Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval
Polynomial interpolation between large numbers of arbitrary nodes does notoriously not, in general, yield useful approximations of continuous functions. Following [1], we suggest to determine for each set of n + 1 nodes another denominator of degree n, and then to interpolate every continuous functi...
Saved in:
| Published in: | Computers & mathematics with applications (1987) Vol. 33; no. 6; pp. 77 - 86 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier Ltd
1997
|
| Subjects: | |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | Polynomial interpolation between large numbers of arbitrary nodes does notoriously not, in general, yield useful approximations of continuous functions. Following [1], we suggest to determine for each set of
n + 1 nodes another denominator of degree
n, and then to interpolate every continuous function by a rational function with that same denominator, so that the resulting interpolation process remains a linear projection. The optimal denominator is chosen so as to minimize the Lebesgue constant for the given nodes. It has to be computed numerically. For that purpose, the barycentric representation of rational interpolants, which displays the linearity of the interpolation and reduces the determination of the denominator to that of the barycentric weights, is used. The optimal weights can then be computed by solving an optimization problem with simple bounds which could not be solved accurately by the first author in [1]. We show here how to do so, and we present numerical results. |
|---|---|
| AbstractList | Polynomial interpolation between large numbers of arbitrary nodes does notoriously not, in general, yield useful approximations of continuous functions. Following [1], we suggest to determine for each set of
n + 1 nodes another denominator of degree
n, and then to interpolate every continuous function by a rational function with that same denominator, so that the resulting interpolation process remains a linear projection. The optimal denominator is chosen so as to minimize the Lebesgue constant for the given nodes. It has to be computed numerically. For that purpose, the barycentric representation of rational interpolants, which displays the linearity of the interpolation and reduces the determination of the denominator to that of the barycentric weights, is used. The optimal weights can then be computed by solving an optimization problem with simple bounds which could not be solved accurately by the first author in [1]. We show here how to do so, and we present numerical results. |
| Author | Mittelmann, H.D. Berrut, J.-P. |
| Author_xml | – sequence: 1 givenname: J.-P. surname: Berrut fullname: Berrut, J.-P. organization: Université de Fribourg, Mathématiques CH-1700 Fribourg, Pérolles, Switzerland – sequence: 2 givenname: H.D. surname: Mittelmann fullname: Mittelmann, H.D. organization: Department of Mathematics, Arizona State University Tempe, AZ 85287-1804, U.S.A |
| BookMark | eNqFkE9LxDAUxIMouLv6EYQc9VBN0jZNTyKL_2DBg3oOafKyRrrJkqQL-ultd8Wrp8c8ZobhN0fHPnhA6IKSa0oov3klohUFZYxets0VIaSsivoIzahoyqLhXByj2Z_lFM1T-hxNVcnIDJkVdJDWA2AdfMrKZ7xx3m3ct_Nr3DsPKuKosgte9dj5DHEb-r3GwU6h7PwQhoTt4PX0TjjsIOL8AQf7TvVn6MSqPsH5712g94f7t-VTsXp5fF7erQpdEZYLVnWdYR1n7ThWVYZYI6BUmhgOmuqWk6oBUnPBS2G6mtnG1MSW1ArKKW_acoHqQ6-OIaUIVm6j26j4JSmREyq5RyUnDrJt5B6VrMfc7SEH47idgyiTduA1GBdBZ2mC-6fhB1DcdQc |
| CitedBy_id | crossref_primary_10_1216_rmjm_1030539685 crossref_primary_10_1016_S0377_0427_97_00147_7 crossref_primary_10_1007_s11075_010_9415_8 crossref_primary_10_1007_s00211_007_0093_y crossref_primary_10_1016_j_apnum_2011_05_001 crossref_primary_10_1007_s40819_020_00891_6 crossref_primary_10_1016_S0377_0427_00_00552_5 crossref_primary_10_1007_s10444_022_09986_8 crossref_primary_10_1016_j_cam_2019_02_034 crossref_primary_10_1007_s00211_011_0442_8 crossref_primary_10_1016_j_cam_2019_112347 crossref_primary_10_1090_S0025_5718_99_01070_4 crossref_primary_10_1109_JSAIT_2021_3085676 crossref_primary_10_1137_110827156 crossref_primary_10_1002_mma_7548 crossref_primary_10_1016_j_cagd_2021_102003 crossref_primary_10_1016_j_jcp_2009_05_045 crossref_primary_10_1016_j_amc_2016_09_003 crossref_primary_10_1016_j_aml_2016_03_007 crossref_primary_10_1016_j_cam_2013_03_044 crossref_primary_10_1007_s10543_011_0357_x crossref_primary_10_1007_s40315_020_00334_9 crossref_primary_10_1016_j_cam_2011_04_004 crossref_primary_10_1137_S0036144502417715 crossref_primary_10_1007_s00211_022_01316_w crossref_primary_10_1109_TMM_2014_2348712 crossref_primary_10_1137_S106482750036615X crossref_primary_10_1016_j_cam_2013_06_030 crossref_primary_10_1051_shsconf_202111907002 crossref_primary_10_1016_j_jat_2013_01_004 crossref_primary_10_1016_j_amc_2017_08_027 crossref_primary_10_1080_10586458_2015_1072862 crossref_primary_10_1088_1361_6501_aca0b3 crossref_primary_10_3390_math8040498 crossref_primary_10_3934_math_2023181 crossref_primary_10_1137_17M114371X crossref_primary_10_1002_num_21864 crossref_primary_10_1007_s11766_024_3711_x crossref_primary_10_1109_TPAMI_2022_3151434 crossref_primary_10_1007_s00211_013_0535_7 crossref_primary_10_1007_s11075_010_9399_4 crossref_primary_10_1016_S0377_0427_96_00163_X |
| Cites_doi | 10.1007/BF01897094 10.1090/S0025-5718-1984-0744931-0 10.1090/S0025-5718-1986-0842136-8 10.2307/2323093 10.1137/0715046 10.1016/0898-1221(88)90067-3 10.1007/BF01396489 |
| ContentType | Journal Article |
| Copyright | 1997 |
| Copyright_xml | – notice: 1997 |
| DBID | 6I. AAFTH AAYXX CITATION |
| DOI | 10.1016/S0898-1221(97)00034-5 |
| DatabaseName | ScienceDirect Open Access Titles Elsevier:ScienceDirect:Open Access CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Computer Science |
| EISSN | 1873-7668 |
| EndPage | 86 |
| ExternalDocumentID | 10_1016_S0898_1221_97_00034_5 S0898122197000345 |
| GroupedDBID | --K --M -~X .DC .~1 0R~ 1B1 1RT 1~. 1~5 29F 4.4 457 4G. 5GY 5VS 6I. 7-5 71M 8P~ 9JN AACTN AAEDT AAEDW AAFTH AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAQXK AAXUO AAYFN ABAOU ABBOA ABFNM ABMAC ABTAH ABVKL ABXDB ABYKQ ACAZW ACDAQ ACGFS ACIWK ACNCT ACNNM ACRLP ACZNC ADBBV ADEZE ADMUD ADTZH AEBSH AECPX AEKER AENEX AEXQZ AFFNX AFKWA AFTJW AGHFR AGUBO AGYEJ AHHHB AHJVU AHZHX AIALX AIEXJ AIGVJ AIKHN AITUG AJBFU AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ AOUOD ARUGR ASPBG AVWKF AXJTR AZFZN BJAXD BKOJK BLXMC CS3 DU5 EBS EFJIC EFLBG EJD EO8 EO9 EP2 EP3 F5P FDB FEDTE FGOYB FIRID FNPLU FYGXN G-2 G-Q GBLVA GBOLZ HVGLF HZ~ IHE IXB J1W JJJVA KOM LG9 M26 M41 MHUIS MO0 N9A NCXOZ O-L O9- OAUVE OK1 OZT P-8 P-9 P2P PC. PQQKQ Q38 R2- RIG RNS ROL RPZ SDF SDG SDP SES SEW SPC SPCBC SST SSV SSW SSZ T5K TAE TN5 WUQ XFK XPP ZMT ZY4 ~G- 0SF AAXKI AAYXX ADVLN AFJKZ AKRWK CITATION |
| ID | FETCH-LOGICAL-c402t-24bbd2b629898a4d0fd8e3ac0d6ec1c96047e0568638db52f7d50f31f81616793 |
| ISSN | 0898-1221 |
| IngestDate | Thu Sep 26 16:58:29 EDT 2024 Fri Feb 23 02:27:09 EST 2024 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 6 |
| Keywords | Lebesgue constant Interpolation Linear interpolation Rational interpolation |
| Language | English |
| License | http://www.elsevier.com/open-access/userlicense/1.0 |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c402t-24bbd2b629898a4d0fd8e3ac0d6ec1c96047e0568638db52f7d50f31f81616793 |
| OpenAccessLink | https://dx.doi.org/10.1016/S0898-1221(97)00034-5 |
| PageCount | 10 |
| ParticipantIDs | crossref_primary_10_1016_S0898_1221_97_00034_5 elsevier_sciencedirect_doi_10_1016_S0898_1221_97_00034_5 |
| PublicationCentury | 1900 |
| PublicationDate | 1997-00-00 |
| PublicationDateYYYYMMDD | 1997-01-01 |
| PublicationDate_xml | – year: 1997 text: 1997-00-00 |
| PublicationDecade | 1990 |
| PublicationTitle | Computers & mathematics with applications (1987) |
| PublicationYear | 1997 |
| Publisher | Elsevier Ltd |
| Publisher_xml | – name: Elsevier Ltd |
| References | Berrut (BIB1) 1994 Henrici (BIB2) 1982 Stoer (BIB14) 1983 Runge (BIB3) 1901; 46 Brutman (BIB7) 1978; 15 Berrut (BIB15) 1988; 15 Werner (BIB13) 1984; 43 Berrut (BIB17) 1989; 54 Rivlin (BIB11) 1981 de Boor (BIB20) 1978; 27 Powell (BIB6) 1981 Berrut, Mittelmann (BIB19) 1994 Rivlin (BIB10) 1974 Schneider, Werner (BIB16) 1986; 47 Erdős, Vértesi (BIB5) 1980; 36 Szabados, Vértesi (BIB9) 1990 Bulirsch, Rutishauser (BIB12) 1968; 141 Szegö (BIB8) 1939; Volume XXIII Epperson (BIB4) 1987; 94 Zhou, Tits (BIB18) 1995 de Boor (10.1016/S0898-1221(97)00034-5_BIB20) 1978; 27 Berrut (10.1016/S0898-1221(97)00034-5_BIB1) 1994 Runge (10.1016/S0898-1221(97)00034-5_BIB3) 1901; 46 Henrici (10.1016/S0898-1221(97)00034-5_BIB2) 1982 Stoer (10.1016/S0898-1221(97)00034-5_BIB14) 1983 Szabados (10.1016/S0898-1221(97)00034-5_BIB9) 1990 Szegö (10.1016/S0898-1221(97)00034-5_BIB8) 1939; Volume XXIII Werner (10.1016/S0898-1221(97)00034-5_BIB13) 1984; 43 Rivlin (10.1016/S0898-1221(97)00034-5_BIB10) 1974 Zhou (10.1016/S0898-1221(97)00034-5_BIB18) 1995 Berrut (10.1016/S0898-1221(97)00034-5_BIB15) 1988; 15 Schneider (10.1016/S0898-1221(97)00034-5_BIB16) 1986; 47 Berrut (10.1016/S0898-1221(97)00034-5_BIB17_1) 1989; 54 (10.1016/S0898-1221(97)00034-5_BIB17_2) 1989; 55 Epperson (10.1016/S0898-1221(97)00034-5_BIB4) 1987; 94 Berrut (10.1016/S0898-1221(97)00034-5_BIB19) 1994 Brutman (10.1016/S0898-1221(97)00034-5_BIB7) 1978; 15 Erdős (10.1016/S0898-1221(97)00034-5_BIB5) 1980; 36 Rivlin (10.1016/S0898-1221(97)00034-5_BIB11) 1981 Bulirsch (10.1016/S0898-1221(97)00034-5_BIB12) 1968; 141 Powell (10.1016/S0898-1221(97)00034-5_BIB6) 1981 |
| References_xml | – volume: 54 start-page: 703 year: 1989 end-page: 718 ident: BIB17 article-title: Barycentric formulae for cardinal (SINC-) interpolants publication-title: Numer. Math. contributor: fullname: Berrut – volume: 36 start-page: 71 year: 1980 end-page: 89 ident: BIB5 article-title: On the almost everywhere divergence of Lagrange interpolatory polynomials for arbitrary systems of nodes publication-title: Acta Math. Acad. Sci. Hungar. contributor: fullname: Vértesi – volume: 15 start-page: 1 year: 1988 end-page: 16 ident: BIB15 article-title: Rational functions for guaranteed and experimentally well-conditioned global interpolation publication-title: Computers Math. Applic. contributor: fullname: Berrut – year: 1981 ident: BIB11 article-title: An Introduction to the Approximation of Functions contributor: fullname: Rivlin – volume: 46 start-page: 224 year: 1901 end-page: 243 ident: BIB3 article-title: Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten publication-title: Z. Math. Phys. contributor: fullname: Runge – year: 1983 ident: BIB14 article-title: Einführung in die Numerische Mathematik I contributor: fullname: Stoer – volume: 27 year: 1978 ident: BIB20 article-title: A Practical Guide to Splines publication-title: Applied Mathematical Sciences contributor: fullname: de Boor – volume: 94 start-page: 329 year: 1987 end-page: 341 ident: BIB4 article-title: On the Runge example publication-title: Amer. Math. Monthly contributor: fullname: Epperson – volume: 15 start-page: 694 year: 1978 end-page: 704 ident: BIB7 article-title: On the Lebesgue function for polynomial interpolation publication-title: SIAM J. Numer. Anal. contributor: fullname: Brutman – volume: 141 start-page: 232 year: 1968 end-page: 319 ident: BIB12 article-title: Interpolation und genäherte Quadratur publication-title: Mathematische Hilfsmittel des Ingenieurs contributor: fullname: Rutishauser – year: 1982 ident: BIB2 article-title: Essentials of Numerical Analysis contributor: fullname: Henrici – volume: 47 start-page: 285 year: 1986 end-page: 299 ident: BIB16 article-title: Some new aspects of rational interpolation publication-title: Math. Comp. contributor: fullname: Werner – year: 1981 ident: BIB6 article-title: Approximation Theory and Methods contributor: fullname: Powell – year: 1990 ident: BIB9 article-title: Interpolation of Functions contributor: fullname: Vértesi – year: 1995 ident: BIB18 article-title: User's guide for FFSQP version 3.5: A FORTRAN code for solving constrained nonlinear (minimax) optimization problems, generating iterates satisfying all inequalitites and linear constraints contributor: fullname: Tits – start-page: 261 year: 1994 end-page: 264 ident: BIB1 article-title: Linear rational interpolation of continuous functions over an interval publication-title: Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics, (Proceedings of Symposia in Applied Mathematics) contributor: fullname: Berrut – year: 1994 ident: BIB19 publication-title: Report 94-4 contributor: fullname: Mittelmann – year: 1974 ident: BIB10 article-title: The Chebyshev Polynomials contributor: fullname: Rivlin – volume: 43 start-page: 205 year: 1984 end-page: 217 ident: BIB13 article-title: Polynomial interpolation: Lagrange versus Newton publication-title: Math. Comp. contributor: fullname: Werner – volume: Volume XXIII year: 1939 ident: BIB8 article-title: Orthogonal Polynomials publication-title: AMS Colloquium Publications contributor: fullname: Szegö – volume: 46 start-page: 224 year: 1901 ident: 10.1016/S0898-1221(97)00034-5_BIB3 article-title: Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten publication-title: Z. Math. Phys. contributor: fullname: Runge – volume: 36 start-page: 71 year: 1980 ident: 10.1016/S0898-1221(97)00034-5_BIB5 article-title: On the almost everywhere divergence of Lagrange interpolatory polynomials for arbitrary systems of nodes publication-title: Acta Math. Acad. Sci. Hungar. doi: 10.1007/BF01897094 contributor: fullname: Erdős – volume: 43 start-page: 205 year: 1984 ident: 10.1016/S0898-1221(97)00034-5_BIB13 article-title: Polynomial interpolation: Lagrange versus Newton publication-title: Math. Comp. doi: 10.1090/S0025-5718-1984-0744931-0 contributor: fullname: Werner – volume: 141 start-page: 232 year: 1968 ident: 10.1016/S0898-1221(97)00034-5_BIB12 article-title: Interpolation und genäherte Quadratur contributor: fullname: Bulirsch – volume: 47 start-page: 285 year: 1986 ident: 10.1016/S0898-1221(97)00034-5_BIB16 article-title: Some new aspects of rational interpolation publication-title: Math. Comp. doi: 10.1090/S0025-5718-1986-0842136-8 contributor: fullname: Schneider – year: 1995 ident: 10.1016/S0898-1221(97)00034-5_BIB18 contributor: fullname: Zhou – year: 1974 ident: 10.1016/S0898-1221(97)00034-5_BIB10 contributor: fullname: Rivlin – year: 1983 ident: 10.1016/S0898-1221(97)00034-5_BIB14 contributor: fullname: Stoer – year: 1990 ident: 10.1016/S0898-1221(97)00034-5_BIB9 contributor: fullname: Szabados – volume: 94 start-page: 329 year: 1987 ident: 10.1016/S0898-1221(97)00034-5_BIB4 article-title: On the Runge example publication-title: Amer. Math. Monthly doi: 10.2307/2323093 contributor: fullname: Epperson – volume: 27 year: 1978 ident: 10.1016/S0898-1221(97)00034-5_BIB20 article-title: A Practical Guide to Splines contributor: fullname: de Boor – start-page: 261 year: 1994 ident: 10.1016/S0898-1221(97)00034-5_BIB1 article-title: Linear rational interpolation of continuous functions over an interval contributor: fullname: Berrut – year: 1982 ident: 10.1016/S0898-1221(97)00034-5_BIB2 contributor: fullname: Henrici – volume: 55 start-page: 747 year: 1989 ident: 10.1016/S0898-1221(97)00034-5_BIB17_2 publication-title: Erratum – year: 1981 ident: 10.1016/S0898-1221(97)00034-5_BIB6 contributor: fullname: Powell – volume: 15 start-page: 694 year: 1978 ident: 10.1016/S0898-1221(97)00034-5_BIB7 article-title: On the Lebesgue function for polynomial interpolation publication-title: SIAM J. Numer. Anal. doi: 10.1137/0715046 contributor: fullname: Brutman – volume: Volume XXIII year: 1939 ident: 10.1016/S0898-1221(97)00034-5_BIB8 article-title: Orthogonal Polynomials publication-title: AMS Colloquium Publications contributor: fullname: Szegö – year: 1981 ident: 10.1016/S0898-1221(97)00034-5_BIB11 contributor: fullname: Rivlin – year: 1994 ident: 10.1016/S0898-1221(97)00034-5_BIB19 contributor: fullname: Berrut – volume: 15 start-page: 1 year: 1988 ident: 10.1016/S0898-1221(97)00034-5_BIB15 article-title: Rational functions for guaranteed and experimentally well-conditioned global interpolation publication-title: Computers Math. Applic. doi: 10.1016/0898-1221(88)90067-3 contributor: fullname: Berrut – volume: 54 start-page: 703 year: 1989 ident: 10.1016/S0898-1221(97)00034-5_BIB17_1 article-title: Barycentric formulae for cardinal (SINC-) interpolants publication-title: Numer. Math. doi: 10.1007/BF01396489 contributor: fullname: Berrut |
| SSID | ssj0004320 |
| Score | 1.7674139 |
| Snippet | Polynomial interpolation between large numbers of arbitrary nodes does notoriously not, in general, yield useful approximations of continuous functions.... |
| SourceID | crossref elsevier |
| SourceType | Aggregation Database Publisher |
| StartPage | 77 |
| SubjectTerms | Interpolation Lebesgue constant Linear interpolation Rational interpolation |
| Title | Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval |
| URI | https://dx.doi.org/10.1016/S0898-1221(97)00034-5 |
| Volume | 33 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://sdu.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1Lj9MwELa63QsX3ojlJR84gFYOqePEyXFhiwpCCGkXaW9RYjuo0raL0oQDv54ZP-KutEKAxCWqIsduZ76Ox5OZbwh52SnkQBEt422jmMiVYK1edEyBL17Ijle8wXrn1Zn8fFGeLsVyNgtNUuO9_6ppuAe6xsrZv9D2NCncgM-gc7iC1uH6R3r_BPLafRttEjp6fsMxsods1j8xJoA-ZdMf9yECuHZdti4nvxEz19fbEfNiccfzaXIgBOug2uE__DcI9Aa-LcTOgmgzscCGsrm9F-SWFOp69OGt6fvRvQ5J2JdkUv96GMzlxjdwXiWnSYxOYM5KDJaFgpmYnWRtWgWHVu6KohPjbG4pMyYL110nGGXHjuHBt29hfdMXv1cXN-4CLiBxNi0Gvw75hivLxsPyuPVNCYl2LA6tpB2UH5BDDqaLz8nhyYflxcdYa5s5qs8wd6wKexMXfFXJ136xm_2dPR_m_C657Q8f9MSh5h6Zme19cidokHo7_4DoACIaQEQjiKgDEQ0gotdARK86GkFEJxBRBBEFcNAAoofk6_vl-bsV8-04mBIpHxj8o1vN2wI5-8tG6LTTpckalerCqIVClh9pwJ8uwaTrNued1HnaZYuuhFNFAcJ8RObbq615TGghqrZrC2nAn0W2oEYYmCqXxlSpUjk_IkkQWv3dsa7UMR0Rlq9RyrXLnchEnR-RMoi29q6jcwlrwMTvH33y748-JbccmzFG5J6R-dCP5jk52OnxhYfNLx7RkGw |
| link.rule.ids | 315,782,786,4028,27932,27933,27934 |
| linkProvider | Elsevier |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Lebesgue+constant+minimizing+linear+rational+interpolation+of+continuous+functions+over+the+interval&rft.jtitle=Computers+%26+mathematics+with+applications+%281987%29&rft.au=Berrut%2C+J.-P.&rft.au=Mittelmann%2C+H.D.&rft.date=1997&rft.pub=Elsevier+Ltd&rft.issn=0898-1221&rft.eissn=1873-7668&rft.volume=33&rft.issue=6&rft.spage=77&rft.epage=86&rft_id=info:doi/10.1016%2FS0898-1221%2897%2900034-5&rft.externalDocID=S0898122197000345 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0898-1221&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0898-1221&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0898-1221&client=summon |