Stability of multidimensional scalar and matrix polynomials

Abstract

A comprehensive study of multidimensional stability and related problems of scalar and matrix polynomials is presented in this survey paper. In particular, applications of this study to stability of multidimensional recursive digital and continuous filters, to synthesis of network with commensurate and noncommensurate transmission lines, and to numerical stability of stiff differential equations are enumerated. A novel approach to the multidimensional stability study is the classification of various regions of analyticity. Various computational tests for checking these regions are presented. These include the classical ones based on inners and symmetric matrix approach, table form, local positivity, Lyapunov test, the impulse response tests, the cepstral method and the graphical ones based on Nyquist-like tests. A thorough discussion and comparison of the computational complexities which arise in the various tests are included. A critical view of the progress made during the last two decades on multidimensional stability is presented in the conclusions. The latter also includes some research topics for future investigations. An extensive list of references constitutes a major part of this survey.