Generalizations of Gershgorin disks and polynomial zeros

Abstract

Abstract. We derive inclusion regions for the eigenvalues of a general complex matrix that are generalizations of Gershgorin disks, along with nonsingularity conditions. We then apply these results to the location of zeros of polynomials.

Inclusion sets for the input and characteristic impedance of lossy microwave lines

“…New circle inclusion sets are computed, these boundaries can be a useful tool for the microwave engineers in order to establish directly the restrictions for the real and imaginary parts of the input and characteristic impedance of a lossy line. The paper is connected to a part of the new trends in pure mathematics [6] that search for different inclusion sets in the complex plane by means of the recent developments in modern software. However in order to visualize the inclusion boundaries in the reflection coefficient's plane one could use the recently proposed 3D microwave chart [7][8].…”

“…1 will be bounded in the complex plane by a circle with the coordinates, radius given x c and r c (5) whereas the position of each point on this circle is given by (6).…”

“…The image of this circle under the transformation (7) is obtained replacing a with r(t) given in (6). The equation of the image circle can be easily obtained computing the image of three different points that belong to the circle defined in (5).…”

“…After applying (6), the circle that bounds hyperbolic tangent in this case becomes: r(t)=15.92+15.92*exp(t). As the length of the line increases, tanh(z) will never lay outside of this circle.…”

Brauer-Type Inclusion Sets of Zeros for Chebyshev Polynomial

“…The inclusion region of polynomial zeros is widely used in the theory of differential equations, the complex functions and the numerical analysis. There are some inclusion regions for polynomial zeros in power basis [1][2][3]. However, the structures of comrade matrix for generalized polynomials are different from this of polynomial in power basis [4], so it is difficult to use them to generalized polynomials such as Chebyshev polynomial and Hermite polynomial, which are widely used in interpolations and numerical fittings.…”