Numerical Interpolation with Polynomials Approximation and Curve Fitting, Focus the MATLAB

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1 IOSR Joural of Mathematcs (IOSR-JM) e-issn: , p-issn: X. Volume 1, Issue 1 Ver. IV (Ja. - Feb. 016), PP Numercal Iterpolato wth Polyomals Appromato ad Curve Fttg, Focus the MATLAB Carlos Fgueroa 1, Raul Rera, Germa Campoy 1 Idustral Egeerg Departmet. DIFUS Uversty of Soora, Meco Abstract: I ths wor we show classcal ad ow forms to carry out umercal terpolato ad curve fttg. Each method s brefly eplaed ad eamples from Lagrage, Newto, Hermte, osculatg polyomal, ad Padé appromato are preseted. I the least-squares le, data learzato method of epoetal ad power fucto eercses are solved. Our dea s to show the advatages of usg MATLAB the study of umercal aalyses ad to verfy the mmal effort requred usg ths program to save tme mag mathematcal demostratos ad developmets essetal the obtag of each method. Keywords: Polyomal appromato, terpolato, regresso. I. Itroducto To appromate fuctos, aother fucto that appromates the orgal s determed. A fucto s bult from a group of ow values or s adjusted to a fuctoal depedecy; classcal techques are the terpolato wth polyomals ad least- squares le. Numercal terpolato cossts estmatg values from ow data that are cosdered accurate, whle the ft curve fttg cosders that data are the result of a epermet ad ca have a type of error. I both cases the dea s to buld a polyomal fucto that matches the ow data so that we have a aalytcal epresso that allows for mapulatos. a P The polyomal terpolato s justfed by the followg theorem of Weerstrass gve polyomal, such that f P (1) there s The cotuous fucto the closed terval ca be appromated through a polyomal each pot of the terval wth a error lower tha. Importat eamples are the Taylor, Lagrage, Chebyshev, ad Hermte polyomals. The dfferece betwee least squares adjustmet ad data terpolato s that the latter, data s assumed accurate; however, we ow errors are commo, the mmum least squares adjustmet a fucto such that,,..., F a1 a a f y 1 () 0 s sought. It s a good appromato whe a mmum ests. Classcal methods are lear regresso, polyomal, multlear, power seres, olear; we are focused o the lear ad olear regresso cases, ecept whe ths ca be reduced to a lear regresso problem, such as epoetal ad power fuctos. Ths approach facltates the dervato of the ecessary calculatos each method. I our curret wor, we aalyze a group of eercses proposed tetboos related to umercal aalyss ad programmg scece ad egeerg courses. Lewse, the mathematcal elemets from where the appled formulas are derved are preseted. For eample, oe boo that accurately covers ths subject s from Vázquez et al.[1]. Smlarly, []s of commo use ad these comprse the ma sources where the proposed eercses were tae ad forms of whch are here developed. Both wors are at uversty level ad are outstadg due to ther clarty of presetato whle respectg the depth the treatmet of the umercal aalyss. I lterature about umercal aalyss usg MATLAB, the wor of Ba Yuad Zhag Meg [3] ca be metoed; ths wor justfes our proposal ad phlosophy to cotrbute wth ddactc ovatos. A breathrough wor s that of Esa Sa [4] whch presets a method for computg the legth of curves usg polyomal terpolato wth a MATLAB algorthm. Aother related wor s that of Cavoretto [5] where he proposed a algorthm for modelg data pots wth MATLAB. Fally, a boo wth appromato theory s that of Robert Plato [6], whch cludes polyomals from Chebyshev ad Fourer ad sples fuctos. Ths wor s more theoretcal. DOI: / Page

2 Numercal Iterpolato wth Polyomals Appromato ad Curve Fttg, Focus the MATLAB We are focused o solvg the ssues rased Vázquez et al [1]; some eamples are avalable o the Iteret such as [7,10,11] ad the boo by Mathews/F [] the tet of whch s to develop our ow soluto ad geerate a dscusso that leads to better results wth MATLAB regardg terpolated ad regresso fuctos. Such a approach ca brg about the best ddactc o ths subject. For a group of pots havg a P such that, f, P II., f (3) Lagrage Iterpolato Lagrage proposed the followg model based o lear terpolato 1 0 P f 0 f 1 (4) The followg s met P 0 f 0 (5) P 1 f 1 The polyomal ca be bult as follows: P L f L f The coeffcets of ths fucto are called Lagrage Coeffcet for the L0 0 1 L1 0 0 (7) L0 1 0 L1 1 1 I geeral for pots f we have (8) whch cotas the Lagrage polyomals of 1 L (9), a polyomal that matches wth these pots s sought; therefore, (6) grade that ca be wrtte as odes ths s suffcet for the frst eercse: Obta the Lagrage polyomal of the followg pots [7] y The terpolatg polyomal of these data s Fgure 1 shows ts graph (10) P L f 0 0, P DOI: / Page

3 Numercal Iterpolato wth Polyomals Appromato ad Curve Fttg, Focus the MATLAB Fgure 1. Lagrage Polyomal The boo of Lus Vázquez et al [1] cotas, the eercse secto, some terestg problems. For eample No. 7, page 115, that attempts to fd a Lagrage terpolatg polyomal for a fucto gve by the followg table f() Usg MATLAB we get that the polyomal results P 1 (11) Fgure shows ts behavor Fgure. Lagrage Polyomal III. Newto Iterpolato Sometmes t s useful to buld several appromate polyomals ad choose the correct oe. Lagrage s ot coveet because each oe s depedet. Those of Newto are estmated usg the recursve system. I fact, P a a P a a a P a a a a such a way that DOI: / Page (1) P P a (13)

4 Numercal Iterpolato wth Polyomals Appromato ad Curve Fttg, Focus the MATLAB Ths s ow as Newto s Dvded Dfferece Polyomal Iterpolato. There s a relatoshp betwee the Lagrage polyomal ad Newto polyomal, that s, t s possble to drectly obta the Lagrage polyomal from Newto s formula from the cocept of dvded dfferece. Ths developmet ca be foud [8]. Equato (4) s geerated from (13). Naamura [9] has a dfferet vew of the study of terpolato; he begs wth lear terpolato, the power seres polyomals ad, from there, arrves at Lagrage; f we add the approach [8], the etre demostratve pcture ca be covered. Eample of the eercses proposed [10]. Calculate the terpolato polyomal for the gve table usg Newto s system ad formula: y From ths calculato we get y Wth the followg graph, Fgure (14) Fgure 3. Newto Iterpolatg Polyomal Hermte Polyomal Taylor polyomal allows appromatg a pot, the fucto ad ts frst dervatves. Lagrage polyomals appromate 1 pots ad oe of ts dervatves. Now the case that s appromated to the fucto ad to ts frst dervatve that correspods to the Hermte polyomals H f H f H (15) H L L H L (16) Oe eercse that we ca dscuss s the followg: Fd the grade three polyomal that passes through the followg pots [11] 0 1/3 1/ 1 4/9 8/9 5/9 6/9 The grade three Hermte polyomal s 4 47 P Wth the followg graph, Fgure 4 (17) DOI: / Page

5 Numercal Iterpolato wth Polyomals Appromato ad Curve Fttg, Focus the MATLAB Fgure 4. Hermte Polyomal Osculatg Polyomal A geeral case s to cosder a polyomal that appromates to the fucto, ad some of ts dervatves, such that d P d f (18) d d A solved eercse Vázquez et al [1] s of our terest, f we complete t usg graphc aalyss. 0 1 y y whch leads to determe that P 4 1 (19) wth the followg graph, Fgure Fgure5. Osculatg Polyomal DOI: / Page

6 Numercal Iterpolato wth Polyomals Appromato ad Curve Fttg, Focus the MATLAB Aother possblty s the ratoal appromato of a fucto, a polyomal rato, ad at least a small porto of the doma. I ca be demostrated that such rato s N for a b (0) Fally, a Padé appromato s gve geerated by the followg eercse proposed the boo of Mathews/F. [] Compare the Padé appromato such that s (1) Fgure 6 cotas both fuctos (dscotuous appromato) R NM, P Q M Fgure 6. Padé Appromato IV. Curve Fttg wth Least-Squares Le The terpolato assumes eact values, however, t s commo to have a epermetal measure wth errors. I ths case, what s sought s a fucto that does ot pass through the pots rather oe that mmzes the error made. For eample, a estmato of the error has the followg form y f () however, the oe that s used most s the mea squared error, equato (). We preset the equatos of the lear regresso, ad eamples of epoetal polyomal regresso ad power fucto. The smplest case, ths case fucto (3) ad the fucto to mmze s the, t s ecessary to solve we arrve at f a, b, a b, F a b a b y 1, F a b (4) a b y 0 a F a, b a b y 0 b 1 1 V. Lear Regresso f s a straght le, therefore (5) (6) DOI: / Page

7 Numercal Iterpolato wth Polyomals Appromato ad Curve Fttg, Focus the MATLAB y y y y a b y a (8) These equatos have ther matr form. Lewse, we have 1 1 y 1 y 1 (9) (7) Polyomal Regresso For the geeral case of polyomal regresso of 1, 1,..., F a a a a a a y 1 grade, the fucto that s mmzed s tag the partal dervatves ad equalg to zero 1 y 1 I a a 1 y (31) a y 1 The matr form s observed. Net, we have problem 18, page 118, from the boo of Vázquez et al [1]. Polyomal regresso of the data to a parabola: y (30) Resultg y wth the followg graph, Fgure 7 (3) Fgure 7. Polyomal Regresso DOI: / Page

8 Epoetal Regresso I ths case the fucto f a, b, Numercal Iterpolato wth Polyomals Appromato ad Curve Fttg, Focus the MATLAB a be b 0 (33) To mae a data learzato the followg s ecessary l f lb a (34) Sce t s a lear regresso, t wll have the form of equatos (5 ) ad (6 ) l y l y a (35) 1 1 For parameter a, equatos (5 ) ad (6 ) are cosdered 1 a b ep l y 1 1 (36) Problem 1 from Vazquez et al [1] Mae a adjustmet of epoetal decle law of the followg data y whose resultg equato s y 9.404e (37) wth the followg graph Fgure 8 Fgure8. Epoetal Regresso For a group of data t s possble to predct whch fucto provdes a good adjustmet. For a epoetal fucto, the represetato of the data wth the as, lear, ad the y as, logarthmc, we see that such appromato s ot accurate, ot coformg to a straght le. Fgure 9 shows ths o-coformty DOI: / Page

9 Numercal Iterpolato wth Polyomals Appromato ad Curve Fttg, Focus the MATLAB Power regresso Fucto f f a,, b, b a (38) has the form Whe the fucto s learzed, we have l f l b al (39) Fgure9. Verfcato of data to a straght le I ths case, coeffcets to verfy are l y l l( y ) l a (40) l l a b ep l y 1 1 (41) Net, we preset the appromato of the data of problem 1, a power y (4) whch has the behavor gve Fgure 10. Fgure10. Power Regresso DOI: / Page

10 Numercal Iterpolato wth Polyomals Appromato ad Curve Fttg, Focus the MATLAB Gve the dffereces, t s ecessary to test to see f the ad y as are logarthmc, we have the result show Fgure 11. Fgure11. Verfcato of data to a straght le We th t s ot a good appromato; wth a degree polyomal we get 4 3 y (44) Sce t appromates to the followg form of Fgure 1 Fgure 1. Polyomal Regresso VI. Dscusso of Results O the bass of the aalyss carred out, we ca establsh that refereces [1] ad [] are complemetary; both wors have detals that ca help uderstad the subject better. We focused more o [1]. I Lagrage polyomals, MATLAB obtas the graph Fgure drectly. Lewse, the Hermte problem of Fgure 4 t s ecessary to use the method of dvded dffereces for ts soluto; aother case s the eample of the osculatg polyomal, solved [1]. They do ot geerate the graph of Fgure 5.Padè appromato of Fgure 6 s a dfferet terval to the oe ased for the boo of Mathew-F [] where t has a good result. Equatos 10, 11,14,ad 17 are cosstet wth the sources. I the topc of least-squares les wth data learzato method, we have Fgure 7 polyomal regresso where we observe a ecellet adjustmet, whle problem 1 cocerg the epoetal decle from Vázquez et al [1] does ot ehbt a good appromato. We beleve that Fgures 8 to 1, a prorty, for our wor the result of Fgure 1 s better. Fally, we acowledge that we dd ot dscuss ad estmate the appromato errors. It s crucal to pot out DOI: / Page

11 Numercal Iterpolato wth Polyomals Appromato ad Curve Fttg, Focus the MATLAB that the deductos of the equatos have to be duly complemeted; otherwse, we oly provde the ma result of the method overloog ts mathematcal justfcato. VII. Coclusos The study of umercal aalyss a scece or egeerg course requres ew ddactc materal. O the Iteret ests a myrad of audovsual lessos of great ddactc value. Our cotrbuto would le to be part of the tred toward the geerato of ew approaches, but wth the characterstc of puttg emphass o the use of MATLAB, ad have tme to carry out mathematcal developmets that demostrate the formula of each method, such as the oes preseted here. We th that the combato of plottg ad calculatg wth software ad mathematcally justfyg everythg, s a effcet way to lear umercal aalyss scece ad egeerg for promotg a deeper uderstadg of the subject of terpolato. Refereces [1]. Lus Vázquez, Salvador Jméez, Carlos Agurre y Pedro J. Pascual.Métodos Numércos para físca e geería.edt. McGraw Hll pp Spa (009) []. Joh H. Mathews ad Kurts D. F. Métodos umércos co MATLAB. Edt. Pretce Hall pp Spa (000). [3]. Ba Yu ad Zhag Meg. Vsual teachg of umercal aalyss based o MATLAB. The 1st Iteratoal coferece of Iformato scece ad Egeerg pp (009) [4]. Esa Sa et al. Numercalcurve legth calculatousg polyomal terpolato.joural of Mathematcal Imagg ad Vso 49 pp (014) [5]. Roberto Cavoretto. A umercal algorthm for multdmesoal modelg of scatterg data pots.computatoal ad Appled Mathematcs.(013) [6]. Robert Plato Cocse Numercal mathematcs. Edt. Amerca Mathematcal Socety. USA (000) [7]. ejerccos de terpolacó y apromacó.umat.et/ejerc/terp/ [8]. Steve C. Chapra ad Raymod P. Caale. Métodos umércos para geeros Edt Mcgraw Hll pp Meco (011) [9]. ShochroNaamura. Aálss umérco y vsualzacó gráfca co MATLAB. Edt. Pretce Hall,pp Meco (1997) [10]. [11]. Jame Álvarez Maldoado DOI: / Page

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