Approximation of the Joint Spectral Radius of a Set of Matrices Using Sum of Squares
نویسندگان
چکیده
We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for a SOS polynomial that proves simultaneous contractibility of a finite set of matrices. We provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices. Additionally, we present a comparison between our approximation scheme and a recent technique due to Blondel and Nesterov, based on lifting of matrices. Theoretical results and numerical investigations show that our approach yields tighter approximations.
منابع مشابه
Joint and Generalized Spectral Radius of Upper Triangular Matrices with Entries in a Unital Banach Algebra
In this paper, we discuss some properties of joint spectral {radius(jsr)} and generalized spectral radius(gsr) for a finite set of upper triangular matrices with entries in a Banach algebra and represent relation between geometric and joint/generalized spectral radius. Some of these are in scalar matrices, but some are different. For example for a bounded set of scalar matrices,$Sigma$, $r_*...
متن کاملApproximation of the joint spectral radius using sum of squares
We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for an SOS polynomial that proves simultaneous contractibility of a finite set of matrices. We provide a bound on the quality of the approximation that unifies several earlier results and is indepe...
متن کاملCartesian decomposition of matrices and some norm inequalities
Let X be an n-square complex matrix with the Cartesian decomposition X = A + i B, where A and B are n times n Hermitian matrices. It is known that $Vert X Vert_p^2 leq 2(Vert A Vert_p^2 + Vert B Vert_p^2)$, where $p geq 2$ and $Vert . Vert_p$ is the Schatten p-norm. In this paper, this inequality and some of its improvements ...
متن کاملSharp Bounds on the PI Spectral Radius
In this paper some upper and lower bounds for the greatest eigenvalues of the PI and vertex PI matrices of a graph G are obtained. Those graphs for which these bounds are best possible are characterized.
متن کاملOn the accuracy of the ellipsoid norm approximation of the joint spectral radius
The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate. We introduce in this paper an approximation ρ̂ that is based on ellipsoid norms, that can be computed by...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007