Abstract
The use of functions, expressible in terms of Lucas polynomials of the second kind, allows us to write down the solution of linear dynamical systems—both in the discrete and continuous case—avoiding the Jordan canonical form of involved matrices. This improves the computational complexity of the algorithms used in literature.
Highlights
Even in recent books, the solution of linear dynamical systems, both in the discrete or continuous time case, is expressed by using all powers of the considered matrix r×r
In order to avoid this serious problem, we propose here an alternative method, based on recursion, using the Fk,n functions, which are essentially linked to Lucas polynomials of the second kind [3], and to the multi-variable Chebyshev polynomials [6]
The use of the Riesz-Fantappiè formula reduces to a finite computation the algorithms used in literature, making use of series expansions, and dramatically improves the computation complexity of the considered problem
Summary
Even in recent books (see e.g. [1] [2]), the solution of linear dynamical systems, both in the discrete or continuous time case, is expressed by using all powers of the considered matrix r×r. In order to avoid this serious problem, we propose here an alternative method, based on recursion, using the Fk,n functions, which are essentially linked to Lucas polynomials of the second kind [3] (i.e. the basic solution of a homogeneous linear recurrence relation with constant coefficients [4] [5]), and to the multi-variable Chebyshev polynomials [6]. Ricci only powers of the considered matrix up to (at most) the order r −1 This is a trivial consequence of the Cayley-Hamilton theorem, and should be used, in our opinion, to reduce the computational cost of solutions. Another shown possibility is the use of the Riesz-Fantappiè formula, by means of which the Taylor expansion of solution is completely avoided. For the connection with Chebyshev polynomials of the second kind in several variables, see [6]
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: 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.
