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Register a new userComputing Simple Multiple Zeros of Polynomial Systems
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Given a polynomial system f associated with a simple multiple zero x of multiplicity {mu}, we give a computable lower bound on the minimal distance between the simple multiple zero x and other zeros of f. If x is only given with limited accuracy, we propose a numerical criterion that f is certified to have {mu} zeros (counting multiplicities) in a small ball around x. Furthermore, for simple double zeros and simple triple zeros whose Jacobian is of normalized form, we define modified Newton iterations and prove the quantified quadratic convergence when the starting point is close to the exact simple multiple zero. For simple multiple zeros of arbitrary multiplicity whose Jacobian matrix may not have a normalized form, we perform unitary transformations and modified Newton iterations, and prove its non-quantified quadratic convergence and its quantified convergence for simple triple zeros.
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In many applications it is important to understand the sensitivity of eigenvalues of a matrix polynomial to perturbations of the polynomial. The sensitivity commonly is described by condition numbers or pseudospectra. However, the computation of pseudospectra of matrix polynomials is very demanding computationally. This paper describes a new approach to computing approximations of pseudospectra of matrix polynomials by using rank-one or projected rank-one perturbations. These perturbations are inspired by Wilkinsons analysis of eigenvalue sensitivity. This approach allows the approximation of both structured and unstructured pseudospectra. Computed examples show the method to perform much better than a method based on random rank-one perturbations both for the approximation of structured and unstructured (i.e., standard) polynomial pseudospectra.
We give a separation bound for an isolated multiple root $x$ of a square multivariate analytic system $f$ satisfying that an operator deduced by adding $Df(x)$ and a projection of $D^2f(x)$ in a direction of the kernel of $Df(x)$ is invertible. We prove that the deflation process applied on $f$ and this kind of roots terminates after only one iteration. When $x$ is only given approximately, we give a numerical criterion for isolating a cluster of zeros of $f$ near $x$. We also propose a lower bound of the number of roots in the cluster.
In this paper, we study how to quickly compute the <-minimal monomial interpolating basis for a multivariate polynomial interpolation problem. We address the notion of reverse reduced basis of linearly independent polynomials and design an algorithm for it. Based on the notion, for any monomial ordering we present a new method to read off the <-minimal monomial interpolating basis from monomials appearing in the polynomials representing the interpolation conditions.
Polynomial preconditioning with the GMRES minimal residual polynomial has the potential to greatly reduce orthogonalization costs, making it useful for communication reduction. We implement polynomial preconditioning in the Belos package from Trilinos and show how it can be effective in both serial and parallel implementations. We further show it is a communication-avoiding technique and is a viable option to CA-GMRES for large-scale parallel computing.
Let $A$ be an algebra and let $f(x_1,...,x_d)$ be a multilinear polynomial in noncommuting indeterminates $x_i$. We consider the problem of describing linear maps $phi:Ato A$ that preserve zeros of $f$. Under certain technical restrictions we solve the problem for general polynomials $f$ in the case where $A=M_n(F)$. We also consider quite general algebras $A$, but only for specific polynomials $f$.
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