Newton's method and the Computational Complexity of the Fundamental Theorem of Algebra

Batra, Prashant

doi:10.1016/j.entcs.2008.03.016

Cited by 7 publications

(5 citation statements)

References 27 publications

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“…, j k ), N 1 and N 0 1 have n À 1 leaves. In all cases, L s ðN 1 Þ ¼ L s ðN 0 1 Þ by Lemmas 32(2), 33(2) and 34 (2), and therefore, by the induction hypothesis, N 1 ffi N 0 1 . Finally, by Remark 11, N and N 0 are obtained from N 1 and N 0 1 by applying the same expansion U À1 , T À1 , or H À1 , and they are isomorphic.…”

Section: Example 29mentioning

confidence: 87%

“…These vectors L(N ) and L s (N ) can be used then to define metrics for fully resolved and arbitrary TCTC-networks, respectively, from metrics for real-valued vectors. The metrics obtained in this way can be understood as generalizations to TCTC n of the (non-splitted or splitted) nodal metrics for phylogenetic trees and they can be computed in low polynomial time if the metric used to compare the vectors can be done so: this is the case, for instance, when this metric is the Manhattan or the Euclidean metric (in the last case, computing the square root with O(10 m+n ) significant digits [2], which should be more than enough).…”

Section: Discussionmentioning

confidence: 99%

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Path lengths in tree-child time consistent hybridization networks

Cardona¹,

Llabrés²,

Rosselló³

et al. 2010

Information Sciences

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a b s t r a c tHybridization networks are representations of evolutionary histories that allow for the inclusion of reticulate events like recombinations, hybridizations, or lateral gene transfers. The recent growth in the number of hybridization network reconstruction algorithms has led to an increasing interest in the definition of metrics for their comparison that can be used to assess the accuracy or robustness of these methods. In this paper we establish some basic results that make it possible the generalization to tree-child time consistent (TCTC) hybridization networks of some of the oldest known metrics for phylogenetic trees: those based on the comparison of the vectors of path lengths between leaves. More specifically, we associate to each hybridization network a suitably defined vector of 'splitted' path lengths between its leaves, and we prove that if two TCTC hybridization networks have the same such vectors, then they must be isomorphic. Thus, comparing these vectors by means of a metric for real-valued vectors defines a metric for TCTC hybridization networks. We also consider the case of fully resolved hybridization networks, where we prove that simpler, 'non-splitted' vectors can be used.

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Section: Example 29mentioning

confidence: 87%

Section: Discussionmentioning

confidence: 99%

Path lengths in tree-child time consistent hybridization networks

Cardona¹,

Llabrés²,

Rosselló³

et al. 2010

Information Sciences

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“…The same holds true if we choose the scaled polynomial p 1 (z) := P 1 (σ 1 z)/σ 2 = d j=0 a j z j . Starting with p 1 we obtain the resultant R = R(s) = res t (G(t, s), H (t, s)) via equations (8), (7), (5), (4), (3) and (2). With a value k 0 such that R(k 0 ) = 0, we may consider the homotopy of square-free polynomials…”

Section: How To Proceed Along the Safe Path?mentioning

confidence: 99%

“…[18]. We avoid a general discussion of other uses of Newton's method in root-finding; the reader may find some pointers to the literature in our survey [2].…”

Section: Our Iteration Along a Homotopymentioning

confidence: 99%

Globally Convergent, Iterative Path-Following for Algebraic Equations

Batra

2010

Math.Comput.Sci.

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Homotopy methods are of great importance for the solution of systems of equations. It is a major problem to ensure well-defined iterations along the homotopy path. Many investigations have considered the complexity of path-following methods depending on the unknown distance of some given path to the variety of ill-posed problems. It is shown here that there exists a construction method for safe paths for a single algebraic equation. A safe path may be effectively determined with bounded effort. Special perturbation estimates for the zeros together with convergence conditions for Newton's method in simultaneous mode allow our method to proceed on the safe path. This yields the first globally convergent, never-failing, uniformly iterative path-following algorithm. The maximum number of homotopy steps in our algorithm reaches a theoretical bound forecast by Shub and Smale i.e., the number of steps is at most quadratic in the condition number. A constructive proof of the fundamental theorem of algebra meeting demands by Gauß, Kronecker and Weierstraß is a consequence of our algorithm.

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“…For p 2, the cost of computing d s p (T 1 , T 2 ), for T 1 , T 2 ∈ T n , as the p-th root of d s p (T 1 , T 2 ) p will depend on the accuracy with which this root is computed. For instance, using the Newton method to compute it with an accuracy of an 1/2 h -th of its value has a cost of O(p 2 log(p) log(hp)); see, for instance, [4]. So, in practice, for small p and not too large h, this step will be dominated by the computation of d s p (T 1 , T 2 ) p , and the total cost will be O(n 2 ) (we understand in this case log(p) as part of the constant factor).…”

Section: ⊓ ⊔mentioning

confidence: 99%

Nodal distances for rooted phylogenetic trees

et al. 2009

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Dissimilarity measures for (possibly weighted) phylogenetic trees based on the comparison of their vectors of path lengths between pairs of taxa, have been present in the systematics literature since the early seventies. For rooted phylogenetic trees, however, these vectors can only separate non-weighted binary trees, and therefore these dissimilarity measures are metrics only on this class of rooted phylogenetic trees. In this paper we overcome this problem, by splitting in a suitable way each path length between two taxa into two lengths. We prove that the resulting splitted path lengths matrices single out arbitrary rooted phylogenetic trees with nested taxa and arcs weighted in the set of positive real numbers. This allows the definition of metrics on this general class of rooted phylogenetic trees by comparing these matrices through metrics in spaces M(n)(R) of real-valued n x n matrices. We conclude this paper by establishing some basic facts about the metrics for non-weighted phylogenetic trees defined in this way using L(p) metrics on M(n)(R), with p [epsilon] R(>0).

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Newton's method and the Computational Complexity of the Fundamental Theorem of Algebra

Cited by 7 publications

References 27 publications

Path lengths in tree-child time consistent hybridization networks

Path lengths in tree-child time consistent hybridization networks

Globally Convergent, Iterative Path-Following for Algebraic Equations

Nodal distances for rooted phylogenetic trees

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