Advertisement

Computing the Dimension of Linear Subspaces

  • Martin Ziegler
  • Vasco Brattka
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1963)

Abstract

Since its very beginning, linear algebra is a highly algorithmic subject. Let us just mention the famous Gauss Algorithm which was invented before the theory of algorithms has been developed. The purpose of this paper is to link linear algebra explicitly to computable analysis, that is the theory of computable real number functions. Especially, we will investigate in which sense the dimension of a given linear subspace can be computed. The answer highly depends on how the linear subspace is given: if it is given by a finite number of vectors whose linear span represents the space, then the dimension does not depend continuously on these vectors and consequently it cannot be computed. If the linear subspace is represented via its distance function, which is a standard way to represent closed subspaces in computable analysis, then the dimension does computably depend on the distance function.

Keywords

Distance Function Linear Algebra Computable Analysis Linear Subspace Turing Machine 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Errett Bishop and Douglas S. Bridges. Constructive Analysis, Springer, Berlin, 1985. 451Google Scholar
  2. 2.
    Vasco Brattka and Klaus Weihrauch. Computability on subsets of Euclidean space I: Closed and compact subsets. Theoretical Computer Science, 219:65–93, 1999. 451, 452, 455, 456, 457zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Xiaolin Ge and Anil Nerode. On extreme points of convex compact Turing located sets. InAnil Nerode and Yu. V. Matiyasevich, editors, Logical Foundations of Computer Science, vol. 813 of LNCS, 114–128, Berlin, 1994. Springer. 457Google Scholar
  4. 4.
    Andrzej Grzegorczyk. On the definitions of computable real continuous functions. Fundamenta Mathematicae, 44:61–71, 1957. 450zbMATHMathSciNetGoogle Scholar
  5. 5.
    Ker-I Ko. Complexity Theory of Real Functions. Progress in Theoretical Computer Science. Birkhäuser, Boston, 1991. 450Google Scholar
  6. 6.
    Daniel Lacombe. Les ensembles récursivement ouverts ou fermés, et leurs applications à l’Analyse récursive. Comp. Rend. Acad. des Sci. Paris, 246:28–31, 1958. 450zbMATHMathSciNetGoogle Scholar
  7. 7.
    Andrè Lieutier. Toward a data type for solid modeling based on domain theory. In K.-I Ko, A. Nerode, M. B. Pour-El, K. Weihrauch, and J. Wiedermann, eds, Computability and Complexity in Analysis, vol. 235 of Informatik Berichte, pages 51–60. FernUniversität Hagen, August 1998. 451, 457Google Scholar
  8. 8.
    Marian B. Pour-El and J. Ian Richards. Computability in Analysis and Physics. Perspectives in Mathematical Logic. Springer, Berlin, 1989. 450Google Scholar
  9. 9.
    Alan M. Turing. On computable numbers, with an application to the “Entschei-dungsproblem”. Proceedings of the London Mathematical Society, 42(2):230–265, 1936. 450Google Scholar
  10. 10.
    Klaus Weihrauch. Computable Analysis. Springer, Berlin, 2000. 450, 451, 452, 453, 456Google Scholar
  11. 11.
    Ning Zhong. Recursively enumerable subsets of Rq in two computing models: Blum-Shub-Smale machine and Turing machine. Theoretical Computer Science, 197:79–94, 1998. 457zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Qing Zhou. Computable real-valued functions on recursive open and closed subsets of Euclidean space. Mathematical Logic Quarterly, 42:379–409, 1996. 457zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Martin Ziegler
    • 1
  • Vasco Brattka
    • 2
  1. 1.Heinz Nixdorf InstituteUniversity of PaderbornPaderbornGermany
  2. 2.Theoretische Informatik IInformatikzentrum FernUniversitätHagenGermany

Personalised recommendations