Abstract
This is an expository account of a constructive theorem on the shortest linear recurrences of a finite sequence over an arbitrary integral domainR. A generalization of rational approximation, which we call “realization”, plays a key role throughout the paper.We also give the associated “minimal realization” algorithm, which has a simple control structure and is division-free. It is easy to show that the number ofR-multiplications required isO(n2), wherenis the length of the input sequence.Our approach is algebraic and independent of any particular application. We view a linear recurring sequence as a torsion element in a naturalR[X]-module. The standardR[X]-module of Laurent polynomials overRunderlies our approach to finite sequences. The prerequisites are nominal and we use short Fibonacci sequences as running examples.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.