Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial has no nonreal roots, but typically nonreal roots are much more numerous than the real ones. We dramatically accelerate the known algorithms in this case by...

We consider the asymptotics of the second-order correlation function of the characteristic polynomial of a random matrix. We show that the known result for a random matrix from the Gaussian Unitary Ensemble essentially continues to hold for a general Hermitian Wigner matrix. Our proofs rely on an explicit formula for the exponential generating function of the second-order correlation function o...

The condition number of a Gram matrix defined by a polynomial basis and a set of points is often used to measure the sensitivity of the polynomial approximation. Given a polynomial basis, we consider the problem of finding a set of points and/or weights which minimizes the condition number of the Gram matrix. We use smoothing methods to solve such minimization problems and show the global conve...

Abstract. We show that in the large matrix limit, the eigenvalues of the normal matrix model for matrices with spectrum inside a compact domain with a special class of potentials homogeneously fill the interior of a polynomial curve uniquely defined by the area of its interior domain and its exterior harmonic moments which are all given as parameters of the potential. Then we consider the ortho...

Rank constraints on matrices emerge in many automatic control applications. In this short document we discuss how to rewrite the constraint into a polynomial equations of the elements in a the matrix. If addition semidefinite matrix constraints are included, the polynomial equations can be turned into an inequality. We also briefly discuss how to implement these polynomial constraints.

Discrete-time symmetric polynomial equations with complex coefficients are studied in the scalar and matrix case. New theoretical results are derived and several algorithms are proposed and evaluated. Polynomial reduction algorithms are first described to study theoretical properties of the equations. Sylvester matrix algorithms are then developed to solve numerically the equations. The algorit...

It is well known from the Cayley Hamilton theorem that every matrix A ∈ Rr×r satisfies its characteristic equation (Gantmacher, 1959), i.e., if p (s) := det (sIr −A) = sr+p1sr−1+· · ·+pr, then p (A) = 0. The Cayley Hamilton theorem is still valid for all cases of matrices over a commutative ring (Atiyah and McDonald, 1964), and thus for multivariable polynomial matrices. Another form of the Cay...

Rank constraints on matrices emerge in many automatic control applications. In this short document we discuss how to rewrite the constraint into a polynomial equations of the elements in a the matrix. If addition semidefinite matrix constraints are included, the polynomial equations can be turned into an inequality. We also briefly discuss how to implement these polynomial constraints.

It is well known that solving polynomial equations, or finding the eigenvalues of matrix polynomials, can be done by transforming to a generalized eigenvalue problem (see for example [10]). In this paper we examine a new way to do this directly from the values of the polynomial or matrix polynomial at distinct evaluation points.

Cardinal’s matrix version of the Sebastiao e Silva polynomial root-finder rapidly approximates the roots as the eigenvalues of the associated Frobenius matrix. We preserve rapid convergence to the roots but amend the algorithm to allow input polynomials with multiple roots and root clusters. As in Cardinal’s algorithm, we repeatedly square the Frobenius matrix in nearly linear arithmetic time p...