13,871 research outputs found

    Novel Approach to Real Polynomial Root-finding and Matrix Eigen-solving

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    Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial has no nonreal roots, but typically nonreal roots are much more numerous than the real ones. We dramatically accelerate the known algorithms in this case by exploiting the correlation between the computations with matrices and polynomials, extending the techniques of the matrix sign iteration, and exploiting the structure of the companion matrix of the input polynomial. We extend some of the proposed techniques to the approximation of the real eigenvalues of a real nonsymmetric matrix.Comment: 17 pages, added algorithm

    A continuous analogue of the tensor-train decomposition

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    We develop new approximation algorithms and data structures for representing and computing with multivariate functions using the functional tensor-train (FT), a continuous extension of the tensor-train (TT) decomposition. The FT represents functions using a tensor-train ansatz by replacing the three-dimensional TT cores with univariate matrix-valued functions. The main contribution of this paper is a framework to compute the FT that employs adaptive approximations of univariate fibers, and that is not tied to any tensorized discretization. The algorithm can be coupled with any univariate linear or nonlinear approximation procedure. We demonstrate that this approach can generate multivariate function approximations that are several orders of magnitude more accurate, for the same cost, than those based on the conventional approach of compressing the coefficient tensor of a tensor-product basis. Our approach is in the spirit of other continuous computation packages such as Chebfun, and yields an algorithm which requires the computation of "continuous" matrix factorizations such as the LU and QR decompositions of vector-valued functions. To support these developments, we describe continuous versions of an approximate maximum-volume cross approximation algorithm and of a rounding algorithm that re-approximates an FT by one of lower ranks. We demonstrate that our technique improves accuracy and robustness, compared to TT and quantics-TT approaches with fixed parameterizations, of high-dimensional integration, differentiation, and approximation of functions with local features such as discontinuities and other nonlinearities

    Discovering the roots: Uniform closure results for algebraic classes under factoring

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    Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the number of roots rr is small but the multiplicities are exponentially large. Our method sets up a linear system in rr unknowns and iteratively builds the roots as formal power series. For an algebraic circuit f(x1,…,xn)f(x_1,\ldots,x_n) of size ss we prove that each factor has size at most a polynomial in: ss and the degree of the squarefree part of ff. Consequently, if f1f_1 is a 2Ξ©(n)2^{\Omega(n)}-hard polynomial then any nonzero multiple ∏ifiei\prod_{i} f_i^{e_i} is equally hard for arbitrary positive eie_i's, assuming that βˆ‘ideg(fi)\sum_i \text{deg}(f_i) is at most 2O(n)2^{O(n)}. It is an old open question whether the class of poly(nn)-sized formulas (resp. algebraic branching programs) is closed under factoring. We show that given a polynomial ff of degree nO(1)n^{O(1)} and formula (resp. ABP) size nO(log⁑n)n^{O(\log n)} we can find a similar size formula (resp. ABP) factor in randomized poly(nlog⁑nn^{\log n})-time. Consequently, if determinant requires nΞ©(log⁑n)n^{\Omega(\log n)} size formula, then the same can be said about any of its nonzero multiples. As part of our proofs, we identify a new property of multivariate polynomial factorization. We show that under a random linear transformation Ο„\tau, f(Ο„xβ€Ύ)f(\tau\overline{x}) completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. This with allRootsNI are the techniques that help us make progress towards the old open problems, supplementing the large body of classical results and concepts in algebraic circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \& Burgisser, FOCS 2001).Comment: 33 Pages, No figure
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