If $N_0$ is a normal matrix, then the Hermitian matrices $\\frac{1}{2}(N_0+N_0^*)$ and $\\frac{i}{2}(N_0^*-N_0)$have same eigenvectors as $N_0$. Their eigenvalues are real part imaginary of $N_0$, respectively.If unitary, only each its sign ofthe needed to completely determine eigenvalue, sincethe sum squares these two parts known be equal $1$.Since unitary upper Hessenberg matrix $U$ has quasi...

Three algorithms providing rigourous bounds for the eigenvalues of a real matrix are presented. The first is an implementation of the bisection algorithm for a symmetric tridiagonal matrix using IEEE floating-point arithmetic. The two others use interval arithmetic with directed rounding and are deduced from the Jacobi method for a symmetric matrix and the Jacobi-like method of Eberlein for an ...

The argument functions of Blaschke products provide a very elegant way of handling non-uniformity of discretizations. In this paper we analyse the efficiency of numerical methods as the bisection method and Newton’s method in the case of calculating non-equidistant discretizations generated by Blaschke products. By taking advantage of the strictly increasing property of argument functions we ma...

A bisection of a graph with n vertices is a partition of its vertices into two sets, each of size n=2. The bisection cost is the number of edges connecting the two sets. Finding the bisection of minimum cost is NP-hard. We present an algorithm that finds a bisection whose cost is within ratio of O(log2 n) from the optimal. For graphs excluding any fixed graph as a minor (e.g. planar graphs) we ...

It is still unclear whether the contralateral line bisection error in unilateral homonymous hemianopia is caused by the visual field defect, strategic oculomotor adaptation or by additional extrastriate brain injury. We therefore simulated hemianopia in healthy participants using a gaze-contingent display paradigm and investigated its effects on manual and ocular line bisection performance and ...

OBJECTIVES To clarify the mechanisms of left unilateral spatial neglect found in the bisection of lines after cueing to the left end point and to determine whether neglect occurs for the mental representation of a line. METHODS A new representational bisection task was developed to eliminate the influence of the right segment of the physical line that would attract attention. Eight patients w...

We present an average-case complexity analysis for the zerofinding problem for functions from C([0, 1]), r > 2 , which change sign at the endpoints. This class of functions is equipped with a conditional r-folded Wiener measure. We prove that the average-case complexity of computing an «-approximation is of order log log( 1 /e), and that a hybrid secant-bisection method with a suitable adaptive...

A method of determining a lower bound for the graph bisection minimization problem is described. The bound is valid for weigthed graphs with edge and node weights. The approach is based on Lagrangian relaxation and was previously used for determining an upper bound on the independence number of a graph. The determination of the lower bound is done by solving a quadratic programming problem. A c...

Let D be a set of n pairwise disjoint unit balls in R and P the set of their center points. A hyperplane H is an m-separator for D if each closed halfspace bounded by H contains at least m points from P . This generalizes the notion of halving hyperplanes, which correspond to n/2-separators. The analogous notion for point sets has been well studied. Separators have various applications, for ins...

I.4: “If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the remaining angles respectively, namely those which the equal sides subtend.” I.9: “To bisect a given rectilinear angle.” I.10: “To bisect a given finite straight line.” I.11: “...