On the bisection method for normal matrices

Bisection Method for Measuring the Distance of a Stable Matri to the Unstable Matrices

W ABSTRACT e describe a bisection method to determine the 2-norm and Frobenius norm-g distance from a given matrix A to the nearest matrix with an eigenvalue on the ima inary axis. If A is stable in the sense that its eigenvalues lie in the open left half e plane, then this distance measures how "nearly unstable" A is. Each step provides ither a rigorous upper bound or a rigorous lower bound on...

On Faster Convergence of the Bisection Method

Let AABC be a triangle with vertices A, B, and C. It is "bisected" as follows: choose a/the longest side (say AB) of AABC, let D be the midpoint of AB, then replace AABC by two triangles aA.DC and ADBC. Let Ag| be a given triangle. Bisect Aqj into two triangles A¡¡ and Aj2. Next bisect each Aj(-, /= 1, 2, forming four new triangles A2j-, i = 1,2, 3,4. Continue thus, forming an infinite sequence...

Optimal Convergence of the Simultaneous Iteration Method for Normal Matrices

Simultaneous iteration methods are extensions of the power method that were devised for approximating several dominant eigenvalues of a matrix and the corresponding eigenvectors. The convergence analysis of these methods has been given for both hermitian and nonhermitian matrices. In this paper we concentrate on normal matrices which include also the hermitian matrices, and provide a rigorous a...

Complexity of the bisection method

The bisection method is the consecutive bisection of a triangle by the median of the longest side. In this paper we prove a subexponential asymptotic upper bound for the number of similarity classes of triangles generated on a mesh obtained by iterative bisection, which previously was known only to be finite. The relevant parameter is γ/σ, where γ is the biggest and σ is the smallest angle of t...

The Bisection Method: Which Root?