11,502 research outputs found
Limit theorems for linear eigenvalue statistics of overlapping matrices
The paper proves several limit theorems for linear eigenvalue statistics of
overlapping Wigner and sample covariance matrices. It is shown that the
covariance of the limiting multivariate Gaussian distribution is diagonalized
by choosing the Chebyshev polynomials of the first kind as the basis for the
test function space. The covariance of linear statistics for the Chebyshev
polynomials of sufficiently high degree depends only on the first two moments
of the matrix entries. Proofs are based on a graph-theoretic interpretation of
the Chebyshev linear statistics as sums over non-backtracking cyclic pathsComment: 44 pages, 4 figures, some typos are corrected and proofs clarified.
Accepted to the Electronic Journal of Probabilit
On Chebyshev polynomials of matrices
The mth Chebyshev polynomial of a square matrix A is the monic polynomial that minimizes the matrix 2-norm of over all monic polynomials of degree m. This polynomial is uniquely defined if m is less than the degree of the minimal polynomial of A. We study general properties of Chebyshev polynomials of matrices, which in some cases turn out to be generalizations of well-known properties of Chebyshev polynomials of compact sets in the complex plane. We also derive explicit formulas of the Chebyshev polynomials of certain classes of matrices, and explore the relation between Chebyshev polynomials of one of these matrix classes and Chebyshev polynomials of lemniscatic regions in the complex plane
Chebyshev polynomials on symmetric matrices
In this paper we evaluate Chebyshev polynomials of the second-kind on a class
of symmetric integer matrices, namely on adjacency matrices of simply laced
Dynkin and extended Dynkin diagrams. As an application of these results we
explicitly calculate minimal projective resolutions of simple modules of
symmetric algebras with radical cube zero that are of finite and tame
representation type
Numerical Experiments for Finding Roots of the Polynomials in Chebyshev Basis
Root finding for a function or a polynomial that is smooth on the interval [a; b], but otherwise arbitrary, is done by the following procedure. First, approximate it by a Chebyshev polynomial series. Second, find the zeros of the truncated Chebyshev series. Finding roots of the Chebyshev polynomial is done by eigenvalues of a nXn matrix such as companion or comrade matrices. There are some methods for finding eigenvalues of these matrices such as companion matrix and chasing procedures.We derive another algorithm by second kind of Chebyshev polynomials.We computed the numerical results of these methods for some special and ill-conditioned polynomials
Wavelets operational methods for fractional differential equations and systems of fractional differential equations
In this thesis, new and effective operational methods based on polynomials and
wavelets for the solutions of FDEs and systems of FDEs are developed. In particular
we study one of the important polynomial that belongs to the Appell family of
polynomials, namely, Genocchi polynomial. This polynomial has certain great
advantages based on which an effective and simple operational matrix of derivative
was first derived and applied together with collocation method to solve some singular
second order differential equations of Emden-Fowler type, a class of generalized
Pantograph equations and Delay differential systems. A new operational matrix of
fractional order derivative and integration based on this polynomial was also
developed and used together with collocation method to solve FDEs, systems of
FDEs and fractional order delay differential equations. Error bound for some of the
considered problems is also shown and proved. Further, a wavelet bases based on
Genocchi polynomials is also constructed, its operational matrix of fractional order
derivative is derived and used for the solutions of FDEs and systems of FDEs. A
novel approach for obtaining operational matrices of fractional derivative based on
Legendre and Chebyshev wavelets is developed, where, the wavelets are first
transformed into corresponding shifted polynomials and the transformation matrices
are formed and used together with the polynomials operational matrices of fractional
derivatives to obtain the wavelets operational matrix. These new operational matrices
are used together with spectral Tau and collocation methods to solve FDEs and
systems of FDEs
Optimization via Chebyshev Polynomials
This paper presents for the first time a robust exact line-search method
based on a full pseudospectral (PS) numerical scheme employing orthogonal
polynomials. The proposed method takes on an adaptive search procedure and
combines the superior accuracy of Chebyshev PS approximations with the
high-order approximations obtained through Chebyshev PS differentiation
matrices (CPSDMs). In addition, the method exhibits quadratic convergence rate
by enforcing an adaptive Newton search iterative scheme. A rigorous error
analysis of the proposed method is presented along with a detailed set of
pseudocodes for the established computational algorithms. Several numerical
experiments are conducted on one- and multi-dimensional optimization test
problems to illustrate the advantages of the proposed strategy.Comment: 26 pages, 6 figures, 2 table
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