Computing the common zeros of two bivariate functions via Bézout resultants

Nakatsukasa, Yuji; Noferini, Vanni; Townsend, Alex

doi:10.1007/s00211-014-0635-z

Cited by 29 publications

(38 citation statements)

References 45 publications

(55 reference statements)

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“…The numerical algorithm that uses MinUnif has complexity O(d 6 ), which is also the complexity of some numerical algorithms for systems of bivariate polynomials that are based on a resultant; see, e.g., [31]. Such an approach is thus not efficient for polynomials of large degree.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Uniform Determinantal Representations

Boralevi

Doornmalen²,

Draisma

et al. 2017

SIAM J. Appl. Algebra Geometry

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Abstract. The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimization, complexity theory, and scientific computing. Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak and Hochstenbach, and we investigate the performance of new representations in their root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research directions.

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Section: Numerical Experimentsmentioning

confidence: 99%

Uniform Determinantal Representations

Boralevi

Doornmalen²,

Draisma

et al. 2017

SIAM J. Appl. Algebra Geometry

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“…One strategy is to first find the Chebyshev interpolant to f (x) on the whole interval, find the roots, and for rootsx i for which We note that a related statement is given in [39], in which subdivision is shown to be important for accuracy when computing common roots of two bivariate functions. In that case subdivision helps even when the polynomial approximant is resampled, as the conditioning depends on the square of the polynomial norms.…”

Section: Cause For Inaccurate Roots and Remedy By Subdivisionmentioning

confidence: 99%

On the stability of computing polynomial roots via confederate linearizations

Nakatsukasa

Noferini

2015

Math. Comp.

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A common way of computing the roots of a polynomial is to find the eigenvalues of a linearization, such as the companion (when the polynomial is expressed in the monomial basis), colleague (Chebyshev basis) or comrade matrix (general orthogonal polynomial basis). For the monomial case, many studies exist on the stability of linearization-based rootfinding algorithms. By contrast, little seems to be known for other polynomial bases. This paper studies the stability of algorithms that compute the roots via linearization in nonmonomial bases, and has three goals. First we prove normwise stability when the polynomial is properly scaled and the QZ algorithm (as opposed to the more commonly used QR algorithm) is applied to a comrade pencil associated with a Jacobi orthogonal polynomial. Second, we extend a result by Arnold that leads to a first-order expansion of the backward error when the eigenvalues are computed via QR, which shows that the method can be unstable. Finally, we focus on the special case of Chebyshev basis, in particular the Chebfun rootfinder: we discuss its stability and describe an optional functionality, made available for improved stability, for computing the roots of a general continuous function f (x), implemented in the recently updated version 5. The main message is that to guarantee backward stability QZ applied to a properly scaled pencil is necessary.

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“…for a certain vector f ∈ S, f = 1; the second-order terms are as in (16). Here σ 1 is the precision used to solve the correction equation and, by assumption, 0 ≤ σ 1 ≤ σ.…”

Section: Propositionmentioning

confidence: 99%

“…For instance, the algorithm [16] from chebfun2 [18] is able to compute all real solutions contained in [−1, 1]×[−1, 1] very efficiently, but it does not compute complex solutions. The best methods at the moment that compute all solutions use continuation method, such as PHCpack.…”

Section: Numerical Experimentsmentioning

confidence: 99%