Related papers: Exact Algorithms for Computing Generalized Eigensp…
Given an $n$ by $n$ matrix $A$ and an $n$-vector $b$, along with a rational function $R(z) := D(z )^{-1} N(z)$, we show how to find the optimal approximation to $R(A) b$ from the Krylov space, $\mbox{span}( b, Ab, \ldots , A^{k-1} b)$,…
An algorithm for constructing a $J$-orthogonal basis of the extended Krylov subspace $\mathcal{K}_{r,s}=\operatorname{range}\{u,Hu, H^2u,$ $ \ldots, $ $H^{2r-1}u, H^{-1}u, H^{-2}u, \ldots, H^{-2s}u\},$ where $H \in \mathbb{R}^{2n \times…
An algorithm to generate a minimal comprehensive Gr\"obner\, basis of a parametric polynomial system from an arbitrary faithful comprehensive Gr\"obner\, system is presented. A basis of a parametric polynomial ideal is a comprehensive…
We present a complete algorithm for finding an exact minimal polynomial from its approximate value by using an improved parameterized integer relation construction method. Our result is superior to the existence of error controlling on…
GMRES is a popular Krylov subspace method for solving linear systems of equations involving a general non-Hermitian coefficient matrix. The conventional bounds on GMRES convergence involve polynomial approximation problems in the complex…
Given a set of $p$ symmetric (real) matrices, the Orthogonal Joint Diagonalization (OJD) problem consists of finding an orthonormal basis in which the representation of each of these $p$ matrices is as close as possible to a diagonal…
We propose new iterative methods for computing nontrivial extremal generalized singular values and vectors. The first method is a generalized Davidson-type algorithm and the second method employs a multidirectional subspace expansion…
Randomized orthogonal projection methods (ROPMs) can be used to speed up the computation of Krylov subspace methods in various contexts. Through a theoretical and numerical investigation, we establish that these methods produce…
In this paper we present an algorithm for computing Groebner bases of linear ideals in a difference polynomial ring over a ground difference field. The input difference polynomials generating the ideal are also assumed to be linear. The…
A new algorithm for real root isolation of polynomial equations based on hybrid computation is presented in this paper. Firstly, the approximate (complex) zeros of the given polynomial equations are obtained via homotopy continuation…
The existing doubling algorithms have been proven efficient for several important nonlinear matrix equations arising from real-world engineering applications. In a nutshell, the algorithms iteratively compute a basis matrix, in one of the…
The eigenvalue shift technique is the most well-known and fundamental tool for matrix computations. Applications include the search of eigeninformation, the acceleration of numerical algorithms, the study of Google's PageRank. The shift…
We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with…
The canonical polyadic decomposition (CPD) is a compact decomposition which expresses a tensor as a sum of its rank-1 components. A common step in the computation of a CPD is computing a generalized eigenvalue decomposition (GEVD) of the…
We use principal angles between two subspaces to define Jordan planes. Jordan planes provide an optimal way to decompose $\mathbb{C}^n$ in relation to given two subspaces. We apply Jordan planes to show that two pairs of of subspaces…
Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role…
Several problems in machine learning, statistics, and other fields rely on computing eigenvectors. For large scale problems, the computation of these eigenvectors is typically performed via iterative schemes such as subspace iteration or…
In this paper we present an algorithm to obtain the parameter planes of families of root-finding methods with several free critical points. The parameter planes show the joint behaviour of all critical points. This algorithm avoids the…
We study the complexity of solving the \emph{generalized MinRank problem}, i.e. computing the set of points where the evaluation of a polynomial matrix has rank at most $r$. A natural algebraic representation of this problem gives rise to a…
The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce a general scaling…