Vector-valued Rational Interpolation 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.

Bivariate Interpolation on the Hypersphere with Thiele Type Rational Interpolants

Interpolation problem on the sphere has applications in geodesy, earth science and other flelds. In this paper, we present an axiomatic approach to vector-valued rational interpolation. In order to interpolate mn+m+n+1 (m;n are even) points on the unit hypersphere S di1 with a bivariate Thiele type rational function, the construction process of it is based on the Samelson inverse for vectors. We generalize Thiele type rational interpolants on the hypersphere of Thierry Gensane and give the numerical example to show validity of interpolant.

A Method of Neville-like Vector-valued Blending Rational Interpolants Based on Continued Fractions

In this work we establish Neville-like vector-valued blending rational interpolants through the Thiele’s vector-valued continued fractions interpolants and the Neville’s algorithm. Especially, our method can offer many styles to solve the vector-valued rational interpolants problem. Using Samelson inverse, we calculate our Neville-like vector-valued blending rational interpolants. We give the numerical examples to display the flexibility and the sneakerization of our method.

A de Montessus Type Convergence Study of a Least-Squares Vector-Valued Rational Interpolation Procedure II

We continue our study of convergence of IMPE, one of the vector-valued rational interpolation procedures proposed by the author in a recent paper, in the context of vector-valued meromorphic functions with simple poles. So far, this study has been carried out in the presence of corresponding residues that are mutually orthogonal. In the present work, we continue to study IMPE in the same context, but in the presence of corresponding residues that are not necessarily orthogonal. Choosing the interpolation points appropriately, we derive de Montessus type convergence results for the interpolants and Konig type results for the poles and residues.

A de Montessus type convergence study of a least-squares vector-valued rational interpolation procedure

In a recent paper of the author [A. Sidi, A new approach to vector-valued rational interpolation, J. Approx. Theory 130 (2004) 177–187], three new interpolation procedures for vector-valued functions F ( z ) , where F : C → C N , were proposed, and some of their algebraic properties were studied. One of these procedures, denoted IMPE, was defined via the solution of a linear least-squares problem. In the present work, we concentrate on IMPE, and study its convergence properties when it is applied to meromorphic functions with simple poles and orthogonal vector residues. We prove de Montessus and Koenig type theorems when the points of interpolation are chosen appropriately.

Algebraic properties of some new vector-valued rational interpolants

In a recent paper of the author [A. Sidi, A new approach to vector-valued rational interpolation, J. Approx. Theory, 130 (2004) 177–187], three new interpolation procedures for vector-valued functions F ( z ) , where F : C → C N , were proposed, and some of their properties were studied. In this work, after modifying their definition slightly, we continue the study of these interpolation procedures. We show that the interpolants produced via these procedures are unique in some sense and that they are symmetric functions of the points of interpolation. We also show that, under the conditions that guarantee uniqueness, they also reproduce F ( z ) in case F ( z ) is a rational function.

A note on vector-valued rational interpolation

Graves-Morris (see [P.R. Graves-Morris, Vector valued interpolants I, Numer. Math. 42 (1983) 331–348 and P.R. Graves-Morris and C.D. Jenkins, Vector valued rational interpolants III, Constr. Approx. 2 (1986) 263–289]) defined the generalized inverse rational interpolants (GIRIs) in the form of R ( x ) = N ( x ) / D ( x ) with the divisibility condition D ( x ) ∣ ∥ N ( x ) ∥ 2 , and proved the Uniqueness Theorem for GIRIs. However, this condition is found not necessary in some cases. In this paper, we remove this divisibility condition, define the extended generalized inverse rational interpolants (EGIRIs) and establish the Uniqueness Theorem for EGIRIs. One can see that the Uniqueness Theorem for GIRIs is the special case of the one for EGIRIs.

A new approach to vector-valued rational interpolation

In this work we propose three different procedures for vector-valued rational interpolation of a function F ( z ) , where F : C → C N , and develop algorithms for constructing the resulting rational functions. We show that these procedures also cover the general case in which some or all points of interpolation coalesce. In particular, we show that, when all the points of interpolation collapse to the same point, the procedures reduce to those presented and analyzed in an earlier paper (J. Approx. Theory 77 (1994) 89) by the author, for vector-valued rational approximations from Maclaurin series of F ( z ) . Determinant representations for the relevant interpolants are also derived.

A note on matrix-valued rational interpolants

In this paper, by defining a kind of transformation from matrix to vector, we succeed in transferring some results on vector-valued rational interpolants to those corresponding to the matrix-valued rational interpolants. Moreover, it is pointed out through a numerical example that the statement of the so-called uniqueness theorem in [4] is incorrect and, what is more, the proof is also wrong. A new uniqueness theorem is given.

Multivariate generalized inverse vector-valued rational interpolants

Bivariate rational interpolating functions of the type introduced in [9, 1] are shown to have a natural extension to the case of rational interpolation of vector-valued quantities using the formalism of Graves-Morris [2]. In this paper, the convergence of Stieltjes-type branched vector-valued continued fractions for two-variable functions are constructed by using the Samelson inverse. Based on them, a kind of bivariate vector-valued rational interpolating function is defined on plane grids. Sufficient conditions for existence, characterisation and uniqueness for the interpolating functions are proved. The results in the paper are illustrated with some examples.

The compass (star) identity for vector-valued rational interpolants

The compass identity (Wynn's five point star identity) for Pade approximants connects neighbouring elements called N, S, E, W and C in the Pade table. Its form has been extended to the cases of rational interpolation of ordinary (scalar) data and interpolation of vector-valued data. In this paper, full specifications of the associated five point identity for the scalar denominator polynomials and the vector numerator polynomials of the vector-valued rational interpolants on real data points are given, as well as the related generalisations of Frobenius' identities. Unique minimal forms of the polynomials constituting the interpolants and results about unattainable points correspond closely to their counterparts in the scalar case.

Divergence of vector-valued rational interpolants to meromorphic functions

Divergence of vector-valued rational interpolants to meromorphic functions

Vector-valued, rational interpolants III

In 1963, Wynn proposed a method for rational interpolation of vector-valued quantities given on a set of distinct interpolation points. He used continued fractions, and generalized inverses for the reciprocals of vector-valued quantities. In this paper, we present an axiomatic approach to vector-valued rational interpolation. Uniquely defined interpolants are constructed for vector-valued data so that the components of the resulting vector-valued rational interpolant share a common denominator polynomial. An explicit determinantal formula is given for the denominator polynomial for the cases of (i) vector-valued rational interpolation on distinct real or complex points and (ii) vector-valued Pade approximation. We derive the connection with thee-algorithm of Wynn and Claessens, and we establish a five-term recurrence relation for the denominator polynomials.