We compare two different root-finding methods, eigenvalue methods and homotopy methods, using three test problems: Mandelbrot polynomials, Fibonacci-Mandelbrot polynomials, and Narayana-Mandelbrot polynomials. For the eigenvalue methods, using both MATLAB and Maple, we computed the eigenvalues of a specialized recursively-constructed, supersparse, upper Hessenberg matrix, inspired by Piers Lawr...

This note documents implementation details of the small bulge, multi-shift QR algorithm with aggressive early deflation that appears as Lapack version 3.1 programs CHSEQR, DHSEQR, SHSEQR and ZHSEQR and the subroutines they call. These codes calculate eigenvalues and optionally a Schur factorization of a Hessenberg matrix. They do the bulk of the work required to calculate eigenvalues and option...

The spectral value set sp ε A of a bounded linear operator A on l2 is the union of the spectra of all operators of the form A+ BKC, where ‖K‖ < ε and B,C are fixed bounded linear operators. It turns out that small changes of ε may cause drastic changes of the set sp ε A. We conjecture that this can never happen if B or C is compact and A is given by an infinite Toeplitz band matrix. In the pres...

In this paper, we present an algorithm for the reduction to block upper-Hessenberg form which can be used to solve the nonsymmetric eigenvalue problem on message-passing multicomputers. On such multicomputers, a nonsymmetric matrix can be distributed across processing nodes logically configured into a two-dimensional mesh using the block-cyclic data distribution. Based on the matrix partitionin...

We consider different phase spaces for the Toda flows (Flaschka, 1974; Moser, 1975) and less familiar SVD (Chu, 1986; Li, 1997). For flow, we handle symmetric non-symmetric matrices with real simple eigenvalues, possibly a given profile. Profiles encode, example, band Hessenberg matrices. assume simplicity of singular values. In all cases, an open cover is constructed, as are corresponding char...

In this manuscript we study type A nilpotent Hessenberg varieties equipped with a natural S1-action using techniques introduced by Tymoczko, Harada-Tymoczko, and Bayegan-Harada, with a particular emphasis on a special class of nilpotent Springer varieties corresponding to the partition λ = (n− 2, 2) for n ≥ 4. First we define the adjacent-pair matrix corresponding to any filling of a Young diag...

Abstract. In this paper, the multi–server queue with general service time distribution and Lebesgue–dominated iid inter–arival times is analyzed. This is done by introducing auxiliary variables for the remaining service times and then examining the embedded Markov chain at arrival instants. The concept of piecewise–deterministic Markov processes is applied to model the inter–arrival behaviour. ...

We investigate polynomials satisfying a three-term recurrence relation of the form Bn(x) = (x − βn)Bn−1(x) − αnxBn−2(x), with positive recurrence coefficients αn+1, βn (n = 1, 2, . . .). We show that the zeros are eigenvalues of a structured Hessenberg matrix and give the left and right eigenvectors of this matrix, from which we deduce Laurent orthogonality and the Gaussian quadrature formula. ...

We show that there are relationships between a generalized Lucas sequence and the permanent and determinant of some Hessenberg matrices.

In this paper we survey several recent results that highlight an interplay between a relatively new class of quasiseparable matrices and polynomials. Quasiseparable matrices generalize two classical matrix classes, Jacobi (tridiagonal) matrices and unitary Hessenberg matrices that are known to correspond to real orthogonal polynomials and Szegö polynomials, resp. The latter two polynomial famil...