Efficient Algorithms for Computing the Nearest Polynomial with Constrained Roots

Continuous changes of the coefficients of a polynomial move the roots continuously. We consider the problem finding the minimal perturbations to the coefficients to move a root to a given locus, such as a single point, the real or imaginary axis, the unit circle, or the right half plane. We measure minimality in both the Euclidean distance to the coefficient vector and maximal coefficient-wise change in absolute value (infinity norm), either with entirely real or with complex coefficients. If the locus is a piecewise parametric curve, we can give efficient, i.e., polynomial time algorithms for the Euclidean norm; for the infinity norm we present an efficient algorithm when a root of the minimally perturbed polynomial is constrained to a single point. In terms of robust control, we are able to compute the radius of stability in the Euclidean norm for a wide range of convex open domains of the complex plane.

• Type: Conference
• Year: 1998
• Where: ISSAC
• Authors: Markus A. Hitz, Erich Kaltofen
• Keywords: Euclidean Norm, Infinity Norm, ISSAC 1998, Mathematics, Single Point