It is described how the Hermite-Padé 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-Padé polynomials. A recursive algorithm is derived, and some numerical examples are given.

1. 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 [1, Section  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 [25], Levin-type sequence transformations [6, 7], the E algorithm [4, 8], and applications on perturbation series of physical problems [7, 9].

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 [10]. Programs for these approximants are available [11]. We summarize those properties that are important for the following.

Consider a function 𝑓 of complex variable 𝑧 with a known (formal) power series𝑓(𝑧)=𝑗=0𝑓𝑗𝑧𝑗.(1.1) The Hermite-Padé polynomials (HPPs) corresponding to a certain algebraic approximant are 𝑁+1 polynomials 𝑃𝑛(𝑧) with degree 𝑝𝑛=deg(𝑃𝑛), 𝑛=0𝑁 such that the order condition𝑁𝑛=0𝑃𝑛(𝑧)𝑓(𝑧)𝑛𝑧=𝑂𝑀(1.2) holds for small 𝑧. Since one of the coefficients of the polynomials can be normalized to unity, the order condition (1.2) gives rise to a system of 𝑀 linear equations for 𝑁+𝑁𝑛=0𝑝𝑛 unknown polynomial coefficients. Thus, the coefficient of 𝑧𝑚 of the Taylor expansion at 𝑧=0 of the left hand side of (1.2) must be zero for 𝑚=0,,𝑀1. In order to have exactly as many equations as unknowns, we choose𝑀=𝑁+𝑁𝑛=0𝑝𝑛(1.3) and assume that the linear system (1.2) has a solution. Then, the HPPs 𝑃𝑛(𝑧) are uniquely defined upon specifying the normalization. The algebraic approximant under consideration then is that pointwise solution 𝑎(𝑧) of the algebraic equation𝑃0(𝑧)+𝑁𝑛=1𝑃𝑛(𝑧)𝑎(𝑧)𝑛=0(1.4) for which the Taylor series of 𝑎(𝑧) coincides with the given power series at least up to order 𝑧𝑀1.

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 (1.4) explicitly, that is for 𝑁4, 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.

2. The Recursive Algorithm

We consider the HPPs𝑃𝑛(𝑧)=𝑝𝑛𝑗=0𝑝𝑛,𝑗𝑧𝑗(2.1) as known. Putting𝑎(𝑧)=𝑘=0𝑎𝑘𝑧𝑘,(2.2) we obtain from  (1.4)𝑝0𝑗=0𝑝0,𝑗𝑧𝑗+𝑁𝑝𝑛=1𝑛𝑗=0𝑝𝑛,𝑗𝑧𝑗𝑘1=0𝑘𝑛=0𝑧𝑘1++𝑘𝑛𝑛𝑚=1𝑎𝑘𝑚=0,(2.3) whence, by equating the coefficient of 𝑧𝐽 to zero, we obtain an infinite set of equations. Due to (1.2), all the equations for 𝐽<𝑀 are satisfied exactly for 𝑎𝑗=𝑓𝑗, 𝑗=0,,𝑀1.

As a first step, we compute 𝑎𝑀. We note that 𝑀>𝑝0. Hence, the coefficient of 𝑧𝑀 does not involve any terms with 𝑝0,𝑗. For this coefficient 𝑅𝑀, we only need to consider terms in (2.3) such that 𝑀=𝑗+𝑘1++𝑘𝑛, and we obtain 𝑅𝑀=0 for𝑅𝑀=𝑁𝑛=1𝑗+𝑘1++𝑘𝑛=𝑀𝑝𝑛𝑛,𝑗𝑚=1𝑎𝑘𝑚.(2.4) The only terms on the RHS involving 𝑎𝑀 are obtained if exactly one of the 𝑘𝑚 is equal to 𝑀, that is, we have 𝑘𝑚=𝑀, 𝑗=0, and 𝑘𝑗=0 for 𝑗𝑚. Thus, we may rewrite all these terms as 𝑎𝑀𝐶, where𝐶=𝑁𝑛=1𝑛𝑝𝑛,0𝑓0𝑛1(2.5) and note that the rest 𝐷𝑀=𝑅𝑀𝑎𝑀𝐶 is independent of 𝑎𝑀. Recalling 𝑅𝑀=0, we obtain𝑎𝑀=𝐷𝑀𝐶.(2.6) Proceeding analogously for 𝐽>𝑀, only terms with 𝐽=𝑗+𝑘1++𝑘𝑛 need to be considered. Hence, 𝑅𝐽=0 for𝑅𝐽=𝑁𝑛=1𝑗+𝑘1++𝑘𝑛=𝐽𝑝𝑛𝑛,𝑗𝑚=1𝑎𝑘𝑚.(2.7) Now, the only terms on the RHS involving 𝑎𝐽 are obtained if exactly one of the 𝑘𝑚 is equal to 𝐽, that is, we have 𝑘𝑚=𝐽, 𝑗=0, and 𝑘𝑗=0 for 𝑗𝑚. Thus, we may rewrite all these terms as 𝑎𝐽𝐶, where 𝐶 is defined above. Proceeding as before, we put 𝐷𝐽=𝑅𝐽𝑎𝐽𝐶 and obtain𝑎𝐽=𝐷𝐽𝐶.(2.8) An equivalent form of the recursive algorithm is obtained in the following way.

Consider for known 𝑃𝑛 and 𝑎0,,𝑎𝐽1 the expression𝑈𝐽=𝑑𝐽𝐽!𝑑𝑧𝐽||||𝑁𝑧=0𝑛=1𝑃𝑛(𝑧)𝐽𝑗=0𝑎𝑗𝑧𝑗𝑛.(2.9) It is easy to see, that this expression is exactly equal to 𝑅𝐽, and hence, is linear in the unknown 𝑎𝐽. Thus, we may compute the quantities 𝐷𝐽 by substituting 𝑎𝐽=0 into 𝑈𝐽, which entails𝐷𝐽=𝑑𝐽𝐽!𝑑𝑧𝐽||||𝑁𝑧=0𝑛=1𝑃𝑛(𝑧)𝐽1𝑗=0𝑎𝑗𝑧𝑗𝑛.(2.10) Thus, starting from 𝐽=𝑀, one may compute all the 𝑎𝐽 consecutively by repeated use of (2.5), (2.10), and (2.8).

This concludes the derivation of the recursive algorithm.

3. 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 so far unused coefficients, for example, 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 so far 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.

4. Examples

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

Example 4.1. As a first example, we consider 𝑁=2, 𝑝0=𝑝1=𝑝2=1, and, hence, 𝑀=5. Since 𝑁=2, we are dealing with a quadratic algebraic approximant. Then, the recursive algorithm is started by 𝑎𝑗=𝑓𝑗, 𝑗=0,,4. For 𝑎5, we obtain 𝑎5𝑝=1,1𝑓4+𝑝2,12𝑓0𝑓4+2𝑓1𝑓3+𝑓22+𝑝2,02𝑓1𝑓4+2𝑓2𝑓3𝑝1,0+2𝑝2,0𝑓0(4.1) and for 𝐽>5, we obtain 𝑎𝐽𝑝=1,1𝑎𝐽1+𝑝2,1𝐽1𝑘=0𝑎𝐽𝑘1𝑎𝑘+𝑝2,0𝐽1𝑘=1𝑎𝑘𝑎𝐽𝑘𝑝1,0+2𝑝2,0𝑓0.(4.2) For 𝑓(𝑧)=(23𝑧)1/2+15𝑧,(4.3) the HPPs are determined to be 𝑃0𝑃(𝑧)=1.1.544503593423590𝑧,1𝑃(𝑧)=.1947992842134984+.06783822675080703𝑧,2(𝑧)=.5044536972622500.01090573365920830𝑧.(4.4) The results for the predicted coefficients given in Table 1.

Example 4.2. As a second example, we consider again 𝑁=2, 𝑝0=𝑝1=𝑝2=1, and 𝑀=5, but now the function 𝑓(𝑧)=17(12𝑧)1/3+𝑧2𝑧(4.5) with the HPPs 𝑃0𝑃(𝑧)=49.523693181668396.946105600281359𝑧,1𝑃(𝑧)=1.+1.695055482965655𝑧,2(𝑧)=.1125387307324166.2732915349762758𝑧.(4.6) The results for the predicted coefficients given in Table 2.

Example 4.3. As a final example, we consider the case 𝑁=𝑝0=𝑝1=𝑝2=2, whence 𝑀=8, and the function 𝑓(𝑧)=exp(𝑧)(23𝑧)1/3+15𝑧.(4.7) The corresponding HPPs are 𝑃0(𝑧)=1.1.027576803009053𝑧+.02070967420422950𝑧2,𝑃1(𝑧)=2.617867885747464.6563757889994458𝑧3.118191126500581𝑧2,𝑃2(𝑧)=3.647182626894738+7.471780741166546𝑧3.356878399103086𝑧2.(4.3) The results for the predicted coefficients are displayed in Table 3.

5. 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 𝑓𝑀, and the next few coefficients.