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

2016
*
arXiv
*
pre-print

We treat

arXiv:1603.08752v1
fatcat:ybntk7oczffcdhdzg4ghgu2jka
*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 ...##
###
Solution of rational interpolation problem via the Hankel polynomial construction

2016
*
Vestnik of Saint Petersburg University Applied Mathematics Computer Science Control Processes
*

Baravy

doi:10.21638/11701/spbu10.2016.403
fatcat:x7ghxmy2vrdj7kbrns5rzj63uy
*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 . ...##
###
Rational Interpolation: Jacobi's Approach Reminiscence

2021
*
Symmetry
*

We treat

doi:10.3390/sym13081401
fatcat:xj5rm5qlyzc2zdgwu3vzchfkmm
*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. ...##
###
The general rational interpolation problem in the scalar case and its Hankel vector

1996
*
Linear Algebra and its Applications
*

This paper deals with

doi:10.1016/0024-3795(94)00223-1
fatcat:hrw7llt5hjfvlkw77qkfbue3ru
*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. ...##
###
A fast Hankel solver based on an inversion formula for Loewner matrices

1998
*
Linear Algebra and its Applications
*

We propose a new O(n 2) algorithm for solving complex n × n linear systems that have

doi:10.1016/s0024-3795(98)10079-4
fatcat:ljf4fszbeffmnounnabmqndr4i
*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. ...##
###
Fast estimates of Hankel matrix condition numbers and numeric sparse interpolation

2011
*
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation - SNC '11
*

We investigate our early termination criterion for sparse

doi:10.1145/2331684.2331704
dblp:conf/snc/KaltofenLY11
fatcat:hrw7hwqqevbkreiiizy6tcyzxa
*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. ...##
###
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

1998
*
Linear Algebra and its Applications
*

certain ways for

doi:10.1016/s0024-3795(97)00346-7
fatcat:zgvki7ajhjhpnhatx7zhyovlju
*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. ...##
###
Hankel matrices, positive functions and related questions

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. ...

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

2017
*
arXiv
*
pre-print

In this paper, we show how it connects to an

arXiv:1712.02922v1
fatcat:wqk2tg5hbjcghnugemmazkmzam
*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. ...##
###
A fast algorithm for rational interpolation via orthogonal polynomials

1989
*
Mathematics of Computation
*

Given

doi:10.1090/s0025-5718-1989-0972369-4
fatcat:idcmbjchvjb3fkmuwelypz5olq
*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. ...##
###
Faulty share detection in Shamir's secret sharing

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. ...

##
###
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

1993
*
Computers and Mathematics with Applications
*

In this way, it can be used to

doi:10.1016/0898-1221(93)90070-c
fatcat:4nbai3tylvd7jpxgtbpdfqtqtq
*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. ...##
###
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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