access icon openaccess Convergence of recursive functions on computers

A theorem is presented which has applications in the numerical computation of fixed points of recursive functions. If a sequence of functions {fn } is convergent on a metric space I ⊆ ℝ, then it is possible to observe this behaviour on the set 𝔻 ⊂ ℚ of all numbers represented in a computer. However, as 𝔻 is not complete, the representation of fn on 𝔻 is subject to an error. Then fn and fm are considered equal when its differences computed on 𝔻 are equal or lower than the sum of error of each fn and fm . An example is given to illustrate the use of the theorem.

Inspec keywords: set theory; convergence of numerical methods; recursive functions

Other keywords: recursive function fixed points; metric space; convergent function sequence; numerical computation; recursive function convergence

Subjects: Other numerical methods; Combinatorial mathematics; Formal logic

References

    1. 1)
    2. 2)
    3. 3)
    4. 4)
      • 3. Garcia-Martinez, M., Campos-Canton, I., Campos-Canton, E., Celikovsky, S.: ‘Difference map and its electronic circuit realization’. Nonlinear Dynamics, November 2013, vol. 74, pp. 819830.
    5. 5)
    6. 6)
      • 4. Glendinning, P.: ‘Stability, instability and chaos: an introduction to the theory of nonlinear differential equations’ (Cambridge University Press, Cambridge, UK, 1994).
    7. 7)
      • 9. Overton, M.L.: ‘Numerical computing with IEEE floating point arithmetic’ (SIAM, Philadelphia, USA, 2001).
    8. 8)
      • 8. Schröder, B.S.: ‘Mathematical analysis: a concise introduction’ (John Wiley & Sons, New Jersey, USA, 2008).
    9. 9)
    10. 10)
      • 7. Rudin, W.: ‘Principles of mathematical analysisInternational Student Edition, (McGraw-Hill, New York, 1976, 3rd edn.).
    11. 11)
    12. 12)
      • 2. Suneel, M.: ‘Electronic circuit realization of the logistic map’. Sadhana-academy Proc. in Engineering Sciences, February 2006, vol. 31, pp. 6978.
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