On the Convergence of the Quasi-Periodic Approximations on a Finite Interval

We investigate the convergence of the quasi-periodic approximations in different frameworks and reveal exact asymptotic estimates of the corresponding errors. The estimates facilitate a fair comparison of the quasi-periodic approximations to other classical well-known approaches. We consider a special realization of the approximations by the inverse of the Vandermonde matrix, which makes it possible to prove the existence of the corresponding implementations, derive explicit formulas and explore convergence properties. We also show the application of polynomial corrections for the convergence acceleration of the quasi-periodic approximations. Numerical experiments reveal the auto-correction phenomenon related to the polynomial corrections so that utilization of approximate derivatives surprisingly results in better convergence compared to the expansions with the exact ones.

On the Convergence of Orbits in Sequence Space l^2

This paper discusses the convergence of orbits for diagonal operators defined on . In particular, the basis elements of are obtained using the linear combinations of the elements of the orbit. Furthermore, via the classical result of the determinant of the Vandermonde matrix, it is shown that, the more the elements of the orbit are used, the faster the convergence of the orbit to the basis elements of . Keywords: Diagonal operators; Convergence of Orbits of operators; Vandermonde matrix; Norm topology

Analysis of Some Nonstationary Iterative Methods Using the Pascal and Vandermonde Linear Systems

In this paper, analysis of some nonstationary iterative methods using the Vandermonde and Pascal linear system is reported. The nonstationary iterative methods selected were GMRES and QMR to assess their performance on the identified linear systems. The paper focused on the convergence relative residual and number of iteration for each type of chosen linear system. The Vandermonde matrix is mostly applied to interpolation of both quadratic and cubic polynomial function. The resulting polynomial has the form: p(x) = an xn + an-1xn-1 +...+ a1 x + a0 . From the numerical experiments conducted using the matlab programming language, the GMRES is recommended when solving the identified linear systems

On the condition number of the Vandermonde matrix of the nth cyclotomic polynomial

Recently, Blanco-Chacón proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number Cond(Vn) of the Vandermonde matrix Vn associated to the nth cyclotomic polynomial. We prove some results on the singular values of Vn and, in particular, we determine Cond(Vn) for n = 2kpℓ, where k, ℓ ≥ 0 are integers and p is an odd prime number.

Numerical Solution of a Certain Class of Singular Fredholm Integral Equations of the First Kind via the Vandermonde Matrix

In this paper, a computational method is presented to solve potential-type Fredholm integral equations of the first kind, equations in which the unknown functions are singular at the endpoints of the integration domain, in addition to the weakly singular logarithmic kernels. This method provides a numerical solution based on the Newton interpolation technique via the Vandermonde matrix, which can accommodate an approximation of the unknown function, in such a manner that its singularity is easily removed, as well as the removal of kernel singularity. In addition, the Gauss–Legendre formula is adapted and applied for the computations of the obtained convergent integrals. Thus, the obtained numerical solution is equivalent to the solution of an algebraic equation in matrix form without applying the collocation method. The numerical solutions of the illustrated example are strongly converging to the exact solution for all values of 1 x  including the end-points 1 whereas the exact solution fails to find the functional values at these end-points; which ensures the powerful and high accuracy of the presented computational technique