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Abstract: We introduce and discuss a new computational model for the Hermite–Lagrange interpolation with nonlinear classes of polynomial interpolants. We distinguish between an interpolation problem and an algorithm that solves it. Our model includes also coalescence phenomena and captures a large variety of known Hermite–Lagrange interpolation problems and algorithms. Like in traditional Hermite–Lagrange interpolation, our model is based on the execution of arithmetic operations (including divisions) in the field where the data (nodes and values) are interpreted and arithmetic operations are counted at unit cost. This leads us to a new view of rational functions and maps defined on arbitrary constructible subsets of complex affine spaces. For this purpose we have to develop new tools in algebraic geometry which themselves are mainly based on Zariski’s Main Theorem and the theory of places (or equivalently: valuations). We finish this paper by exhibiting two examples of Lagrange interpolation problems with nonlinear classes of interpolants, which do not admit efficient interpolation algorithms (one of these interpolation problems requires even an exponential quantity of arithmetic operations in terms of the number of the given nodes in order to represent some of the interpolants). In other words, classic Lagrange interpolation algorithms are asymptotically optimal for the solution of these selected interpolation problems and nothing is gained by allowing interpolation algorithms and classes of interpolants to be nonlinear. We show also that classic Lagrange interpolation algorithms are almost optimal for generic nodes and values. This generic data cannot be substantially compressed by using nonlinear techniques. We finish this paper highlighting the close connection of our complexity results in Hermite–Lagrange interpolation with a modern trend in software engineering: architecture tradeoff analysis methods (ATAM). [Copyright &y& Elsevier]

Abstract: The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of homogeneous equations in variables is satisfiable (the resultant is a polynomial in the system’s coefficients which vanishes if and only if the system is satisfiable). In this paper, we investigate the complexity of computing the multivariate resultant. First, we study the complexity of testing the multivariate resultant for zero. Our main result is that this problem is -hard under deterministic reductions in any characteristic, for systems of low-degree polynomials with coefficients in the ground field (rather than in an extension). In null characteristic, we observe that this problem is in the Arthur–Merlin class if the generalized Riemann hypothesis holds true, while the best known upper bound in positive characteristic remains . Second, we study the classical algorithms to compute the resultant. They usually rely on the computation of the determinant of an exponential-size matrix, known as Macaulay matrix. We show that this matrix belongs to a class of succinctly representable matrices, for which testing the determinant for zero is proved -complete. This means that improving Canny’s upper bound requires either to look at the fine structure of the Macaulay matrix to find an ad hoc algorithm for computing its determinant, or to use altogether different techniques. [Copyright &y& Elsevier]

Abstract: Many methods to compute the winding number of plane curves have been proposed, often with the aim of counting the number of roots of polynomials (or, more generally, zeros of analytic functions) inside some region by using the principle of argument. In this paper we propose another method, which are not based on numerical integration, but on discrete geometry. We give conditions that ensure its correct behavior, and a complexity bound based on the distance of the curve to singular cases. Besides, we provide a modification of the algorithm that detects the proximity to a singular case of the curve. If this proximity is such that the number of operations required grows over certain threshold, set by the user, the modified algorithm returns without winding number computation, but with information about the distance to singular case. When the method is applied to polynomials, this information refers to the localization of the roots placed near the curve. [Copyright &y& Elsevier]

Abstract: This paper examines worst-case evaluation bounds for finding weak minimizers in unconstrained optimization. For the cubic regularization algorithm, Nesterov and Polyak (2006) and Cartis et al. (2010) show that at most iterations may have to be performed for finding an iterate which is within of satisfying second-order optimality conditions. We first show that this bound can be derived for a version of the algorithm, which only uses one-dimensional global optimization of the cubic model and that it is sharp. We next consider the standard trust-region method and show that a bound of the same type may also be derived for this method, and that it is also sharp in some cases. We conclude by showing that a comparison of the bounds on the worst-case behaviour of the cubic regularization and trust-region algorithms favours the first of these methods. [Copyright &y& Elsevier]

Abstract: In this paper we study planar polynomial differential systems of this form: where and , , and. A lot of properties of planar polynomial differential systems are related to irreducible Darboux polynomials of the corresponding derivation: . Darboux polynomials are usually computed with the method of undetermined coefficients. With this method we have to solve a polynomial system. We show that this approach can give rise to the computation of an exponential number of reducible Darboux polynomials. Here we show that the Lagutinskii–Pereira algorithm computes irreducible Darboux polynomials with degree smaller than , with a polynomial number, relatively to , and , binary operations. We also give a polynomial-time method to compute, if it exists, a rational first integral with bounded degree. [Copyright &y& Elsevier]

Abstract: In DeVore et al. (2011) we considered smooth functions on which depend on a much smaller number of variables or continuous functions which can be approximated by such functions. We were interested in approximating those functions when we can calculate point values at points of our choice. The number of points we needed for non-adaptive algorithms was higher than that in the adaptive case. In this paper we improve on DeVore et al. (2011) and show that in the non-adaptive case one can use the same number of points (up to a multiplicative constant depending on ) that we need in the adaptive case. [Copyright &y& Elsevier]

Abstract: In this paper we present explicit formulae for the multivariate Lagrange–Newton transformation and its inverse with respect to points  , where , and belong to the field . Moreover, we derive fast algorithms for computing these transformations. The running time of them is base operations from . [Copyright &y& Elsevier]

We study the complexity of Fredholm problems (I−Tk)u=f of the second kind on Id=[0,1]d, where Tk is an integral operator with kernel k. Previous work on the complexity of this problem has assumed either that we had complete information about k or that k and f had the same smoothness. In addition, most of this work has assumed that the information about k and f was exact. In this paper, we assume that k and f have different smoothness; more precisely, we assume that f∈Wr,p(Id) with r>d/p and that k∈Ws,∞(I2d) with s>0. In addition, we assume that our information about k and f is contaminated by noise. We find that the nth minimal error is Θ(n−μ+δ), where μ=min{r/d,s/(2d)} and δ is a bound on the noise. We prove that a noisy modified finite element method has nearly minimal error. This algorithm can be efficiently implemented using multigrid techniques. We thus find tight bounds on the ϵ-complexity for this problem. These bounds depend on the cost c(δ) of calculating a δ-noisy information value. As an example, if the cost of a δ-noisy evaluation is proportional to δ−t, then the ϵ-complexity is roughly (1/ϵ)t+1/μ. [Copyright &y& Elsevier]