Special inclusion sets for matrices

Abstract

This thesis investigates various methods of producing spectral inclusion sets of matrices. The purpose of this investigation is threefold. First of all, various methods will be compared in order to give insight into the fastest and sharpest method of producing spectral inclusion sets for various types of matrices (chapter eight). Secondly, new types of inclusion sets will be introduced by intersecting inclusions sets that are produced using very simple calculations (chapter nine). Finally, new results will be presented and a new method will be introduced for producing spectral inclusion sets of certain types of Toeplitz matrices (chapter ten). Methods of producing spectral inclusion sets of matrices and operators have been developed, primarily, because of two shortcomings in the exact calculations of the eigenvalues. First of all, actual calculations of the spectrum for large matrices can take a great deal of time even on fast computers. Secondly, such calculations may produce erroneous results due to round-off errors. Such erroneous results are particularly prevalent when attempting to calculate the spectrum of ill-conditioned matrices. All of the methods examined in this thesis avoid one or both of these problems. Two groups of methods will be considered in this thesis. One group is classified as "simple" (chapters one through three). The "simple" methods are those methods which are limited to adding, subtracting, multiplying, dividing and raising matrix elements to powers. Such methods are both fast and avoid round-off errors. These "simple" methods include what are called in this paper "pre-Gerschgorin," Gerschgorin's method and Parker's second theorem. A second group of methods considered in this paper are classified as "involved" (chapters four, five, and seven). Such methods utilize extensive searching, large numbers of similarity transformations, and/or a large number of incremental calculations. These methods avoid round-off error and produce very small inclusion sets but may require considerable calculation time. Among the "involved" methods are Cassini, Brualdi, minimal Gerschgorin, the numerical range, and the pseudospectra. No one will be surprised that the major drawback with all of these methods is that they produce sets that are only guaranteed to include the spectrum. While each method produces a set that includes the eigenvalues, the set produced is usually somewhat larger than the actual spectrum of the matrix or operator. Furthermore, each method, except perhaps the pseudospectra, have varying, and sometimes unpredictable, degrees of "sharpness" depending upon the application. In chapters seven and eight it will be shown that the pseudospectra is the most powerful of the "involved" methods. In most cases, the pseudospectra will produce a significantly small spectra inclusion set than any other method. Even in those few instances in which another method produces a small inclusion set, that set will only be slightly smaller than the pseudospectra's set. Therefore, it can be said that the pseudospectra is the only method that consistently produces small spectral inclusion sets. In chapter nine it will also be shown that it is possible to produce relatively sharp spectral inclusion sets by intersecting sets produced by the "simple" methods. This means that small spectral inclusion sets may be produced by using a minimal amount of calculation time. Therefore, this thesis will establish new methods of producing sharp spectral inclusion sets very quickly through the intersection of sets. In the chapter ten of this thesis, two new theorems will be presented and a new method will be introduced for producing spectral inclusion sets of certain types of Toeplitz matrices. The new theorems will be based on Gerschgorin's theorem and the minimal Gerschgorin theorem. It will be demonstrated that the minimal Gerschgorin set can be used, in a new way, to very quickly produce relatively small inclusion set for Toeplitz matrices.