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Solution of Interpolation Problems via the Hankel Polynomial Construction [article]

Alexei Yu. Uteshev, Ivan Baravy
2016 arXiv   pre-print
We treat the interpolation problem {f(x_j)=y_j}_j=1^N for polynomial and rational functions.  ...  Developing the approach by C.Jacobi, we represent the interpolants by virtue of the Hankel polynomials generated by the sequences {∑_j=1^N x_j^ky_j/W^'(x_j) }_k∈ N and {∑_j=1^N x_j^k/(y_jW^'(x_j)) }_k∈  ...  Introduction The title of the paper describes ends and means of the present research: solution of the univariate interpolation problems with the aid of the Hankel polynomials, i.e. polynomials in the variable  ... 
arXiv:1603.08752v1 fatcat:ybntk7oczffcdhdzg4ghgu2jka

Solution of rational interpolation problem via the Hankel polynomial construction

Alexei Yu. Uteshev, St. Petersburg State University, Ivan I. Baravy, St. Petersburg State University
2016 Vestnik of Saint Petersburg University Applied Mathematics Computer Science Control Processes  
Baravy SOLUTION OF THE RATIONAL INTERPOLATION PROBLEM VIA THE HANKEL POLYNOMIAL CONSTRUCTION St. Petersburg State University, 7-9, Universitetskaya nab., St.  ...  Jacobi, the interpolant is represented as a ratio of two Hankel polynomials, i. e. polynomials of the form H K (x) = det[c i+j−1 − c i+j−2 x] K i,j=1 .  ... 
doi:10.21638/11701/spbu10.2016.403 fatcat:x7ghxmy2vrdj7kbrns5rzj63uy

Rational Interpolation: Jacobi's Approach Reminiscence

Alexei Uteshev, Ivan Baravy, Elizaveta Kalinina
2021 Symmetry  
We treat the interpolation problem {f(xj)=yj}j=1N for polynomial and rational functions. Developing the approach originated by C.  ...  Jacobi, we represent the interpolants by virtue of the Hankel polynomials generated by the sequences of special symmetric functions of the data set like {∑j=1Nxjkyj/W′(xj)}k∈N and {∑j=1Nxjk/(yjW′(xj))}  ...  Acknowledgments: The authors wish to thank the anonymous reviewers for their very constructive comments. Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/sym13081401 fatcat:xj5rm5qlyzc2zdgwu3vzchfkmm

The general rational interpolation problem in the scalar case and its Hankel vector

Gong-ning Chen, Bin Zhao, Hui-pin Zhang
1996 Linear Algebra and its Applications  
This paper deals with the general rational interpolation problem (GRIP) in the scalar case.  ...  It turns out that given a fixed GRIP, we can readily obtain a unique veetor, called the Hankel vector, such that both of the allowable MeMillan degrees of interpolants to this GRIP and the corresponding  ...  The authors acknowledge with many thanks the helpful comments and suggestions of the referees.  ... 
doi:10.1016/0024-3795(94)00223-1 fatcat:hrw7llt5hjfvlkw77qkfbue3ru

A fast Hankel solver based on an inversion formula for Loewner matrices

Peter Kravanja, Marc Van Barel
1998 Linear Algebra and its Applications  
We propose a new O(n 2) algorithm for solving complex n × n linear systems that have Hankel structure. Via FFTs the Hankel system is transformed into a Loewner system.  ...  We present an O(n 2) algorithm to solve these interpolation problems. One of the advantages of this algorithm is that it incorporates pivoting. We have implemented our I-I~nkel solver in Fortran 90.  ...  Acknowledgements The authors thank Georg Heinig for providing them with Refs. [11,25,28,30,29,27 ]. They also want to thank the referees for their helpful suggestions and comments.  ... 
doi:10.1016/s0024-3795(98)10079-4 fatcat:ljf4fszbeffmnounnabmqndr4i

Fast estimates of Hankel matrix condition numbers and numeric sparse interpolation

Erich L. Kaltofen, Wen-shin Lee, Zhengfeng Yang
2011 Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation - SNC '11  
We investigate our early termination criterion for sparse polynomial interpolation when substantial noise is present in the values of the polynomial.  ...  We harness the Gohberg-Semencul formula for the inverse of a Hankel matrix to compute estimates for the structured condition numbers of all arising Hankel matrices in quadratic arithmetic time overall,  ...  Note added on Jan. 18, 2012: Changed dimensions of first singular leading principle Hankel matrix in Introduction on p. 131 to (t + 1) × (t + 1) from t × t.  ... 
doi:10.1145/2331684.2331704 dblp:conf/snc/KaltofenLY11 fatcat:hrw7hwqqevbkreiiizy6tcyzxa

Page 2272 of Mathematical Reviews Vol. , Issue 2002C [page]

2002 Mathematical Reviews  
It is shown that the “classical” Hankel matrix constructed by using the Markov parameters of the transformed system coincides with the Hankel matrix of the original system in the generalized orthonormal  ...  The parametrization is obtained via a finite-dimensional convex optimization problem dual to the infinite-dimensional primal problem of maximizing a suitably de- fined entropy criterion.  ... 

The general rational interpolation problem and its connection with the Nevanlinna-Pick interpolation and power moment problem

Gong-ning Chen
1998 Linear Algebra and its Applications  
certain ways for the discussion and solution of the NP problem and its relatives on the basis of the theories of the rational interpolation and "algebraic moment problems,  ...  The Hankel vector approach in a recent work of the author with Zhao and Zhang on the general rational interpolation problem (GRIP) allows the problem to be reduced to what amounts to a limiting case, i.e  ...  via the Hankel vector approach.  ... 
doi:10.1016/s0024-3795(97)00346-7 fatcat:zgvki7ajhjhpnhatx7zhyovlju

Hankel matrices, positive functions and related questions

Yves V. Genin
2000 Linear Algebra and its Applications  
The Hankel matrix extension problem is discussed and solved in a simple manner as an illustration of the proposed techniques.  ...  This paper is concerned with the subclass of positive functions that can be associated to nonnegative definite Hankel matrices and with its extension to the subclass of appropriate pseudo-positive functions  ...  Acknowledgement The author would like to express his gratitude to A. Bultheel and P. Delsarte for their helpful comments and constructive suggestions on a preliminary version of this paper.  ... 
doi:10.1016/s0024-3795(00)00121-x fatcat:wi4uidt6zzb4lksrfymyb4x7nq

Real Zero Polynomials and A. Horn's Problem [article]

Lei Cao, Hugo J. Woerdeman
2017 arXiv   pre-print
In this paper, we show how it connects to an interpolation problem for two variable real zero polynomials and a tracial moment problem. In addition, we outline an algorithm to construct a pair (A,B).  ...  Horn's problem concerns find two self adjoint matrices A, B, so that A, B, and A+B have prescribed spectrum.  ...  Acknowledgement We thank the referee for careful reading and useful suggestions which helped in identifying an oversight and led to significant improvement of the paper.  ... 
arXiv:1712.02922v1 fatcat:wqk2tg5hbjcghnugemmazkmzam

A fast algorithm for rational interpolation via orthogonal polynomials

{Ömer Eğecioğlu, Çetin K. Koç
1989 Mathematics of Computation  
Given the data set, the algorithm generates a set of orthogonal polynomials by the classical threeterm recurrence relation and then uses Newton interpolation to find the numerator and the denominator polynomials  ...  A variant of this algorithm that avoids Newton interpolation can be used to construct all rational interpolants using only 0(N2) arithmetic operations.  ...  We are grateful to the anonymous referee whose comments and suggestions have greatly improved the readability of this paper.  ... 
doi:10.1090/s0025-5718-1989-0972369-4 fatcat:idcmbjchvjb3fkmuwelypz5olq

Faulty share detection in Shamir's secret sharing

Alexei Yu. Uteshev, St. Petersburg State University, Aleksei V. Marov, RAIDIX
2019 Vestnik of Saint Petersburg University Applied Mathematics Computer Science Control Processes  
The authors thank the referees for valuable suggestions that helped to improve the quality of the paper.  ...  In the framework of this algorithm, the polynomial (9) is referred to as the error locator polynomial, and is found via the solution of the rational interpolation problem for the data set (1) .  ...  Computation of the error locator polynomial (9) via its representation in the Hankel polynomial form (7) is a part of that algorithm.  ... 
doi:10.21638/11702/spbu10.2019.210 fatcat:nhnulibaufbktp3ggrcoln3wza

Page 4911 of Mathematical Reviews Vol. , Issue 99g [page]

1999 Mathematical Reviews  
The heart of our Hankel solver is a superfast algorithm to solve these interpolation problems.  ...  This explicit formula for the inverse of a Loewner matrix contains certain parameters that are computed by solving two linearized rational interpolation problems on the unit circle.  ... 

Rational interpolation via orthogonal polynomials

Luca Gemignani
1993 Computers and Mathematics with Applications  
In this way, it can be used to construct all existing rational interpolants with the cost of 0(N2) arithmetic operations where N + 1 is the number of data points.  ...  Relations between rational interpolants and Hankel matrices are investigated. A modification of a Jacobi-like algorithm for rational interpolation proposed in [l] is developed.  ...  If n = 0, the problem is reduced to polynomial interpolation and the solution exists and is unique.  ... 
doi:10.1016/0898-1221(93)90070-c fatcat:4nbai3tylvd7jpxgtbpdfqtqtq

Page 882 of Mathematical Reviews Vol. , Issue 83b [page]

1983 Mathematical Reviews  
A polynomial formulation and the properties of the optimal Hankel norm approximations are then shown to result directly from the general form of the solution of the interpolation problems considered.”  ...  The authors deal with the computation of state space realizations via the theory of linear systems over rings.  ... 
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