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interpolation and extrapolation
Numerical Analysis 2000 Vol. II: Interpolation and extrapolation
Vol. II: Interpolation and extrapolation
This volume is dedicated to two closely related subjects: interpolation and extrapolation. The papers can be divided into three categories: historical papers, survey papers and papers presenting new developments.
Interpolation is an old subject since, as noticed in the paper by M. Gasca and T. Sauer, the term was coined by John Wallis in 1655. Interpolation was the rst technique for obtaining an approximation of a function. Polynomial interpolation was then used in quadrature methods and
methods for the numerical solution of ordinary differential equations.
Obviously, some applications need interpolation by functions more complicated than polynomials. The case of rational functions with prescribed poles is treated in the paper by G. Muhlbach. He gives a survey of interpolation procedures using Cauchy{Vandermonde systems. The well-known formulae of Lagrange, Newton and Neville{Aitken are generalized. The construction of rational B-splines is discussed.
Trigonometric polynomials are used in the paper by T. Strohmer for the reconstruction of a signal from non-uniformly spaced measurements. They lead to a well-posed problem that preserves some important structural properties of the original infnite dimensional problem.
More recently, interpolation in several variables was studied. It has applications in fnite differences and fnite elements for solving partial differential equations. Following the pioneer works of P. de Casteljau and P. Bezier, another very important domain where multivariate interpolation plays a fundamental role is computer-aided geometric design (CAGD) for the approximation of surfaces.
The history of multivariate polynomial interpolation is related in the paper by M. Gasca and T. Sauer.
The paper by R.A. Lorentz is devoted to the historical development of multivariate Hermite interpolation by algebraic polynomials.
In his paper, G. Walz treats the approximation of multivariate functions by multivariate Bernstein polynomials. An asymptotic expansion of these polynomials is given and then used for building, by extrapolation, a new approximation method which converges much faster.
Extrapolation is based on interpolation. In fact, extrapolation consists of interpolation at a point outside the interval containing the interpolation points. Usually, this point is either zero or infnity. Extrapolation is used in numerical analysis to improve the accuracy of a process depending of
a parameter or to accelerate the convergence of a sequence. The most well-known extrapolation processes are certainly Romberg's method for improving the convergence of the trapezoidal rule for the computation of a de nite integral and Aitken's process which can be found in any textbook of numerical analysis.
An historical account of the development of the subject during the 20th century is given in the paper by C. Brezinski.
The theory of extrapolation methods lays on the solution of the system of linear equations corresponding to the interpolation conditions. In their paper, M. Gasca and G. Muhlbach show, by using elimination techniques, the connection between extrapolation, linear systems, totally positive matrices and CAGD.
Journal of Computational and Applied Mathematics Copyright © 2003 Elsevier B.V. Volume 122, Issues 1-2, Pages 1-357 (1 October 2000)
Convergence acceleration during the 20th century, Pages 1-21
C. Brezinski
On the history of multivariate polynomial interpolation, Pages 23-35
Mariano Gasca and Thomas Sauer
Elimination techniques: from extrapolation to totally positive matrices and CAGD, Pages 37-50
M. Gasca and G. Mühlbach
The epsilon algorithm and related topics, Pages 51-80
P. R. Graves-Morris, D. E. Roberts and A. Salam
Scalar Levin-type sequence transformations, Pages 81-147
Herbert H. H. Homeier
Vector extrapolation methods. Applications and numerical comparison, Pages 149-165
K. Jbilou and H. Sadok
Multivariate Hermite interpolation by algebraic polynomials: A survey, Pages 167-201
R. A. Lorentz
Interpolation by Cauchy–Vandermonde systems and applications, Pages 203-222
G. Mühlbach
The E-algorithm and the Ford–Sidi algorithm, Pages 223-230
Naoki Osada
Diophantine approximations using Padé approximations, Pages 231-250
M. Prévost
The generalized Richardson extrapolation process GREP(1) and computation of derivatives of limits of sequences
with applications to the d(1)-transformation, Pages 251-273
Avram Sidi
Matrix Hermite–Padé problem and dynamical systems, Pages 275-295
Vladimir Sorokin and Jeannette Van Iseghem
Numerical analysis of the non-uniform sampling problem, Pages 297-316
Thomas Strohmer
Asymptotic expansions for multivariate polynomial approximation, Pages 317-328
Guido Walz
Prediction properties of Aitken's iterated 2 process, of Wynn's epsilon algorithm, and of Brezinski's iterated
theta algorithm, Pages 329-356
Ernst Joachim Weniger
Index, Page 357
Numerical Analysis 2000 Vol. II: Interpolation and extrapolation
Vol. II: Interpolation and extrapolation
This volume is dedicated to two closely related subjects: interpolation and extrapolation. The papers can be divided into three categories: historical papers, survey papers and papers presenting new developments.
Interpolation is an old subject since, as noticed in the paper by M. Gasca and T. Sauer, the term was coined by John Wallis in 1655. Interpolation was the rst technique for obtaining an approximation of a function. Polynomial interpolation was then used in quadrature methods and
methods for the numerical solution of ordinary differential equations.
Obviously, some applications need interpolation by functions more complicated than polynomials. The case of rational functions with prescribed poles is treated in the paper by G. Muhlbach. He gives a survey of interpolation procedures using Cauchy{Vandermonde systems. The well-known formulae of Lagrange, Newton and Neville{Aitken are generalized. The construction of rational B-splines is discussed.
Trigonometric polynomials are used in the paper by T. Strohmer for the reconstruction of a signal from non-uniformly spaced measurements. They lead to a well-posed problem that preserves some important structural properties of the original infnite dimensional problem.
More recently, interpolation in several variables was studied. It has applications in fnite differences and fnite elements for solving partial differential equations. Following the pioneer works of P. de Casteljau and P. Bezier, another very important domain where multivariate interpolation plays a fundamental role is computer-aided geometric design (CAGD) for the approximation of surfaces.
The history of multivariate polynomial interpolation is related in the paper by M. Gasca and T. Sauer.
The paper by R.A. Lorentz is devoted to the historical development of multivariate Hermite interpolation by algebraic polynomials.
In his paper, G. Walz treats the approximation of multivariate functions by multivariate Bernstein polynomials. An asymptotic expansion of these polynomials is given and then used for building, by extrapolation, a new approximation method which converges much faster.
Extrapolation is based on interpolation. In fact, extrapolation consists of interpolation at a point outside the interval containing the interpolation points. Usually, this point is either zero or infnity. Extrapolation is used in numerical analysis to improve the accuracy of a process depending of
a parameter or to accelerate the convergence of a sequence. The most well-known extrapolation processes are certainly Romberg's method for improving the convergence of the trapezoidal rule for the computation of a de nite integral and Aitken's process which can be found in any textbook of numerical analysis.
An historical account of the development of the subject during the 20th century is given in the paper by C. Brezinski.
The theory of extrapolation methods lays on the solution of the system of linear equations corresponding to the interpolation conditions. In their paper, M. Gasca and G. Muhlbach show, by using elimination techniques, the connection between extrapolation, linear systems, totally positive matrices and CAGD.
Journal of Computational and Applied Mathematics Copyright © 2003 Elsevier B.V. Volume 122, Issues 1-2, Pages 1-357 (1 October 2000)
Convergence acceleration during the 20th century, Pages 1-21
C. Brezinski
On the history of multivariate polynomial interpolation, Pages 23-35
Mariano Gasca and Thomas Sauer
Elimination techniques: from extrapolation to totally positive matrices and CAGD, Pages 37-50
M. Gasca and G. Mühlbach
The epsilon algorithm and related topics, Pages 51-80
P. R. Graves-Morris, D. E. Roberts and A. Salam
Scalar Levin-type sequence transformations, Pages 81-147
Herbert H. H. Homeier
Vector extrapolation methods. Applications and numerical comparison, Pages 149-165
K. Jbilou and H. Sadok
Multivariate Hermite interpolation by algebraic polynomials: A survey, Pages 167-201
R. A. Lorentz
Interpolation by Cauchy–Vandermonde systems and applications, Pages 203-222
G. Mühlbach
The E-algorithm and the Ford–Sidi algorithm, Pages 223-230
Naoki Osada
Diophantine approximations using Padé approximations, Pages 231-250
M. Prévost
The generalized Richardson extrapolation process GREP(1) and computation of derivatives of limits of sequences
with applications to the d(1)-transformation, Pages 251-273
Avram Sidi
Matrix Hermite–Padé problem and dynamical systems, Pages 275-295
Vladimir Sorokin and Jeannette Van Iseghem
Numerical analysis of the non-uniform sampling problem, Pages 297-316
Thomas Strohmer
Asymptotic expansions for multivariate polynomial approximation, Pages 317-328
Guido Walz
Prediction properties of Aitken's iterated 2 process, of Wynn's epsilon algorithm, and of Brezinski's iterated
theta algorithm, Pages 329-356
Ernst Joachim Weniger
Index, Page 357