Rational Interpolation Method Research Articles

Bounded Property and Point Control of a Bivariate Rational Interpolating Surface

A bivariate rational interpolation method with parameters was created in an earlier work which was based on function values only. This paper will deal with the bounded property and the point control method of the interpolating surface. It is proved that the values of the interpolating function in the interpolation region are bounded no matter what the parameters might be; this is called the bounded property of the interpolation. Also, the approximation expressions of the interpolation are derived; they do not depend on the parameters. More important is that the value of the interpolating function at any point in the interpolating region can be modified under the condition that the interpolating data are not changed by selecting suitable parameters, so the interpolation surface may be modified for the given interpolation data when needed in practical design. In the special case, the “Central Point-Mean Value” control is studied, and an example is given to show the control.

Nonhyperbolic moveout analysis in VTI media using rational interpolation

Anisotropic velocity analysis using qP-waves in transversely isotropic media with a vertical symmetry axis (VTI) usually is done by inferring the anellipticity parameter [Formula: see text] and the normal moveout velocity [Formula: see text] from the nonhyperbolic character of the moveout. Several approximations explicit in these parameters exist with varying degrees of accuracy. Here, we present a rational interpolation approach to nonhyperbolic moveout analysis in the [Formula: see text] domain. This method has no additional computational overhead compared to using expressions explicit in [Formula: see text] and [Formula: see text]. The lack of such overhead stems from the observation that, for fixed [Formula: see text] and zero-offset two-way traveltime [Formula: see text], the moveout curve for different values of [Formula: see text] can be calculated by simple stretching of the offset axis. This observation is based on the assumptions that the traveltimes of qP-waves in transversely isotropic media mainly depend on [Formula: see text] and [Formula: see text], and that the shear-wave velocity along the symmetry axis has a negligibleinfluence on these traveltimes. The accuracy of the rational interpolation method is as good as that of these approximations. The method can be tuned accurately to any offset range of interest by increasing the order of the interpolation. We test the method using both synthetic and field data and compare it with the nonhyperbolic moveout equation of Alkhalifah and Tsvankin (1995) and the shifted hyperbola equation of Fomel (2004). Both data types confirm that for [Formula: see text], our method significantly outperforms the nonhyperbolic moveout equation in terms of combined unbiased parameter estimation with accurate moveout correction. Comparison with the shifted hyperbola equation of Fomel for Greenhorn-shale anisotropy establishes almost identical accuracy of the rational interpolation method and his equation. Even though the proposed method currently deals with homogeneous media only, results from application to synthetic and field data confirm the applicability of the proposed method to horizontally layered VTI media.

Thiele-type and Lagrange-type generalized inverse rational interpolation for rectangular complex matrices

A variety of matrix rational interpolation problems include the partial realization problem for matrix power series and the minimal rational interpolation problem for general matrix functions. Different from the previous work, in this paper we consider a new method of matrix rational interpolation, with rectangular real or complex interpolated matrices and distinct real or complex interpolation points. Based on an axiomatic definition for the generalized inverse matrix rational interpolants (GMRI), GMRI are constructed in the following two forms: (i) Thiele-type continued fraction expression; (ii) an explicit determinantal formula for the denominator scalar polynomials and for the numerator matrix polynomials, which are of Lagrange-type expression. As a direct application of GMRI, a matrix rational extrapolation is introduced.

Frequency response computation via rational interpolation

A simple rational interpolation method is given for estimating a frequency response matrix. The method is based on a generalization of the resolvent identity that avoids the potential inaccuracies introduced by the subtraction of nearly equal values in the calculation of the difference terms. When coupled with a pole cancellation method, the resulting interpolation algorithm is accurate and efficient. The error in this procedure has the form of a modified response matrix, which means that the interpolation algorithm can be used to approximate both the response matrix and the error. >

A Recursive Computation Scheme for Multivariate Rational Interpolants

We derive here a recursive computation scheme for the rational interpolation method introduced in [7]. Explicit formulas for these multivariate rational interpolants are repeated in § 2 while the recursive algorithm is described in § 3. A number of interesting special cases such as the univariate rational interpolation problem and the multivariate Padé approximants introduced in [6] and [10] are dealt with in § 4. For some of these rational approximants other recursive schemes were described previously. Finally § 5 contains the numerical results: the multivariate rational interpolants described here are compared with multivariate polynomial interpolants, interpolating branched continued fractions introduced by Cuyt and Verdonk [8], interpolating branched continued fractions introduced by Siemaszko [12] and several multivariate Padé approximants [6], [3], [10]. Besides the fact that our multivariate rational interpolants allow a large degree of freedom in the choice for the numerator and denominator in order to fit the function to be approximated as well as possible, they also produce very accurate numerical results.

Remark on “Algorithm 631: Finding a Bracketed Zero by Larkin's Method of Rational Interpolation”

Remark on “Algorithm 631: Finding a Bracketed Zero by Larkin's Method of Rational Interpolation”

Permutation method of ladder synthesis, rational interpolation, and Zolotarev (elliptic) filters

Lin, Tokad, and Szentirmai have discussed in this TRANSACTIONS an algorithm proposed by the author for the synthesis of ladders. In this correspondence, details are given on the relations of this algorithm with rational interpolation, on the pass band case, and on the remarkable case of Zolotarev (elliptic) filters.