Series Prediction based on Algebraic Approximants

It is described how the Hermite-Pad\'e polynomials corresponding to an algebraic approximant for a power series may be used to predict coefficients of the power series that have not been used to compute the Hermite-Pad\'e polynomials. A recursive algorithm is derived and some numerical examples are given.


Introduction
Using sequence transformation and extrapolation algorithms for the prediction of further sequence elements from a finite number of known sequence elements is a topic of growing importance in applied mathematics. For a short introduction see the book of Brezinski and Redivo Zaglia [3,Sec. 6.8]. We mention theoretical work on prediction properties of Padé approximants and related algorithms like the epsilon algorithm, and the iterated Aitken and Theta algorithms [2,5,7,11], Levin-type sequence transformations [8,9], the E algorithm [2,10], and applications on perturbation series of physical problems [6,8].
Here, we will concentrate on a different class of approximants, namely, the algebraic approximants. For a general introduction to these approximants and the related Hermite-Padé polynomials see [1]. Programs for these approximants are available [4]. We summarize those properties that are important for the following.
Consider a function f of complex variable z with a known (formal) power series The Hermite-Padé polynomials (HPPs) corresponding to a certain algebraic approximant are N + 1 polynomials P n (z) with degree p n = deg(P n ), n = 0..N such that the order condition holds for small z. Since one of the coefficients of the polynomials can be normalized to unity, the order condition (2) gives rise to a system of M linear equations for N + N n=0 p n unknown polynomial coefficients. Thus, the coefficient of z m of the Taylor expansion at z = 0 of the left hand side of Eq. (2) must be zero for m = 0, . . . , M −1. In order to have exactly as many equations as unknowns, we choose and assume that the linear system (2) has a solution. Then, the HPPs P n (z) are uniquely defined upon specifying the normalization. The algebraic approximant under consideration then is that pointwise solution a(z) of the algebraic equation for which the Taylor series of a(z) coincides with the given power series at least up to order z M −1 .
We note that for N = 1, the algebraic approximants are nothing but the well-known Padé approximants.
Although we assumed that the power series of f is known, quite often in practice, only a finite number of coefficients is really known. These coefficients then may be used to compute the Hermite-Padé polynomials and the algebraic approximant under consideration.
We note that the higher coefficients of the Taylor series of a(z) may be considered as predictions for the higher coefficients of the power series. The latter are also of interest in applications.
The question then arises how to compute the Taylor series of a(z). If it is possible to solve the equation (4) explicitly, i.e. for N ≤ 4, a computer algebra system may be used to do the job. But even then, a recursive algorithm for the computation of the coefficients of the Taylor series would be preferable in order to reduce computational efforts.
In the following section, such a recursive algorithm is obtained. In a further section, we will present numerical examples.

The recursive algorithm
We consider the HPPs as known. Putting we obtain from Eq. (4) whence, by equating the coefficient of z J to zero, we obtain an infinite set of equations. Due to Eq. (2), all the equations for J < M are satisfied exactly for a j = f j , j = 0, . . . , M − 1.
As a first step, we compute a M . We note that M > p 0 . Hence, the coefficient of z M does not involve any terms with p 0,j . For this coefficient R M , we only need to consider terms in Eq. (7) such that M = j + k 1 + . . . + k n and we Now, the only terms on the RHS involving a J are obtained if exactly one of the k m is equal to J, i.e., we have k m = J, j = 0 and k j = 0 for j = m. Thus, we may rewrite all these terms as a J C where C is defined above. Proceeding as before, we put D J = R J − a J C and obtain An equivalent form of the recursive algorithm is obtained in the following way: Consider for known P n and a 0 , . . . , a J−1 the expression It is easy to see, that this expression is exactly equal to R J , and hence, is linear in the unknown a J . Thus, we may compute the quantities D J by substituting a J = 0 into U J , which entails Thus, starting from J = M, one may compute all the a J consecutively by repeated use of Eqs. (9), (14), and (12).
This concludes the derivation of the recursive algorithm.

Modes of application
Basically, there are two modes of application. a) one computes a sequence of HPPs and for the resulting algebraic approximants, one predicts a fixed number of sofar unused coefficients, e.g., only one new coefficient. This mode is mainly for tests. b) one computes from all available coefficients certain HPPs. For the best HPPs one computes a larger number of predictions for sofar unused coefficients.
In the following examples, we concentrate on mode b). Here, it is to be expected that the computed values have the larger errors the higher coefficients are predicted.

Examples
The examples serve to introduce to the approach. All numerical calculations in this section were done using Maple (Digits=16).

Example 1
As a first example, we consider N = 2, p 0 = p 1 = p 2 = 1 and, hence, M = 5. Since N = 2, we are dealing with a quadratic algebraic approximant. Then, the recursive algorithm is started by a j = f j , j = 0, . . . , 4. For a 5 , we obtain and for J > 5, we obtain the HPPs are determined to be The results for the predicted coefficients given in Table 1. Table 1 The case of N = 2, p 0 = p 1 = p 2 = 1 for Eq. (17). Displayed are the coefficients of the Taylor series, the predicted coefficients and absolute and relative errors of the predicted coefficients.

Example 2
As a second example, we consider again N = 2, p 0 = p 1 = p 2 = 1, and M = 5, but now the function with the HPPs The results for the predicted coefficients given in Table 2. Table 2 The case of N = 2, p 0 = p 1 = p 2 = 1 for Eq. (21). Displayed are the coefficients of the Taylor series, the predicted coeffizients and absolute and relative errors of the predicted coefficients. Example 3 As a final example, we consider the case N = p 0 = p 1 = p 2 = 2, whence M = 8, and the function The corresponding HPPs are The results for the predicted coefficients are displayed in Table 3. Table 3 The case of N = 2, p 0 = p 1 = p 2 = 2 for Eq. (25). Displayed are the coefficients of the Taylor series, the predicted coeffizients and absolute and relative errors of the predicted coefficients.

Conclusions
It is seen that even rather low-order algebraic approximants, or HPPs, respectively, can lead to quite accurate predictions of the unknown coefficients of the power series, especially for f M , and the next few coefficients.