James Lucien Howland

The sign matrix and the separation of matrix eigenvalues

Abstract The sign matrices uniquely associated with the matrices ( M − ζ j I ) 2 , where ζ j are the corners of a rectangle oriented at π /4 to the axes of a Cartesian coordinate system, may be used to compute the number of eigenvalues of the arbitrarily chosen matrix M which lie within the rectangle, and to determine the left and right invariant subspaces of M associated with these eigenvalues. This paper is concerned with the proof of this statement, and with the details of the computation of the required sign matrices.

Attractive cycles in the iteration of meromorphic functions

SummaryThe existence of attractive cycles constitutes a serious impediment to the solution of nonlinear equations by iterative methods. This problem is illustrated in the case of the solution of the equationz tanz=c, for complex values ofc, by Newtons method. Relevant results from the theory of the iteration of rational functions are cited and extended to the analysis of this case, in which a meromorphic function is iterated. Extensive numerical results, including many attractive cycles, are summarized.

A constructive method for the solution of the stability problem

SummaryIn this paper it is shown that the problem of solving the Liapounov matrix equationSM +MTS = −I is greatly simplified when the given real matrixM is in upper Hessenberg form. The solution is obtained as a linear combinationS = ΣpiSi ofn linearly independent symmetric matricesSi, whereSiM +MTSi =2Di and ΣpiDi = −1/2I. Explicit formulae are given for the elements of theSi, andDi while determination of thepi requires the solution of ann ×n linear system.

Selective solutions to transcendental equations

Abstract This paper presents a combination of global iterative methods, based on the Fatou-Julia theory, and local methods to find selected roots of elementary transcendental equations, z = F ( z , c ), z and c complex, that occur in complex Sturm-Liouville eigenvalue problems, in dielectric spectroscopy and in orbit determination. Suitable starting values for the iteration function, z n +1 = F ( z n , c ), and appropriate regions for each determination of the inverse iteration function, z n = F −1 ( z n , c ), are presented. Convergence criteria are derived from the facts that F has very few attractive fixed points and that the attractive fixed points of F −1 have relatively large basins of attraction in the above-mentioned regions. Certain fixed points of F can be reached quickly by means of a local method, like Newtons method. It is shown by means of digital figures that, in general, Newtons method may lead to attractive cycles or to unpredictable roots when the starting values are near the Fatou-Julia set. Convergence to parasitic roots or even to strange attractors may occur with iterative methods of higher order.

Global Iterative Solutions of Elementary Transcendental Equations

Recent progress in the Julia-Fatou theory of iteration of entire and meromorphic functions throws new insight in the global iterative solution of elementary transcendental equations which occur frequently in the boundary conditions of Sturm-Liouville problems resulting from the separation of variables of boundary value problems for partial differential equations. A particular example is treated in detail by iterating an elementary entire function and its inverse to obtain any root in a preassigned order according to the fundamental regions of the given entire function.

Rational transformation from Schur to Jordan form

Abstract This paper presents a constructive proof of the existence of the Jordan canonical form of an arbitrary lower triangular complex matrix by means of similarities given by rational transformations on rows and columns. The general procedure is illustrated by worked out examples.