Jieqing Tan

Exploring multivariate Padé approximants for multiple hypergeometric series

We investigate the approximation of some hypergeometric functions of two variables, namely the Appell functions Fi, i = 1,...,4, by multivariate Padé approximants. Section 1 reviews the results that exist for the projection of the Fi onto ϰ=0 or y=0, namely, the Gauss function 2F1(a, b; c; z), since a great deal is known about Padé approximants for this hypergeometric series. Section 2 summarizes the definitions of both homogeneous and general multivariate Padé approximants. In section 3 we prove that the table of homogeneous multivariate Padé approximants is normal under similar conditions to those that hold in the univariate case. In contrast, in section 4, theorems are given which indicate that, already for the special case F1(a, b, b′; c; x; y) with a = b = b′ = 1 and c = 2, there is a high degree of degeneracy in the table of general multivariate Padé approximants. Section 5 presents some concluding remarks, highlighting the difference between the two types of multivariate Padé approximants in this context and discussing directions for future work.

Bivariate composite vector valued rational interpolation

In this paper we point out that bivariate vector valued rational interpolants (BVRI) have much to do with the vector-grid to be interpolated. When a vector-grid is well-defined, one can directly design an algorithm to compute the BVRI. However, the algorithm no longer works if a vector-grid is ill-defined. Taking the policy of divide and conquer, we define a kind of bivariate composite vector valued rational interpolant and establish the corresponding algorithm. A numerical example shows our algorithm still works even if a vector-grid is ill-defined.

On the finite sum representations of the Lauricella functions F D .

Abstract By using divided differences, we derive two different ways of representing the Lauricella function of n variables FD(n)(a,b1,b2,. . .,bn;c;x1,x2,. . .,xn) as a finite sum, for b1,b2,. . .,bn positive integers, and a,c both positive integers or both positive rational numbers with c−a a positive integer.

Composite schemes for multivariate blending rational interpolation

It is demonstrated that Newtons interpolation polynomials and Thieles interpolating continued fractions can be incorporated to generate various interpolation schemes based on rectangular grids, among them are two kinds of bivariate blending rational interpolants. However, blending rational interpolants strongly depend on the existence of so-called blending differences, which means that for some grids of data, one may fail to find out the corresponding rational interpolants as a whole. In this paper, we offer a solution scheme by adopting composite interpolation over triangular sub-grids. Characterization theorem is given, error estimation is worked out and vector valued case as well as matrix valued case is discussed.

A finite sum representation of the Appell series F 1 ( a,b,b';c;x,y )

Abstract We use Picards integral representation of the Appell series F 1 (a,b,b′; c; x,y) for Re (a)>0, Re (c−a)>0 to obtain a finite sum algebraic representation of F1 in the case when a, b, b′ and c are positive integers with c>a. The series converges for |x| |y| and we show that F 1 (a,b,b′; c; x,y) has two overlaying singularities at each of the points x=1 and y=1, one polar and one logarithmic in nature, when a, b, b′, c∈ N with c>a.

General order multivariate Padé approximants for Pseudo-multivariate functions. II

Although general order multivariate Pade approximants were introduced some decades ago, very few explicit formulas for special functions have been given. We explicitly construct some general order multivariate Pade approximants to the class of so-called pseudo-multivariate functions, using the Pade approximants to their univariate versions. We also prove that the constructed approximants inherit the normality and consistency properties of their univariate relatives, which do not hold in general for multivariate Pade approximants. Examples include the multivariate forms of the exponential and the q-exponential functions.

Generalized B-splines’ geometric iterative fitting method with mutually different weights

Abstract In this paper, a novel geometric iterative fitting method is presented which has a set of mutually different weights. It possesses the advantages of least square progressive iterative approximation method (abbr. LSPIA method) which can handle point sets of large sizes and adjust the number of control points and knot vector flexibly. The presenting method degrades into LSPIA method with appropriate choices of weights, and it illustrates better effects for the previous iteration steps comparing with the LSPIA method. Also, this method is further applied to generalized B-splines which have changing core functions (the mentioned generalized B-spline is a special generalization of classical B-spline with linear core function). Combining the advantages of generalized B-splines and choice of different weights, it can handle much more complicated practical problems. Detailed discussion about the choosing of core functions and weights is also given. Plentiful numerical examples are also presented to show the effectiveness of the method.

Smooth orientation interpolation using parametric quintic-polynomial-based quaternion spline curve

Abstract In this paper, a G 2 continuous quintic-polynomial-based unit quaternion interpolation spline curve with tension parameters is presented to interpolate a given sequence of solid orientations. The curve in unit quaternion space S 3 is an extension of the quintic polynomial interpolation spline curve in Euclidean space. It preserves the interpolatory property and G 2 continuity. Meanwhile, the unit quaternion interpolation spline curve possesses the local shape adjustability due to the presence of tension parameters. The change of one tension parameter will only affect the adjacent two pieces of curves. Compared with the traditional B-spline unit quaternion interpolation curve and v -spline unit quaternion interpolation curve, the proposed curve can automatically interpolate the given data points, without solving the nonlinear system of equations over quaternions to obtain the control points, which greatly improves the computational efficiency. Simulation results demonstrate the effectiveness of the proposed scheme.

A combined approximating and interpolating ternary 4-point subdivision scheme

Abstract In this paper, a new combined approximating and interpolating ternary 4-point subdivision scheme with multiple parameters is proposed. A set of nice properties, such as support, continuity and polynomial generation, are briefly discussed. The new combined scheme not only contains a lot of classical ternary schemes as special cases, but also generates brand-new ternary schemes. Compared to other approximating subdivision schemes, limit curves generated by the new scheme are more consistent with the corresponding control polygons and keep detail features better. Examples are given to show the effectiveness of the scheme. Furthermore, fractal property is analyzed and fractal curves are also given.