Analysis of Lagrange Interpolation Formula
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1 P IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue, December 4. ISS Aalyss of Lagrage Iterpolato Formula Vjay Dahya PDepartmet of MathematcsMaharaja Surajmal Isttute Abstract: Ths work presets a theoretcal aalyss of Lagrage Iterpolato Formula. I order to aalyze the method, power seres, bass fucto ad quadratc terpolato usg bass fucto ad cubc terpolato are chose. Also to check the performace of the cosdered method a error assocated wth Lagrage terpolato has cosdered. Errors are aalyzed by comparg the actual sampled values wth the values obtaed by Lagrage s terpolato formula. Keywords: Iterpolato, cubc terpolato, bass fucto quadratc terpolato, ut step fucto.. Itroducto: From very acet tme terpolato s beg used for varous purposes. Sr Edmud Whttaker, a professor of umercal mathematcs at the Uversty of Edburgh from 93 to 93, observed The most commo form of terpolato occurs whe we seek data from a table whch does ot have the exact values we wat. Lu Zhuo used the equvalet of secod order Gregory- ewto terpolato to costruct a Imperal Stadard Caledar. I 65 AD, Ida astroomer ad mathematca Brahmagupta troduced a method for secod order terpolato of the se fucto ad, later o, a method for terpolato of uequal-terval data. umerous researchers study the possblty of terpolato based o the Fourer trasformer, the Hartley trasformer ad the dscrete cose trasform. I 983, Parker et al. publshed a frst comparso of terpolato techque medcal mage processg. They faled, however to mplemet cubc B-sple terpolato correctly ad arrve at erroeous cocluso cocerg ths techque[]. I preset days, several algorthms are used for mage reszg [] based o Lagrage s Iterpolato Formula[3]. I ths paper, Lagrage s terpolato formula s used for recostructg power seres fttg to aalyze ts performace. Ths paper s orgazed as follows: secto, we expla the mathematcal prcpal of Lagrage s Iterpolato method. The mplemetato s devsed secto 3. I secto 4 the expermetal results are gve. The cocluso s summarzed secto 5. 69
2 IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue, December 4. ISS Theory of Iterpolato ad cosdered fuctos. Lagrage Iterpolato: We cosder the problem of approxmatg a gve fucto by a class of smpler fuctos, maly polyomals. There are two ma uses of terpolato or terpolatg polyomals. The frst use s recostructg the fucto f(x whe t s ot gve explctly ad oly the values of f(x ad/or ts certa order dervatves at a set of pots, called odes, tabular pots or argumets are kow. The secod use s to replace the fucto f(x by a terpolatg polyomal p(x so that may commo operatos such as determato of roots, dfferetato ad tegrato etc. whch are teded for the fucto f(x may be performed usg p(x. A polyomal p(x s called a terpolatg polyomal f the value of p(x ad/or ts certa order dervatves cocdes wth those of f(x ad/or ts same order dervatves at oe or more tabular pots. I geeral, f there are + dstct pots a x < x < x < x3 <... < x b, the the problem of terpolato s to obta p(x satsfyg the codtos px ( f( x,,,..., ( Substtutg the codtos, we obta the system of equatos a + ax + ax ax f( x a + ax + ax ax f( x a + ax + ax ax f( x ( Ths system of equato has a uque soluto. The terpolato pots or odes are gve as: x f( x f x f( x f x f( x f (3 6
3 P P IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue, December 4. There exsts oly P expressed as a power seres: Where arr th ISS degree polyomal that passes through a gve set of + pots. Its form s g x a ax ax a x ( (4 ukow coeffcets,,,3,,. th It does ot matter how we defe the P degree polyomal whether by Fttg power seres, Lagrage Iterpolato fuctos or by ewto forward or backward terpolato. The resultg polyomal wll always be the same.. Power Seres Fttg to Defe Lagrage Iterpolato: g(x must match f(x at the selected data pots. g( x f a + ax + ax ax f( x g( x f a + ax + ax ax f( x g x f a + ax + ax + + ax f x (... ( (5 Solve the smultaeous equato we get: x x... x a f... a x x x f x x... x a f (6 It s relatvely computatoally costly to solve the coeffcets of the terpolatg fucto g(x..3 Lagrage Iterpolato Usg Bass Fuctos: We ote that geeral g(xrr frr. Let g( x fv( x (7 6
4 xrr IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue, December 4. ISS Where vrr(x polyomal of degree assocated wth each ode such that j v( xj j (8 For example we have 5 terpolato pots or odes, the g( x f v ( x + f v ( x + f v ( x + f v ( x + f v ( x ( Usg the defto for v( x j, we have gx ( 3 f3..e the sum of polyomal of degree s also a polyomal of degree. g(x s equvalet to fttg the power seres ad computg coeffcet a, a, a,..., a. 3. Aalyss ad Implemetato 3. Lagrage Lear Iterpolato Usg Bass Fuctos: Lear Lagrage ( s the smplest form of Lagrage terpolato: g( x fv ( x That s g( x fv ( x + fv ( x ( ( x x ( x x Where v( x ad v( x. ( x x ( x x Example: Gve the followg data:, frr.5 x RR 5, frr 4. fd the lear terpolatg fucto g(x. Soluto: Lagrage bass fuctos are: (5 x ( x v( x ad v( x 3 3 The terpolatg fucto gx (.5 v( x + 4. v( x. 6
5 IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue, December 4. ISS Lagrage Quadratc Iterpolato Usg Bass Fuctos: For Quadratc Lagrage terpolato, g( x fv ( x That s g( x fv( x + fv( x + fv( x ( Where v x ( x x ( x x (, ( x x( x x v x ( x x ( x x (, ( x x( x x ( x x( x x v( x ( x x ( x x. ote that locato of the roots of v( x, v( xad v( x are defed such that the basc premse of terpolato s satsfed, amely that gx ( f. 4. Errors Assocated wth Lagrage Iterpolato Usg Taylor seres aalyss, the error ca be show to be gve by: E(x f(x g(x ( + Ex ( Lx ( f ( ξ, x ξ x ( Where ( + f ( ξ + dervatves of f w.r.t. x evaluated at ξ. ( x x( x x...( x x ad Lx ( th a +P P degree polyomal. ( +! ow f f(x s a polyomal of degree M, the ( + f ( x Ex ( for all x. Therefore g(x wll be exact represetato of f(x. 5. Cocluso I ths paper, accordg to the aalyss the performace of Lagrage terpolato formula o dfferet types of fucto s preseted. Expermetal 63
6 IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue, December 4. ISS results show that t works better for a fucto whose value wll crease or rema costat wth the depedet varable. 6. Refereces []. Mejerg, Erk. A Chroology of Iterpolato: From Acet Astroomy to Moder Sgal ad Image Processg. Proceedgs of the IEEE. vol. 9, o. 3, pp March []. Japg Xao, Xuecheg Zou, Zhegl Lu, Xu Guo, "Adaptve Iterpolato Algorthm for Real-tme Image Reszg," ccc, vol., pp.-4, Frst Iteratoal Coferece o Iovatve Computg, Iformato ad Cotrol - Volume II (ICICIC'6, 6 [3]. umercal Methods for Egeers, by Steve C. Chapra ad Raymod P. Caale, Forth Edto, Chapter 8, Page o:
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