Certified Real Eigenvalue Location

The location of real eigenvalues provides critical insights into the stability and resonance properties of physical systems. This paper presents a hybrid symbolic numeric approach for certified real eigenvalue localization. Our method combines Gershgorin disk analysis with Hermite matrix certification to compute certified intervals that enclose the real eigenvalues. These intervals can be further refined through bisectionlike procedures to achieve the desired precision. The proposed approach delivers reliable interval certifications while preserving computational efficiency. The effectiveness of the framework is demonstrated through a concise, fully worked computational example.

💡 Research Summary

The paper introduces a hybrid symbolic‑numeric framework for certified localization of real eigenvalues of a real (or real‑coefficient) matrix. The authors start by computing the characteristic polynomial of the matrix using La Budde’s algorithm, a numerically stable method that reduces the matrix to Hessenberg form via orthogonal similarity transformations and then recursively builds the polynomial coefficients. This avoids explicit determinant calculations and preserves the polynomial’s conditioning, achieving O(n³) complexity with only real arithmetic.

Once the characteristic polynomial p(x) is obtained, the problem of finding real eigenvalues reduces to isolating the real roots of p(x). To obtain coarse spectral bounds, the classical Gershgorin Disk Theorem is applied: for each row i of the matrix A, a disk D_i centered at a_{ii} with radius R_i = Σ_{j≠i}|a_{ij}| is constructed. Every eigenvalue of A lies in the union of these disks, and the intersection of each disk with the real axis yields candidate intervals for real eigenvalues.

The novelty lies in certifying whether a given Gershgorin disk (or a real interval) actually contains a real root. This is achieved through generalized Hermite matrices. For a univariate polynomial p(x) and an auxiliary polynomial q(x), the Hermite matrix H_q(p) = V^T D_q V is defined, where V is the Vandermonde matrix of the (unknown) roots and D_q is diagonal with entries q(z_i). The matrix is real, symmetric, and has a Hankel structure, so its eigenvalues are real. Hermite’s theorem states that the signature σ(H_q(p)) = N_+ – N_–, where N_+ (resp. N_–) counts real roots where q is positive (resp. negative).

Three choices of q(x) are used: (1) q(x)=1, whose signature directly yields the total number of real roots; (2) q(x)=(x−a)(x−b), which tests a real interval