Zeros, eigenvalues and arithmetic

Correlations of Eigenvalues and Riemann Zeros

Precise Zeros of Analytic Functions Using Interval Arithmetic

An interval arithmetic algorithm for the computation of the zeros of an analytic function inside a given rectangle is presented. It is based on the argument principle in the set of complex numbers C, is guaranteed to converge, and delivers its answers to a prespecified accuracy. The precision of computation is varied dynamically to maximize efficiency. Предлагается алгоритм на базе интервальной...

Accurate simple zeros of polynomials in floating point arithmetic

In the paper, we examine the local behavior of Newton’s method in floating point arithmetic for the computation of a simple zero of a polynomial assuming that an good initial approximation is available. We allow an extended precision (twice the working precision) in the computation of the residual. We prove that, for a sufficient number of iterations, the zero is as accurate as if computed in t...

Arithmetic Properties of Eigenvalues of Generalized Harper Operators on Graphs

Let Q denote the field of algebraic numbers in C. A discrete group G is said to have the σ-multiplier algebraic eigenvalue property, if for every matrix A ∈ Md(Q(G,σ)), regarded as an operator on l(G), the eigenvalues of A are algebraic numbers, where σ ∈ Z(G,U(Q)) is an algebraic multiplier, and U(Q) denotes the unitary elements of Q. Such operators include the Harper operator and the discrete...

Representations and Bounds for Zeros of Orthogonal Polynomials and Eigenvalues of Sign-Symmetric Tri-diagonal Matrices

where P ,(.\-) = 0, P,,(.\-) = I. So without loss of generality MC can take the simpler recurrence formula ( I .2), where c’,, is real and i,, , :> 0 (,I > 0). ;I\ ;I starting point for our analysis. Note that the value ol’ i, i4 irrelevant: it bill bc convenient. however. to assume i , = I. It is well known (see [3]) that P,,(.\) has II real. distinct /eras I,,, c Y,~’ < < I,,,,. Moreover. the...