ELM Numercl Anlss Dr Muhrrem Mercmek INTEPOLATION ELM Numercl Anlss Some of the contents re dopted from Lurene V. Fusett, Appled Numercl Anlss usng MATLAB. Prentce Hll Inc., 999
ELM Numercl Anlss Dr Muhrrem Mercmek Tod s lecture Polnoml Interpolton Lgrnge Interpolton Newton Interpolton Dffcultes wth Polnoml Interpolton Hermte Interpolton tonl-functon Interpolton
ELM Numercl Anlss Dr Muhrrem Mercmek Lgrnge Interpolton Polnomls Bsc concept The Lgrnge nterpoltng polnoml s the polnoml of degree n- tht psses through the n ponts. Usng gven severl pont, we cn fnd Lgrnge nterpolton polnoml.
Generl Form of Lgrnge The generl form of the polnoml s p = L + L + + L n n where the gven ponts re,,.., n, n. The equton of the lne pssng through two ponts, nd, s The equton of the prbol pssng through three ponts,,,, nd, s p p............ n k k k k k k n k k k L ELM Numercl Anlss Dr Muhrrem Mercmek
ELM Numercl Anlss Dr Muhrrem Mercmek 5 Lgrnge Interpolton Emple :, =-,,, =0,,, =,8 p 0 0 8 0 0 0 0 8 8 8
ELM Numercl Anlss Dr Muhrrem Mercmek 6 Lgrnge Interpolton We cn represent the Lgrnge polnoml wth coeffcent c k. p = c N + c N + + c n N n c k N k k k... k k k k... k n... k k... n
ELM Numercl Anlss Dr Muhrrem Mercmek 7 Hgher Order Interpolton Polnomls Emple : Hgher order nterpolton polnomls = [ - - 0 ], = [ -5 0 0-0 5]
evew nd Dscusson In Lgrnge nterpolton polnoml, t lws go through gven ponts. It s less convenent thn the Newton form when ddtonl dt ponts m be dded to the problem. p 8 ELM Numercl Anlss Dr Muhrrem Mercmek
-9 Newton form of the equton of strght lne pssng through two ponts, nd, s Newton form of the equton of prbol pssng through three ponts,,,, nd, s The generl form of the polnoml pssng through n ponts,,, n, n s Newton Interpolton Polnomls p p...... n n p 9 ELM Numercl Anlss Dr Muhrrem Mercmek
Substtutng, nto Substtutng, nto Substtutng, nto p Consder prbol equton obtned usng three ponts 0 ELM Numercl Anlss Dr Muhrrem Mercmek Newton Interpolton Polnomls
Emple : Pssng through the ponts, =-,,, =0,, nd, =, 8. The equtons s Where the coeffcents re 0 p 0 0 0 8 p ELM Numercl Anlss Dr Muhrrem Mercmek Newton Interpolton Polnomls
Pssng through the ponts, =-,,, =0,, nd, =, 8. p p ELM Numercl Anlss Dr Muhrrem Mercmek Newton Interpolton Polnomls
ELM Numercl Anlss Dr Muhrrem Mercmek Addtonl Dt Ponts Emple : ddng the ponts, = -, - nd 5, 5 =, to the prevous dt p
ELM Numercl Anlss Dr Muhrrem Mercmek Hgher Order Interpolton Polnomls Emple 5: Consder gn the dt wth Lgrnge form. = [ - - 0 ], = [ -5 0 0-0 5] Do t gn wth Newton form. p 55 6 the polnoml s cubc
ELM Numercl Anlss Dr Muhrrem Mercmek 5 Hgher Order Interpolton Polnomls Emple 6: If the vlues re modfed slghtl, the dvded-dfference tble shows the smll contrbuton from the hgher degree terms: N.5 5.95.0 0.067 0.00 0.0007
ELM Numercl Anlss Dr Muhrrem Mercmek 6 Dffcultes: Humped nd Flt Dt Emple 7: The dt = [ - -.5 - -0.5 0 0.5.5 ] = [ 0 0 0 0.87 0.87 0 0 0] llustrte the dffcult wth usng hgher order polnomls to nterpolte modertel lrge number of ponts.
ELM Numercl Anlss Dr Muhrrem Mercmek 7 Dffcultes: Nos Strght Lne Emple 8: The dt = [ 0.00 0.0 0.80.00.0.90.00.0.95.00] = [ 0.0 0. 0.76.0.8.9.0.08.90.95] Not good wth nos strght lne. Newton polnoml coeffcents
ELM Numercl Anlss Dr Muhrrem Mercmek 8 Dffcultes: unge Functon Emple 9: f The functon 5 s n emple of the fct tht polnoml nterpolton does not produce good ppromton for some functons usng more functon vlues t evenl spced vlues does not necessrl mprove the stuton. = [ - -0.5 0.0 0.5.0 ] = [0.085 0.79.0000 0.79 0.085]
ELM Numercl Anlss Dr Muhrrem Mercmek 9 Dffcultes: unge Functon Emple 0: = [-.000-0.750-0.500-0.50 0.000 0.50 0.500 0.750.000 ] = [0.085 0.066 0.8 0.90.000 0.90 0.8 0.066 0.085] The nterpolton polnoml overshoots the true polnoml much more severel thn the polnoml formed b usng onl fve ponts. -9
ELM Numercl Anlss Dr Muhrrem Mercmek 0 Hermte Interpolton tonl-functon Interpolton Some of the contents re dopted from Lurene V. Fusett, Appled Numercl Anlss usng MATLAB. Prentce Hll Inc., 999
ELM Numercl Anlss Dr Muhrrem Mercmek Hermte Interpolton Hermte nterpolton llows us to fnd plonoml tht mtched both functon vlue nd some of the dervtve vlues
ELM Numercl Anlss Dr Muhrrem Mercmek Hermte Interpolton Emple :
ELM Numercl Anlss Dr Muhrrem Mercmek Dffcult Dt Emple : As wth lower order polnoml nterpolton, trng to nterpolte n humped nd flt regons brngs overshootngs.
ELM Numercl Anlss Dr Muhrrem Mercmek tonl-functon Interpolton Wh use rtonl-functon nterpolton? Some functons re not well ppromted b polnomls.rungefuncton but re well ppromted b rtonl functons, tht s quotents of polnomls.
Bulrsch-Stoer lgorthm Bulrsch-Stoer lgorthm The pproch s recursve, Gven set of m+ dt ponts,,, m+, m+, we seek n nterpolton functon of the form 5 ELM Numercl Anlss Dr Muhrrem Mercmek........................ m m m m m m m m m Bulrsch-Stoer pttern
ELM Numercl Anlss Dr Muhrrem Mercmek 6 Bulrsch-Stoer lgorthm Bulrsch-Stoer method for three dt ponts
Bulrsch-Stoer method for fve dt ponts Thrd stge 5 5 5 5 5 5 5 5 5 5 5 5 5 = 5 5 5 = = = = Second stge Frst stge dt Bulrsch-Stoer lgorthm 7 ELM Numercl Anlss Dr Muhrrem Mercmek
ELM Numercl Anlss Dr Muhrrem Mercmek 8 Bulrsch-Stoer lgorthm Forth stge Ffth stge 5 5 5 5 5 5 5 5 5 5 5 5
ELM Numercl Anlss Dr Muhrrem Mercmek 9 tonl-functon nterpolton Emple : dt ponts: = [- -0.5 0.0 0.5.0] = [0.085 0.79.0000 0.79 0.085]