Multivariate approximants with Levin-like transforms

doi:10.1016/j.cpc.2005.06.004

Computer Physics Communications

Volume 172, Issue 1, 15 October 2005, Pages 1-18

https://doi.org/10.1016/j.cpc.2005.06.004 Get rights and content

Abstract

It is well known that Levin-like univariate approximants work better than Padé approximants for a large class of functions. We propose to investigate the Levin-like multivariate approximants. It is shown that a simple modification of the qd-algorithm makes it possible to construct the homogeneous multivariate Padé approximants of order $[n, n]$ and $[n, n + 1]$ . We demonstrate how to construct the homogeneous and nested Levin-like approximants and compare their effectiveness with the homogeneous and nested Padé approximants which exhaust the same number of terms of a series in more than one variable. It is found that the Levin-like multivariate approximants are much more effective for a large class of functions.

Introduction

The concept of Padé approximants is essentially a century-old problem. However, the natural problem of the generalization of the rational approximants to more than one variable dates only from the early seventies of the last century. During the last few decades a number of papers [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11] have appeared which try to apply the techniques developed for the univariate Padé approximants to the case of N variables, in particular to the case of two variables. However, as in the univariate case, the concept of degree is not clear any more and many choices are possible for developing the multivariate Padé approximants.

The most systematic development on the subject is known as the Canterbury approximant or generalized Chisholm approximant, and the simplest of these is the Chisholm approximant [4]. Karlsson and Wallin [3] proposed a different choice and a similar type of approximant, the so-called homogeneous approximants, was proposed by Cuyt [7]. Another definition was given by Hughes Jones [1] and a general definition was given by Levin [5]. A different class of Padé approximants was introduced by Chaffy-Camas [12], [13] and this consists in computing first a Padé approximant with respect to the variable x and then computing the Padé approximant with respect to the variable y in the case of a bivariate approximant, and is known as the nested Padé approximant. Murphy and O'Donohoe [14] proposed a two variable generalization of the Stieltjes-type continued fraction. A number of theorems have been proved for the multivariate Padé approximants.

It is well known that the nonlinear sequence transform ɛ is closely related to the Padé approximants [15], [16]. Starting with the work of Levin [17] and Sidi [18] a class of transforms has been proposed in the recent past which are much more effective then the ɛ transform in summing a wide class of convergent and divergent sequences. The transform proposed by Sidi has been extensively used by Weniger et al. [19], [20], [21], [22], [23], [24], [25] and these transforms can sum a class of wildly divergent perturbation series which are not summable by the Padé approximants. Roy et al. [26], [27], [28], [29], [30] studied rational approximants based on the Levin and Sidi transforms and compared their performances with those of the Padé approximants and have applied these approximants to a number of physical problems. It was found that the approximants based on the Levin-like transforms perform much better than the Padé approximants, at least for a large class of functions. It is therefore tempting to see the performance of these transforms for functions of more than one variable.

In the next section we discuss the general multivariate Padé approximants, especially, the homogeneous and the nested Padé approximants. We then demonstrate that a class of homogeneous multivariate approximants can be simply constructed with a simple modification of the qd-algorithm [31], [32]. In the subsequent section we briefly discuss the univariate Levin-like approximants and generalize these to the multivariate case. In the final section we compare the performances of the homogeneous and nested Levin-like multivariate approximants with the homogeneous and nested Padé approximants which use the same number of terms of the Taylor series. It is found that the Levin-like multivariate approximants perform better than Padé approximants for a significant class of functions.

Section snippets

General multivariate Padé approximants

As the problem associated with many variables has the same kind of solution as for the two variable problem, we confine our discussion to the bivariate approximants. Let a bivariate function $f (x, y)$ be given by the formal Taylor series expansion $f (x, y) = \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} c_{i j} x^{i} y^{j}$ where $c_{i j} = \frac{1}{i!} \frac{1}{j!} {(\frac{\partial^{i + j}}{\partial x^{i} \partial y^{j}})}_{(0, 0)} .$ To obtain a bivariate approximant one has to define the lattice N and D and polynomials $p (x, y)$ and $q (x, y)$ $p (x, y) = \sum_{(i j) \in N} a_{i j} x^{i} y^{j}, q (x, y) = \sum_{(i j) \in D} b_{i j} x^{i} y^{j}$ such that $f (x, y) = \frac{p (x, y)}{q (x, y)} + \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} e_{i j} x^{i}$

Multivariate approximants with Levin-like transforms

In the following we briefly discuss the Levin-like approximants for a function of one variable and then extend these to the multivariate case. Let ${S_{n}}$ be a sequence of complex numbers tending to the limit S. An associated sequence $g_{n}$ is defined by the following relation $S = S_{n} + g_{n} ω_{n},$ where $S_{n}$ is the nth term of the sequence ${S_{n}}$ and $ω_{n} = Δ S_{n}$ or $Δ S_{n - 1}$ . Here Δ is the usual forward difference operator. Thus the problem of evaluating the limit of a sequence boils down to the estimation of the remainder

Comparative study of Padé and Levin-like approximants

Example 1

As the first example we consider the simple function $f (x, y) = \frac{1}{(1 - x) (1 + y)} .$ It may be noted that the function given above is the Appell hypergeometric function $F (1, 1, 1; 1; x, - y)$ . To obtain the homogeneous Padé approximants we note that $C_{l} (x, y) = \sum_{i = 0}^{l} {(- 1)}^{i} x^{l - i} y^{i} .$ Written explicitly, $C_{0} = 1$ , $C_{1} = x - y$ , $C_{2} = x^{2} - x y + y^{2}$ and so on. We construct the qd-table as follows: $\begin{matrix} x - y \\ 0 & \frac{x y}{x - y} \\ \frac{x^{2} - x y + y^{2}}{x - y} & \frac{- x y}{x - y} \\ 0 & \frac{- x^{2} y^{2}}{(x - y) (x^{2} - x y + y^{2})} \\ \frac{x^{3} - x^{2} y + x y^{2} - y^{3}}{x^{2} - x y + y^{2}} \\ 0 \\ ⋮ & ⋮ \end{matrix}$ Thus ${}^{h}{p_{01}} = \frac{1}{1 - x + y},$ ${}^{h}{p_{11}} = \frac{1}{1 - \frac{x - y}{1 - \frac{x y}{x - y}}} = \frac{x - y - x y}{x - y - x^{2} + x y - y^{2}},$ ${}^{h}{p_{12}} = \frac{1}{1 - \frac{\frac{x y}{x - y}}{1 + \frac{x y}{x - y}}} =$

Conclusions

We have discussed how to construct some rational approximants for functions of more than one variable, or multivariate approximants, using Levin and Sidi transforms. The standard work on multivariate Padé approximants is relatively recent, and the more useful multivariate Padé approximants seem to be the homogeneous and the nested approximants. We construct homogeneous and nested approximants with Levin-like transforms for a few functions and test their effectiveness against the ones obtained

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Computer Physics Communications

Multivariate approximants with Levin-like transforms

Abstract

Introduction

Section snippets

General multivariate Padé approximants

Multivariate approximants with Levin-like transforms

Comparative study of Padé and Levin-like approximants

Conclusions

J. Math. Anal. Appl.

J. Comp. Appl. Math.

J. Comp. Appl. Math.

SIAM J. Numer. Anal.

C. R. Acad. Sci. Paris Ser. I

Comp. Phys. Commun.

Int. J. Quan. Chem.

Int. J. Quan. Chem.

Ann. Phys.

Comp. Phys. Commun.

Chem. Phys. Lett.

Numer. Math.

Comp. Phys. Commun.

J. Approx. Theory

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J. Inst. Math. Appl.

BIT

Padé Approximants