Computing with Locally Effective Matrices

  1. Rubio, J. 1
  2. Sergeraert, F. 2
  1. 1 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

  2. 2 Institut Fourier
    info

    Institut Fourier

    Saint-Martin-d’Hères, Francia

    ROR https://ror.org/05rwrfh97

Journal:
International Journal of Computer Mathematics

ISSN: 0020-7160

Year of publication: 2005

Volume: 82

Issue: 10

Pages: 1177-1189

Type: Article

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DOI: 10.1080/00207160512331323326 SCOPUS: 2-s2.0-31744444693 WoS: WOS:000232207600001 GOOGLE SCHOLAR lock_openOpen access editor

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Abstract

In this work, we start from the naive notion of integer infinite matrix (i.e., the functions of the set ℤ<sup>ℕ × ℕ</sup> = {f: ℕ × ℕ → ℤ}). Then, several undecidability results are established, leading to a convenient data structure for effective machine computations. We call this data structure a locally effective matrix. We study when (and how) the standard matrix calculus (Ker and CoKer computations) can be extended to the infinite case. We find again several undecidability barriers. When these limitations are overcome, we describe effective procedures for computing in the locally effective case. Finally, the role played by these data structures in the development of real symbolic computation systems for algebraic topology (based on the effective homology notion) is illustrated.