Computing the roots of a scalar polynomial, or the eigenvalues of a matrix polynomial, expressed in the Chebyshev basis {Tk(x)} is a fundamental problem that arises in many applications. In this work, we analyze the backward stability of the polynomial rootfinding problem solved with colleague matrices. In other words, given a scalar polynomial p(x) or a matrix polynomial P (x) expressed in the...

0957-4174/$ see front matter 2013 Elsevier Ltd. A http://dx.doi.org/10.1016/j.eswa.2013.01.045 ⇑ Corresponding author. Tel.: +86 13060687155. E-mail addresses: [email protected], ynzh In view of the great potential in parallel processing and ready implementation via hardware, neural networks are now often employed to solve online nonlinear matrix equation problems. Recently, a novel clas...

Consider I ⊂ C[x1, . . . , xm], a zero dimensional complete intersection ideal, with I = (f1, . . . , fm). Assume that I has clusters of roots, each cluster of radius at most ε in the ∞-norm. We compute the “approximate radical” of I, i.e. an ideal which contains one root for each cluster, corresponding to the center of gravity of the points in the cluster, up to an error term asymptotically bo...

A unified deflating subspace approach is presented for the solution of a large class of matrix equations, including Lyapunov, Sylvester, Riccati and also some higher order polynomial matrix equations including matrix m-th roots and matrix sector functions. A numerical method for the computation of the desired deflating subspace is presented that is based on adapted versions of the periodic QZ a...