^{1,2}

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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.

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 [

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 [

Consider a function

We note that for

Although we assumed that the power series of

We note that the higher coefficients of the Taylor series of

The question then arises how to compute the Taylor series of

In the following section, such a recursive algorithm is obtained. In a further section, we will present numerical examples.

We consider the HPPs

As a first step, we compute

Consider for known

This concludes the derivation of the recursive algorithm.

Basically, there are two modes of application:

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,

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.

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

As a first example, we consider

The case of

Relative error (%) | ||||
---|---|---|---|---|

5 | .001 | .18 | ||

6 | .001 | .38 | ||

7 | .002 | .58 | ||

8 | .004 | .76 | ||

9 | .006 | .93 | ||

10 | .008 | 1.10 |

As a second example, we consider again

The case of

Relative error (%) | ||||
---|---|---|---|---|

5 | 67.938 | 68.212 | .274 | .40 |

6 | 120.739 | 122.291 | 1.552 | 1.29 |

7 | 218.459 | 224.194 | 5.735 | 2.62 |

8 | 400.498 | 418.053 | 17.555 | 4.38 |

9 | 741.657 | 790.063 | 48.406 | 6.53 |

10 | 1384.425 | 1509.437 | 125.012 | 9.03 |

As a final example, we consider the case

The case of

Relative error (%) | ||||
---|---|---|---|---|

8 | 3.888956 | 3.878509 | .010447 | .27 |

9 | 5.356681 | 5.301047 | .055634 | 1.04 |

10 | 7.451679 | 7.275227 | .176452 | 2.37 |

11 | 10.447061 | 10.006950 | .440111 | 4.21 |

12 | 14.739132 | 13.781978 | .957155 | 6.49 |

13 | 20.903268 | 18.995972 | 1.907297 | 9.12 |

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