1,721,056 research outputs found
Pascal matrix, classical polynomial and difference equations
No full textAlthough the Pascal matrix is one of the oldest in the history of Mathematics, owing to both its utility in many applications and its countless properties, it continues to create interest. In this paper we review some recent works on the Pascal matrix by focusing our attention on its relations with linear algebra, difference equations and classical polynomials, such as the Legendre, Bernestein and Laguerre polynomials
Conservation of polynomial type and the stability problem for Linear Multistep Methods
No full textThe study of the linear stability for linear multistep methods leads to consider the location of the zeros of the associated characteristic polynomial with respect to the unit circle in the complex plane. It is known that if the discrete problem is an initial value one, it is sufficient to determine when all the roots are inside the unit disk. The choice to fix all the conditions at the beginning of the interval of integration leads, however, to severe restrictions on the order of the methods (Dahlquist barriers). To overcome this drawback one can use a linear multistep method coupled with boundary conditions (BVMs). In the BVMs approach the classical stability conditions are generalized. In this talk, a rigorous analysis of the linear stability for some classes of BVMs is presented. The necessary information on the coefficients of the methods are obtained by an extensive use of properties of the Pascal matrix
On the periodic solutions of discrete Hamiltonian systems
No full textAlmost all numerical methods for solving conservative problems cannot avoid a more or less perceptible drift phenomenon. Considering that the drift would be absent on a periodic or quasi-periodic solution, one way to eliminate such unpleasant phenomenon is to look for discrete periodic or quasi-periodic solutions. It is quite easy to show that only symmetric methods are able to provide solutions having such behavior. The open problem is to find the suitable stepsize and to be sure that the obtained periodic solution is stable. In the preliminary results here presented we show that this problem is strongly connected with a classical problem of evolution of planar polygons already discussed by Schoenberg in [5, 61 and more recently treated in [2]
Special polynomials as continuous dynamical systems
No full textThe special polynomials play a central role in the applications of mathematics. This number has notably increased during the years. Nevertheless, seen from the point of view of dynamical systems, they display an incredible number of common properties. At the core of this unifying approach is the Pascal matrix. Some examples will be discussed in the paper
One parameter family of linear difference equations and the stability problem for the numerical solution of ODEs
No full text The study of the stability properties of numerical methods leads to considering linear difference equations depending on a complex parameter q. Essentially, the associated characteristic polynomial must have constant type for q ∈ ℂ-. Usually such request is proved with the help of computers. In this paper, by using the fact that the associated polynomials are solutions of a "Legendre-type" difference equation, a complete analysis is carried out for the class of linear multistep methods having the highest possible order.</p
On the properties of matrices defining some classes of BVMs
No full textIt is known that the matrices defining the discrete problem generated by a k-step Boundary Value Method (BVM) have a quasi-Toeplitz band structure. In particular, when the boundary conditions are skipped, they become Toeplitz matrices. In this paper, by introducing a characterization of positive definiteness for such matrices, we shall prove that the Toeplitz matrices which arise when using the methods in the classes of BVMs known as Generalized BDF and Top Order Methods have such property
High-order symmetric schemes for the energy conservation of polynomial Hamiltonian problems
No full textBOUNDARY-VALUE METHODS AND BV-STABILITY IN THE SOLUTION OF INITIAL-VALUE PROBLEMS
No full textIn this paper we consider boundary value techniques based on a three-term numerical method for solving initial value problems. The notions of BV-stability and BV-relative stability are introduced in order to clarify the conditions that a three-term scheme must satisfy for solving efficiently initial value problems. In particular we investigate the BV-stability of boundary value methods based on the mid-point rule, on the Simpson method, and on an Adams-type method. The problem of approximating the solution at the final point is approached and an error estimate at this point is given. Among the main features of the boundary value methods studied there is the possibility of employing the same method for an initial value problem with increasing and decreasing modes and the possibility of implementing efficiently boundary value methods on parallel computers
A PROJECTION METHOD FOR THE NUMERICAL-SOLUTION OF LINEAR-SYSTEMS IN SEPARABLE STIFF DIFFERENTIAL-EQUATIONS
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