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...

We generalize a “one eigenstate” theorem of Levy, Perdew and Sahni (LPS) [1] to the case of densities coming from eigenmixture density operators. The generalization is of a special interest for the radial density functional theory (RDFT) for nuclei [2], a consequence of the rotational invariance of the nuclear Hamiltonian; when nuclear ground states (GSs) have a finite spin, the RDFT uses eigen...

This paper is triggered by the preprint [P. Jain, C. Jin, S.M. Kakade, and P. Netrapalli. Computing matrix squareroot via non convex local search. Preprint, arXiv:1507.05854, 2015.], which analyzes gradient-descent for computing the square root of a positive definite matrix. Contrary to claims of Jain et al., the author’s experiments reveal that Newton-like methods compute matrix square roots r...

Combined with strong tracking filter (STF) theory, the Strong Tracking Square-Root Unscented Kalman Filter (UKF)-based satellite attitude determination algorithm is proposed in this paper. QR decomposition and Cholesyk decomposition are introduced in this paper, which improves the stability of filter. In addition, by introduced adaptive fading factor, the prediction error covariance matrix can ...