Global polynomial interpolants can have extremely undesirable properties if the sample points are not chosen correctly.
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- Ezra Park
- 5 years ago
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1 Ruge Pheomeo Global polyomial iterpolats ca have extremely udesirable properties if the sample poits are ot chose correctly. Example: Iterpolate the fuctio f(x) = over the iterval [, ] at the equally spaced + 6x2 data sites x j = + j, j =,,..., Data to iterpolate, =
2 Ruge Pheomeo Global polyomial iterpolats ca have extremely udesirable properties if the sample poits are ot chose correctly. Example: Iterpolate the fuctio f(x) = over the iterval [, ] at the equally spaced + 6x2 data sites x j = + j, j =,,..., Polyomial iterpolat, =.5 Iterpolat - - -
3 Ruge Pheomeo Global polyomial iterpolats ca have extremely udesirable properties if the sample poits are ot chose correctly. Example: Iterpolate the fuctio f(x) = over the iterval [, ] at the equally spaced + 6x2 data sites x j = + j, j =,,..., Data to iterpolate, =
4 Ruge Pheomeo Global polyomial iterpolats ca have extremely udesirable properties if the sample poits are ot chose correctly. Example: Iterpolate the fuctio f(x) = over the iterval [, ] at the equally spaced + 6x2 data sites x j = + j, j =,,..., Polyomial iterpolat, =2.5 Iterpolat - - -
5 Ruge Pheomeo Global polyomial iterpolats ca have extremely udesirable properties if the sample poits are ot chose correctly. Example: Iterpolate the fuctio f(x) = over the iterval [, ] at the equally spaced + 6x2 data sites x j = + j, j =,,..., Data to iterpolate, =
6 Ruge Pheomeo Global polyomial iterpolats ca have extremely udesirable properties if the sample poits are ot chose correctly. Example: Iterpolate the fuctio f(x) = over the iterval [, ] at the equally spaced + 6x2 data sites x j = + j, j =,,..., Polyomial iterpolat, =4.5 Iterpolat - - -
7 Ruge Pheomeo Global polyomial iterpolats ca have extremely udesirable properties if the sample poits are ot chose correctly. Example: Iterpolate the fuctio f(x) = over the iterval [, ] at the equally spaced + 6x2 data sites x j = + j, j =,,..., Polyomial iterpolat, =8.5 Iterpolat - - -
8 Ruge Pheomeo Global polyomial iterpolats ca have extremely udesirable properties if the sample poits are ot chose correctly. Example: Iterpolate the fuctio f(x) = over the iterval [, ] at the equally spaced + 6x2 data sites x j = + j, j =,,..., 5 Absolute value of the differece betwee p ad f, =
9 Ruge Pheomeo The wildly oscillatory behavior (large errors) we see i the iterpolatig polyomial ear the boudaries is called the Ruge Pheomeo. It is a geeral pheomeo for polyomial iterpolatio that is associated with a poor choice of samplig poits, i this case equally-spaced. There is a beautiful theory that predicts whe this pheomeo will occur ad how to fix it: See Approximatio Theory ad Approximatio Practice, SIAM 23, by L.N. Trefethe Our focus will just be o the fix.
10 Defeatig the Ruge Pheomeo <latexit sha_base64="cwwzoh5kvzjpb6xpixp+ydlarkm=">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</latexit> <latexit 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sha_base64="3pqplgz/bp9yfgpecgaffylg6e=">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</latexit> <latexit sha_base64="3pqplgz/bp9yfgpecgaffylg6e=">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</latexit> Polyomial iterpolat, =2.5 Iterpolat Idea: Icrease the desity of the samplig poits ear the boudaries to damp out the oscillatios. Choose the samplig odes x j, j =,..., accordig to Z xj j = c ( x 2 ), j =,..., As goes from to the samplig odes will cluster more ad more towards the boudaries.
11 Defeatig the Ruge Pheomeo <latexit sha_base64="ndbgsqbrjbj6lnjdmlr+wf8hsio=">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</latexit> <latexit sha_base64="ndbgsqbrjbj6lnjdmlr+wf8hsio=">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</latexit> <latexit sha_base64="ndbgsqbrjbj6lnjdmlr+wf8hsio=">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</latexit> <latexit sha_base64="ndbgsqbrjbj6lnjdmlr+wf8hsio=">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</latexit> <latexit sha_base64="hapmyje+cfsd7nzimstxqmw/s+i=">aaacixicbvdltgjbejxfrcqx6nhlrmkcf7jrii8birepmlhaaotmdg2mzm5uzovsuefvorzr/fkvbl/xufxelcstirv3emu8ipbntrot5xa2nxkb2d2srt7+wehufxrtyexyucxuisq4vmngkvwkkoarqsabr6auj+8fbrt6ad+udjinob7qvey8zikaqrz3l4ui8kys4jwcge524cigcq7esvd6oysdkaietrpute2e6oqs4etlktwene2zd2owmopahodjj7d2kfgavr9jlsqi9u/9ujdtqehz4zjkgoncrva4x68zy++6xazxqiszq/ymfjae+922ugkeygkz4uzxmw2oogxnqtmwbhof7omgasgm8jl3zz2kxhk4xhi94g+mgrncu5qtuveuyjdlnz7cqfsxksyisfklbsjs65ihdyrkveii4/khbysn+vd+ra+ra/5ampa7bytjvg/v+ccoy8=</latexit> <latexit sha_base64="hapmyje+cfsd7nzimstxqmw/s+i=">aaacixicbvdltgjbejxfrcqx6nhlrmkcf7jrii8birepmlhaaotmdg2mzm5uzovsuefvorzr/fkvbl/xufxelcstirv3emu8ipbntrot5xa2nxkb2d2srt7+wehufxrtyexyucxuisq4vmngkvwkkoarqsabr6auj+8fbrt6ad+udjinob7qvey8zikaqrz3l4ui8kys4jwcge524cigcq7esvd6oysdkaietrpute2e6oqs4etlktwene2zd2owmopahodjj7d2kfgavr9jlsqi9u/9ujdtqehz4zjkgoncrva4x68zy++6xazxqiszq/ymfjae+922ugkeygkz4uzxmw2oogxnqtmwbhof7omgasgm8jl3zz2kxhk4xhi94g+mgrncu5qtuveuyjdlnz7cqfsxksyisfklbsjs65ihdyrkveii4/khbysn+vd+ra+ra/5ampa7bytjvg/v+ccoy8=</latexit> <latexit sha_base64="hapmyje+cfsd7nzimstxqmw/s+i=">aaacixicbvdltgjbejxfrcqx6nhlrmkcf7jrii8birepmlhaaotmdg2mzm5uzovsuefvorzr/fkvbl/xufxelcstirv3emu8ipbntrot5xa2nxkb2d2srt7+wehufxrtyexyucxuisq4vmngkvwkkoarqsabr6auj+8fbrt6ad+udjinob7qvey8zikaqrz3l4ui8kys4jwcge524cigcq7esvd6oysdkaietrpute2e6oqs4etlktwene2zd2owmopahodjj7d2kfgavr9jlsqi9u/9ujdtqehz4zjkgoncrva4x68zy++6xazxqiszq/ymfjae+922ugkeygkz4uzxmw2oogxnqtmwbhof7omgasgm8jl3zz2kxhk4xhi94g+mgrncu5qtuveuyjdlnz7cqfsxksyisfklbsjs65ihdyrkveii4/khbysn+vd+ra+ra/5ampa7bytjvg/v+ccoy8=</latexit> <latexit sha_base64="hapmyje+cfsd7nzimstxqmw/s+i=">aaacixicbvdltgjbejxfrcqx6nhlrmkcf7jrii8birepmlhaaotmdg2mzm5uzovsuefvorzr/fkvbl/xufxelcstirv3emu8ipbntrot5xa2nxkb2d2srt7+wehufxrtyexyucxuisq4vmngkvwkkoarqsabr6auj+8fbrt6ad+udjinob7qvey8zikaqrz3l4ui8kys4jwcge524cigcq7esvd6oysdkaietrpute2e6oqs4etlktwene2zd2owmopahodjj7d2kfgavr9jlsqi9u/9ujdtqehz4zjkgoncrva4x68zy++6xazxqiszq/ymfjae+922ugkeygkz4uzxmw2oogxnqtmwbhof7omgasgm8jl3zz2kxhk4xhi94g+mgrncu5qtuveuyjdlnz7cqfsxksyisfklbsjs65ihdyrkveii4/khbysn+vd+ra+ra/5ampa7bytjvg/v+ccoy8=</latexit> Example: Iterpolat of the fuctio f(x) = for = 2 as varies over [, ]. + 6x2 p 6 (x)
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