Chapter Spline Method of Interpolation More Examples Computer Engineering
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1 Chpter. Splne Metho of Interpolton More Emples Computer Engneerng Emple A root rm wth rp lser snner s ong quk qulty hek on holes rlle n " " retngulr plte. The enters of the holes n the plte esre the pth the rm nees to tke, n the hole enters re lote on Crtesn oornte system (wth the orgn t the ottom left orner of the plte gven y the speftons n Tle. Tle The oorntes of the holes on the plte. (n. y (n Fgure Loton of the holes on the retngle plte...
2 .. Chpter. If the lser s trversng from. to. n lner pth, wht s the vlue of y t. usng lner splnes? Fn the pth of the root f t follows lner splnes. Fn the length of the pth trverse y the root followng lner splnes. Soluton Sne we wnt to fn the vlue of y t., we nee to hoose the two t ponts losest to. tht lso rket. to evlute t. The two ponts re. n.. Then., ( 7. y., ( 7. y gves y( y( y( y( ( (... Hene y ( 7..(.,.. At, y (. 7..(.. 7. n. We hve lrey otne the lner splne onnetng. n.. y ( 7..(.,.. ( The rest of the splnes re s follows: y ( 7. ( (.,.. (. 6. y( 6. ( (., (.. y (. ( ( 7.8, (.. y (. ( ( 9., (
3 Splne Metho of Interpolton-More Emples: Computer Engneerng.. The length of the root s pth n e foun y smply ng the length of the lne segments together. The length of strght lne from one pont, to nother pont ( y ( y, s gven y ( ( y y. Hene, the length of the lner splnes from. to. 6 s L (.. (7. 7. (.. (6. 7. (7.8. (. 6. ( (.. ( (.. Lner splne nterpolton s no fferent from lner polynoml nterpolton. Lner splnes stll use t only from the two onseutve t ponts. Also t the nteror ponts of the t, the slope hnges ruptly. Ths mens tht the frst ervtve s not ontnuous t these ponts. So how o we mprove on ths? We n o so y usng qurt splnes. Emple A root rm wth rp lser snner s ong quk qulty hek on holes rlle n " " retngulr plte. The enters of the holes n the plte esre the pth the rm nees to tke, n the hole enters re lote on Crtesn oornte system (wth the orgn t the ottom left orner of the plte gven y the speftons n Tle. Tle The oorntes of the holes on the plte. (n. y (n Fn the length of the pth trverse y the root usng qurt splnes. Compre the nswer from prt ( to the lner splne result n ffth orer polynoml result. Soluton Sne there re s t ponts, fve qurt splnes pss through them. y(,..,..,. 7. 8, , 9.. 6
4 .. Chpter. The equtons re foun s follows.. Eh qurt splne psses through two onseutve t ponts. psses through. n.. (. (. 7. (. (. 7. (. (. psses through. n.. 7. (. (. 6. (. (. psses through. n ( 7.8 (7.8. ( 7.8 (7.8 psses through 7. 8 n 9.. ( 9. (9. ( 9. (9... psses through 9. n. 6.. (.6 (.6.. Qurt splnes hve ontnuous ervtves t the nteror t ponts. At. (. (. At. (. (. At 7. 8 (7.8 (7.8 At 9. (9. (9. ( ( ( ( ( (6 (7 (8 (9 ( ( ( ( (. Assumng the frst splne s lner, (
5 Splne Metho of Interpolton-More Emples: Computer Engneerng Solvng the ove equtons gves the unknowns s Therefore, the splnes re gven y 7.889,. ( y...777, , , ,
6 ..6 Chpter. Fgure Ffth orer polynoml to trverse ponts of root pth (usng ret metho of nterpolton. The length of funton y f ( from to s gven y y L In ths se, f ( s efne y fve seprte funtons. to. 6 y , , , , , 9..6 L y y y y 7.8. y
7 Splne Metho of Interpolton-More Emples: Computer Engneerng We n fn the length of the ffth orer polynoml result n smlr fshon to the qurt splnes. In ths se, we o not nee to rek the ntegrls nto fve ntervls. The ffth orer polynoml result through the s ponts s gven y y( ,..6 Therefore, L.6. y The solute reltve ppromte error qurt splne s % The solute reltve ppromte error polynoml n qurt splne s % otne etween the results from the lner n otne etween the results from the ffth orer
Chapter Spline Method of Interpolation More Examples Electrical Engineering
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