Point equation of the boundary of the numerical range of a matrix polynomial

Enclosure of the Numerical Range of a Class of Non-selfadjoint Rational Operator Functions

“…The geometric properties of the numerical range of matrix polynomials and rational matrix functions have been studied extensively [3,14] and it is possible to numerically approximate the shape of the numerical range of matrix polynomials [6]. However, as matrix functions generated by a discretization of a differential equation are very large, the available algorithms 152 C. Engström, A. Torshage IEOT are very time consuming.…”

Numerical approximation of the boundary of numerical range of matrix polynomials

“…(1) with numerical range W (P ) as in (2). In the following result, we formulate the point equation of the curve ∂W (P ), [1].…”

“…This method is the first method for the estimation of the numerical range of a general matrix polynomial besides the application of the definition. The algorithm described in the next section is based on a recent theoretical result of Chien, Nakazato and Psarrakos, [1], which yields an algebraic curve of degree at most 2n(n − 1)m that contains ∂W (P ). Illustrative examples are given in Section 3.…”

The q-numerical range and the Davis-Wielandt shell of reducible 3 × 3 matrices

“…Some are curiosity driven and some are driven by its ubiquitous nature in pure and applied mathematics. One of the most challenging problems is the characterization of the boundary ∂F q (A) and the construction of its equation [2,3,4,8,9]. Since F q (A) is a special C-numerical range (see [15,18] for definitions and details), its boundary lies on an algebraic curve in the complex plane.…”

“…If A 1 = a 1,1 a 1,2 a 2,1 a 2,2 , then by the first part of Property (P 2 ), F (A 0 ) coincides with the convex hull of the union F (A 1 ) ∪ {a 3,3 }, where F (A 1 ) is an elliptical disk with foci at the eigenvalues of A 1 [8]. On the other hand, by the second part of the same property, it is clear that we cannot have a similar conclusion for F q (A 0 ) when 0 ≤ q < 1.…”