Remarks on the root-clustering of a polynomial in a certain region in the complex plane

Abstract

Abstract.A general formulation for the root clustering of a polynomial is given. An attempt has been made to answer an open question raised by Kalman. Introduction.Since 1852, when Hermite first established the connection between the number of the roots of a polynomial in an arbitrary half-plane and the signature of a certain quadratic form, the root-clustering problem has been investigated by many mathematicians, physicists and engineers. Recently Kalman [1] showed a general formulation which includes all the previously known results as particular cases. His formulation applies to the region r in the complex plane given by

Robust matrix root-clustering analysis in a union of ?-subregions

“…Transformability of D ensures the existence of a unique and non-singular solution P. This conjecture is not available for DˆH described in (10) with a < b. The arbitrary choice of Q cannot guarantee the existence of a single and non-singular solution P. The reader may ® nd more details about those problems of existence in the papers by Howland (1971), Ahn (1974) and Gutman and Jury (1981). &…”

“…To enable this choice of Q, a restriction to transformable regions is assumed. Further details about that problem can be found in the work of Howland (1971), Jury and Ahn (1974) and Gutman and Jury (1981).…”

On Generalised Stein's Inequalites and the Inertias of Their Solutions: Application to Pencil Finite Root‐Clustering

“…Since matrix inequalities, even when linear, have resisted to numeric tools for a long time, Lyapunov's equality ( A ′ P + PA = − Q for some Q > 0) and Stein's equality (− P + A ′ PA = − Q for some Q > 0) were preferred from a computational point of view. Indeed, it can be proven that Q can be arbitrarily chosen , making the equalities more tractable from a computational point of view.…”