio.3 [] Lagrage iterpolatio polyomial is i the form where ( ) ) ( ) ( j j j j L f P = = ( ) j L j k j j i j :, = = Π. ca be writte as P() ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( )( ) ( ) y y y y P ) )( (... ) )( (... ) (... ) ( ) ( = 7