Convergence of recursive functions on computers

• Published in: Journal of engineering (Stevenage, England) Vol. 2014; no. 10; pp. 560 - 562
• Main Author: Nepomuceno, Erivelton Geraldo
• Format: Journal Article
• Language: English
• Published: The Institution of Engineering and Technology 01.10.2014; Wiley
• Subjects: convergence of numerical methods; convergent function sequence; metric space; numerical computation; recursive function convergence; recursive function fixed points; recursive functions; set theory; convergent function sequence; metric space; numerical computation; recursive functions; convergence of numerical methods; set theory; recursive function fixed points; recursive function convergence
• ISSN: 2051-3305; 2051-3305
• DOI: 10.1049/joe.2014.0228

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.