### Abstract

Classical rational interpolation is known to suffer from several drawbacks, such as unattainable points and randomly located poles for a small number of nodes, as well as an erratic behavior of the error as this number grows larger. In a former article, we have suggested to obtain rational interpolants by a procedure that attaches optimally placed poles to the interpolating polynomial, using the barycentric representation of the interpolants. In order to improve upon the condition of the derivatives in the solution of differential equations, we have then experimented with a conformal point shift suggested by Kosloff and Tal-Ezer. As it turned out, such shifts can achieve a spectacular improvement in the quality of the approximation itself for functions with a large gradient in the center of the interval. This leads us to the present work which combines the pole attachment method with shifts optimally adjusted to the interpolated function. Such shifts are also constructed for functions with several shocks away from the extremities of the interval.

Original language | English (US) |
---|---|

Pages (from-to) | 81-92 |

Number of pages | 12 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 164 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2004 |

### Fingerprint

### Keywords

- Interpolation
- Optimal interpolation
- Point shifts
- Rational approximation

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Numerical Analysis

### Cite this

**Adaptive point shifts in rational approximation with optimized denominator.** / Berrut, J. P.; Mittelmann, Hans.

Research output: Contribution to journal › Article

*Journal of Computational and Applied Mathematics*, vol. 164, no. 1, pp. 81-92. https://doi.org/10.1016/S0377-0427(03)00485-0

}

TY - JOUR

T1 - Adaptive point shifts in rational approximation with optimized denominator

AU - Berrut, J. P.

AU - Mittelmann, Hans

PY - 2004/3/1

Y1 - 2004/3/1

N2 - Classical rational interpolation is known to suffer from several drawbacks, such as unattainable points and randomly located poles for a small number of nodes, as well as an erratic behavior of the error as this number grows larger. In a former article, we have suggested to obtain rational interpolants by a procedure that attaches optimally placed poles to the interpolating polynomial, using the barycentric representation of the interpolants. In order to improve upon the condition of the derivatives in the solution of differential equations, we have then experimented with a conformal point shift suggested by Kosloff and Tal-Ezer. As it turned out, such shifts can achieve a spectacular improvement in the quality of the approximation itself for functions with a large gradient in the center of the interval. This leads us to the present work which combines the pole attachment method with shifts optimally adjusted to the interpolated function. Such shifts are also constructed for functions with several shocks away from the extremities of the interval.

AB - Classical rational interpolation is known to suffer from several drawbacks, such as unattainable points and randomly located poles for a small number of nodes, as well as an erratic behavior of the error as this number grows larger. In a former article, we have suggested to obtain rational interpolants by a procedure that attaches optimally placed poles to the interpolating polynomial, using the barycentric representation of the interpolants. In order to improve upon the condition of the derivatives in the solution of differential equations, we have then experimented with a conformal point shift suggested by Kosloff and Tal-Ezer. As it turned out, such shifts can achieve a spectacular improvement in the quality of the approximation itself for functions with a large gradient in the center of the interval. This leads us to the present work which combines the pole attachment method with shifts optimally adjusted to the interpolated function. Such shifts are also constructed for functions with several shocks away from the extremities of the interval.

KW - Interpolation

KW - Optimal interpolation

KW - Point shifts

KW - Rational approximation

UR - http://www.scopus.com/inward/record.url?scp=1442265047&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1442265047&partnerID=8YFLogxK

U2 - 10.1016/S0377-0427(03)00485-0

DO - 10.1016/S0377-0427(03)00485-0

M3 - Article

AN - SCOPUS:1442265047

VL - 164

SP - 81

EP - 92

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1

ER -