arXiv subjects

Marc Van Barel

Publications and source records attributed to Marc Van Barel.

28 records · Page 2Linked to original sources

arXiv2020

A Robust Numerical Path Tracking Algorithm for Polynomial Homotopy Continuation

We propose a new algorithm for numerical path tracking in polynomial homotopy continuation. The algorithm is `robust' in the sense that it is designed to prevent path jumping and in many cases, it can be used in (only) double precision arithmetic. It is based on an adaptive stepsize predictor that uses Padé techniques to detect local difficulties for function approximation and danger for path jumping. We show the potential of the new path tracking algorithm through several numerical examples and compare with existing implementations.

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arXiv2020

Robust Numerical Tracking of One Path of a Polynomial Homotopy on Parallel Shared Memory Computers

We consider the problem of tracking one solution path defined by a polynomial homotopy on a parallel shared memory computer. Our robust path tracker applies Newton's method on power series to locate the closest singular parameter value. On top of that, it computes singular values of the Hessians of the polynomials in the homotopy to estimate the distance to the nearest different path. Together, these estimates are used to compute an appropriate adaptive stepsize. For n-dimensional problems, the cost overhead of our robust path tracker is O(n), compared to the commonly used predictor-corrector methods. This cost overhead can be reduced by a multithreaded program on a parallel shared memory computer.

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arXiv2020

Backward Error Measures for Roots of Polynomials

We analyze different measures for the backward error of a set of numerical approximations for the roots of a polynomial. We focus mainly on the element-wise mixed backward error introduced by Mastronardi and Van Dooren, and the tropical backward error introduced by Tisseur and Van Barel. We show that these measures are equivalent under suitable assumptions. We also show relations between these measures and the classical element-wise and norm-wise backward error measures.

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arXiv2020

Min-Max Elementwise Backward Error for Roots of Polynomials and a Corresponding Backward Stable Root Finder

A new measure called min-max elementwise backward error is introduced for approximate roots of scalar polynomials $p(z)$. Compared with the elementwise relative backward error, this new measure allows for larger relative perturbations on the coefficients of $p(z)$ that do not participate much in the overall backward error. By how much these coefficients can be perturbed is determined via an associated max-times polynomial and its tropical roots. An algorithm is designed for computing the roots of $p(z)$. It uses a companion linearization $C(z) = A-zB$ of $p(z)$ to which we added an extra zero leading coefficient, and an appropriate two-sided diagonal scaling that balances $A$ and makes $B$ graded in particular when there is variation in the magnitude of the coefficients of $p(z)$. An implementation of the QZ algorithm with a strict deflation criterion for eigenvalues at infinity is then used to obtain approximations to the roots of $p(z)$. Under the assumption that this implementation of the QZ algorithm exhibits a graded backward error when $B$ is graded, we prove that our new algorithm is min-max elementwise backward stable. Several numerical experiments show the superior performance of the new algorithm compared with the MATLAB \texttt{roots} function. Extending the algorithm to polynomial eigenvalue problems leads to a new polynomial eigensolver that exhibits excellent numerical behaviour compared with other existing polynomial eigensolvers, as illustrated by many numerical tests.

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arXiv2019

Biorthogonal Extended Krylov Subspace Methods

A general framework for oblique projections of nonhermitian matrices onto rational Krylov subspaces is developed. To obtain this framework we revisit the classical rational Krylov subspace algorithm and prove that the projected matrix can be written efficiently as a structured pencil, where the structure can take several forms, such as Hessenberg or inverse Hessenberg. One specific instance of the structures appearing in this framework for oblique projections is a tridiagonal pencil. This is a direct generalization of the classical biorthogonal Krylov subspace method where the projection becomes a single nonhermitian tridiagonal matrix and of the Hessenberg pencil representation for rational Krylov subspaces. Based on the compact storage of this tridiagonal pencil in the biorthogonal setting, we can develop short recurrences. Numerical experiments confirm the validity of the approach.

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arXiv2018

Truncated Normal Forms for Solving Polynomial Systems: Generalized and Efficient Algorithms

We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. Normal form algorithms provide an algebraic approach to solve this problem. The framework presented in Telen et al. (2018) uses truncated normal forms (TNFs) to compute the algebra structure of R/I and the solutions of I. This framework allows for the use of much more general bases than the standard monomials for R/I. This is exploited in this paper to introduce the use of two special (nonmonomial) types of basis functions with nice properties. This allows, for instance, to adapt the basis functions to the expected location of the roots of I. We also propose algorithms for efficient computation of TNFs and a generalization of the construction of TNFs in the case of non-generic zero-dimensional systems. The potential of the TNF method and usefulness of the new results are exposed by many experiments.

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arXiv2018

Optimal Lebesgue constants for least squares polynomial approximation on the (hyper)sphere

We investigate the uniform approximation provided by least squares polynomials on the unit Euclidean sphere $\mathbb{S}^q$ in $\mathbb{R}^{q+1}$, with $q\ge 2$. Like any other polynomial projection, the study concerns the growth, as the degree $n$ tends to infinity, of the associated Lebesgue constant, i.e., of the uniform norm of the least squares operator. If the least squares polynomial of degree $n$ is based on a set of points, which are nodes of a positive weighted quadrature rule of degree of exactness $2n$, then we state two different sufficient conditions for having an optimal Lebesgue constant that increases with $n$ at the minimal projections order. Hence, under our assumptions least squares and hyperinterpolation polynomials provide a comparable approximation with respect to the uniform norm.

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arXiv2018

Uniform approximation on the sphere by least squares polynomials

The paper concerns the uniform polynomial approximation of a function $f$, continuous on the unit Euclidean sphere of ${\mathbb R}^3$ and known only at a finite number of points that are somehow uniformly distributed on the sphere. First we focus on least squares polynomial approximation and prove that the related Lebesgue constants w.r.t. the uniform norm grow at the optimal rate. Then, we consider delayed arithmetic means of least squares polynomials whose degrees vary from $n-m$ up to $n+m$, being $m=\lfloor θn\rfloor$ for any fixed parameter $0<θ<1$. As $n$ tends to infinity, we prove that these polynomials uniformly converge to $f$ at the near-best polynomial approximation rate. Moreover, for fixed $n$, by using the same data points we can further improve the approximation by suitably modulating the action ray $m$ determined by the parameter $θ$. Some numerical experiments are given to illustrate the theoretical results.

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arXiv2018

A Stabilized Normal Form Algorithm for Generic Systems of Polynomial Equations

We propose a numerical linear algebra based method to find the multiplication operators of the quotient ring $\mathbb{C}[x]/I$ associated to a zero-dimensional ideal $I$ generated by $n$ $\mathbb{C}$-polynomials in $n$ variables. We assume that the polynomials are generic in the sense that the number of solutions in $\mathbb{C}^n$ equals the Bézout number. The main contribution of this paper is an automated choice of basis for $\mathbb{C}[x]/I$, which is crucial for the feasibility of normal form methods in finite precision arithmetic. This choice is based on numerical linear algebra techniques and governed by the numerical properties of the given generators of $I$.

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arXiv2017

Solving Polynomial Systems via a Stabilized Representation of Quotient Algebras

We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. We propose a general algebraic framework to find the solutions and to compute the structure of the quotient ring R/I from the null space of a Macaulay-type matrix. The affine dense, affine sparse, homogeneous and multi-homogeneous cases are treated. In the presented framework, the concept of a border basis is generalized by relaxing the conditions on the set of basis elements. This allows for algorithms to adapt the choice of basis in order to enhance the numerical stability. We present such an algorithm and show numerical results.

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