Rational Interpolation Problem Research Articles

On the Univariate Vector-Valued Rational Interpolation and Recovery Problems

In this paper, we consider a novel vector-valued rational interpolation algorithm and its application. Compared to the classic vector-valued rational interpolation algorithm, the proposed algorithm relaxes the constraint that the denominators of components of the interpolation function must be identical. Furthermore, this algorithm can be applied to construct the vector-valued interpolation function component-wise, with the help of the common divisors among the denominators of components. Through experimental comparisons with the classic vector-valued rational interpolation algorithm, it is found that the proposed algorithm exhibits low construction cost, low degree of the interpolation function, and high approximation accuracy.

Balanced Curves and Minimal Rational Connectedness on Fano Hypersurfaces

Abstract On a general hypersurface $X$ of degree $n$ (resp. $<n$) in ${\mathbb {P}}^n$ and any $e\geq n-1$ (resp. arbitrarily large $e$), we construct families of rational curves of degree $e$ going through the maximal number of general points. This solves in many cases the problem of rational curve interpolation on $X$.

Minimal solutions of the rational interpolation problem

We explore connections between the approach of solving the rational interpolation problem via resolutions of ideals and syzygies, and the standard method provided by the Extended Euclidean Algorithm (EEA). As a consequence, we obtain explicit descriptions for solutions of minimal degrees in terms of the degrees of elements appearing in the EEA. This result allows us to describe the minimal degree in a μ-basis of a polynomial planar parametrization in terms of a critical degree arising in the EEA.

The multipoint Padé approximation problem and its Hankel vector

The multipoint Padé approximation problem and the structure of multipoint Padé approximation table have been investigated thoroughly in the literature. In this paper, we show that each multipoint Padé approximation problem can be reduced to what amounts to a certain Padé approximation problem at infinity, by using the so-called Hankel vector method that was introduced in Chen et al. (1996) for the general rational interpolation problem. This connection offers a new way to study the properties of multipoint Padé approximants and the structure and computation of entries of multipoint Padé approximation table based on the characteristic polynomials of Hankel matrices.

Accurate computations for eigenvalues of products of Cauchy-polynomial-Vandermonde matrices

In this paper, we consider the product eigenvalue problem for the class of Cauchy-polynomial-Vandermonde (CPV) matrices arising in a rational interpolation problem. We present the explicit expressions of minors of CPV matrices. An algorithm is designed to accurately compute the bidiagonal decomposition for strictly totally positive CPV matrices and their additive inverses. We then illustrate the sign regularity of the bidiagonal decomposition to show that all the eigenvalues of a product involving such matrices are computed to high relative accuracy. Numerical experiments are given to confirm the claimed high relative accuracy.

The set of unattainable points for the Rational Hermite Interpolation Problem

We describe geometrically and algebraically the set of unattainable points for the Rational Hermite Interpolation Problem (i.e. those points where the problem does not have a solution). We show that this set is a union of equidimensional complete intersection varieties of odd codimension, the number of them being equal to the minimum between the degrees of the numerator and denominator of the problem. Each of these equidimensional varieties can be further decomposed as a union of as many rational (irreducible) varieties as input data points. We exhibit algorithms and equations defining all these objects.

On the decay of the off-diagonal singular values in cyclic reduction

It was recently observed in [10] that the singular values of the off-diagonal blocks of the matrix sequences generated by the Cyclic Reduction algorithm decay exponentially. This property was used to solve, with a higher efficiency, certain quadratic matrix equations encountered in the analysis of queuing models. In this paper, we provide a theoretical bound to the basis of this exponential decay together with a tool for its estimation based on a rational interpolation problem. Numerical experiments show that the bound is often accurate in practice. Applications to solving n×n block tridiagonal block Toeplitz systems with n×n quasiseparable blocks and certain generalized Sylvester equations in O(n2log⁡n) arithmetic operations are shown.

Computing minimal interpolation bases

We consider the problem of computing univariate polynomial matrices over a field that represent minimal solution bases for a general interpolation problem, some forms of which are the vector M-Padé approximation problem in Van Barel and Bultheel (1992) and the rational interpolation problem in Beckermann and Labahn (2000). Particular instances of this problem include the bivariate interpolation steps of Guruswami–Sudan hard-decision and Kötter–Vardy soft-decision decodings of Reed–Solomon codes, the multivariate interpolation step of list-decoding of folded Reed–Solomon codes, and Hermite–Padé approximation.In the mentioned references, the problem is solved using iterative algorithms based on recurrence relations. Here, we discuss a fast, divide-and-conquer version of this recurrence, taking advantage of fast matrix computations over the scalars and over the polynomials. This new algorithm is deterministic, and for computing shifted minimal bases of relations between m vectors of size σ it uses O ˜(mω−1(σ+|s|)) field operations, where ω is the exponent of matrix multiplication, and |s| is the sum of the entries of the input shift s, with min⁡(s)=0. This complexity bound improves in particular on earlier algorithms in the case of bivariate interpolation for soft decoding, while matching fastest existing algorithms for simultaneous Hermite–Padé approximation.

Solution of rational interpolation problem via the Hankel polynomial construction

Solution of rational interpolation problem via the Hankel polynomial construction

Computing Eigenvalues of Real Symmetric Matrices with Rational Filters in Real Arithmetic

Powerful algorithms have recently been proposed for computing eigenvalues of large matrices by methods related to contour integrals; best known are the works of Sakurai and coauthors and Polizzi and coauthors. Even if the matrices are real symmetric, most such methods rely on complex arithmetic, leading to expensive linear systems to solve. An appealing technique for overcoming this starts from the observation that certain discretized contour integrals are equivalent to rational interpolation problems, for which there is no need to leave the real axis. Investigation shows that using rational interpolation per se suffers from instability; however, related techniques involving real rational filters can be very effective. This article presents a technique of this kind that is related to previous work published in Japanese by Murakami.

Subresultants, Sylvester sums and the rational interpolation problem

We present a solution for the classical univariate rational interpolation problem by means of (univariate) subresultants. In the case of Cauchy interpolation (interpolation without multiplicities), we give explicit formulas for the solution in terms of symmetric functions of the input data, generalizing the well-known formulas for Lagrange interpolation. In the case of the osculatory rational interpolation (interpolation with multiplicities), we give determinantal expressions in terms of the input data, making explicit some matrix formulations that can independently be derived from previous results by Beckermann and Labahn.

Padé–type rational and barycentric interpolation

In this paper, we consider the particular case of the general rational Hermite interpolation problem where only the value of the function is interpolated at some points, and where the function and its first derivatives agree at the origin. Thus, the interpolants constructed in this way possess a Pade---type property at 0. Numerical examples show the interest of the procedure. The interpolation procedure can be easily modified to introduce a partial knowledge on the poles and the zeros of the function to approximated. A strategy for removing the spurious poles is explained. A formula for the error is proved in the real case. Applications are given.

Fast and Stable Rational Interpolation in Roots of Unity and Chebyshev Points

A new method for interpolation by rational functions of prescribed numerator and denominator degrees is presented. When the interpolation nodes are roots of unity or Chebyshev points, the algorithm is particularly simple and relies on discrete Fourier transform matrices, which results in a fast implementation using the fast Fourier transform. The method is generalized for arbitrary grids, which requires the construction of polynomials orthogonal on the set of interpolation nodes. The appearance of common factors in the numerator and denominator due to finite-precision arithmetic is explained by the behavior of the singular values of the linear system associated with the rational interpolation problem. The new algorithm has connections with other methods, particularly the work of Jacobi and Kronecker, Berrut and Mittelmann, and Eğecioğlu and Koç. Short MATLAB codes and numerical experiments are included.

Comments on “A Simple and Accurate Algorithm for Barycentric Rational Interpolation”

For original paper see L. Knockaert , ibid., vol.15, p.154-7, (2008). It is pointed out that the null space of Lowner matrices for the Barycentric rational interpolation can also be solved by an algebraic algorithm based on the Grobner basis theory. This algorithm is known for solving minimal rational interpolation problems in the context of system and coding theory.

Rational interpolation and mixed inverse spectral problem for finite CMV matrices

For finite-dimensional CMV matrices the mixed inverse spectral problem of reconstructing the matrix by its submatrix and a part of its spectrum is considered. A general rational interpolation problem which arises in solving the mixed inverse spectral problem is studied, and the description of the space of its solutions is given. We apply the developed technique to give sufficient conditions for the uniqueness of the solution of the mixed inverse spectral problem.

Efficient order basis computation (abstract only)

Order basis (also known as 3/4-basis, minimal approximant basis), originally developed by Beckermann and Labahn in 1994 for rational approximation and interpolation problems, has recently been applied to many other important problems in polynomial matrix computation. These include column reduction, matrix inverse, determinant, and null space basis computation. Storjohann, in 2006, provided an efficient way to compute a part of order basis that is within a given degree bound. We extend Storjohann's result by providing a way to compute a complete order basis with a similar computational cos.

Moment Matching Model Order Reduction in Time Domain via Laguerre Series

A new time-domain model order reduction method based on the Laguerre function expansion of the impulse response is presented. The Laguerre coefficients of the impulse response of the reduced-order model, which is calculated using a projection whose matrices form basis of appropriate Krylov subspaces, match, up to a given order, those of the original system. In addition, it is shown that the obtained reduced-order model in time-domain, is equivalent to the one obtained by the classical moment matching around a single expansion point in frequency-domain. Accordingly, a new time-domain interpretation for the rational interpolation problem is deduced.

On the denominator values and barycentric weights of rational interpolants

We improve upon the method of Zhu and Zhu [A method for directly finding the denominator values of rational interpolants, J. Comput. Appl. Math. 148 (2002) 341–348] for finding the denominator values of rational interpolants, reducing considerably the number of arithmetical operations required for their computation. In a second stage, we determine the points (if existent) which can be discarded from the rational interpolation problem. Furthermore, when the interpolant has a linear denominator, we obtain a formula for the barycentric weights which is simpler than the one found by Berrut and Mittelmann [Matrices for the direct determination of the barycentric weights of rational interpolation, J. Comput. Appl. Math. 78 (1997) 355–370]. Subsequently, we give a necessary and sufficient condition for the rational interpolant to have a pole.

Werner-type matrix valued rational interpolation and its recurrence algorithms

In this paper, a practical Werner-type continued fraction method for solving matrix valued rational interpolation problem is provided by using a generalized inverse of matrices. In order to reduce the continued fraction form to rational function form of the in-terpolants, an efficient forward recurrence algorithm is obtained.

Generalized Rational Interpolation Over Commutative Rings and Remainder Decoding

We propose a new decoding procedure for Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon (RS) codes over Z/sub m/ where m is a product of prime powers. Our method generalizes the remainder decoding technique for RS codes originally introduced by Welch and Berlekamp and retains its key feature of not requiring the prior evaluation of syndromes. It thus represents a significant departure from other algorithms that have been proposed for decoding linear block codes over integer residue rings. Our decoding procedure involves a Welch-Berlekamp (WB)-type algorithm for solving a generalized rational interpolation problem over a commutative ring R with identity. The solution to this problem includes as a special case, the solution to the WB key equation over R which is central to our decoding procedure. A remainder decoding approach for decoding cyclic codes over Z/sub m/ up to the Hartmann-Tzeng bound is also presented.