A method for the inverse numerical range problem

Abstract

Abstract. For a given complex square matrix A, we develop, implement and test a fast geometric algorithm to find a unit vector that generates a given point in the complex plane if this point lies inside the numerical range of A, and if not, the method determines that the given point lies outside the numerical range.

Numerical Range for the Matrix Exponential Function

“…In this section, we give three matrices and plot the numerical range of the corresponding matrix exponential function for each case. The plots are obtained using an inverse numerical range Matlab file based on the algorithm described in [3]. Figure 5.1.…”

Continuous selections of the inverse numerical range map

“…3 ) from w − j to w + j opposite x 0 . By [4,Theorem 11], strong continuity holds at all points in F (A (j) 3 ), except possibly one exceptional point. This will only be a problem if that one point happens to be either w + j or w − j , since in that case it may not be possible to find a continuous path γ j : [t − j , t + j ] → CS n such that γ j (t ± j ) = y(t ± j ), and such that f A (γ j (t)) parametrizes the arc of the boundary of F (A 3 ) would not prevent finding a continuous path γ j (t), unless the eigenvalue is either w ± j .…”

“…In fact, f −1 A (z) contains a set of n linearly independent vectors for every z ∈ int F (A) [2,Theorem 1]. Algorithms for computing at least one element of f −1 A (z) are presented in [2,3,13,17]. As a multivalued map, there are several possible notions of continuity that could apply to the inverse numerical range map f −1 A .…”

Analytic bounds on amplifier gain-bandwidth product from complex power flow

“…Following Thornton [6], we denote being able to generate power at all phase angles as "unrestricted power". Practical algorithms exist to check whether a given point is in the numerical range (and moreover, have been coded as MATLAB functions that are freely available) [12] [13].…”