Nonlinear Equations - UFRJ

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154 [CH. 10: HOMOTOPY 2. The condition length L(f t , z t ; 0, 1) is bounded by a uniform polynomial in n, D, dim H d . It is unknown how to do that in general. A deterministic algorithm producing such paths within expected polynomial time would provide an affirmative answer for Smale’s 17 th problem. Here is a possibility: pick a fixed initial zero (say X 0 = Z 0 = e 0 ), a fixed initial polynomial having Z 0 as a root, and follow a linear path. For instance, ⎡ √ ) ⎤ d1 (X d1−1 0 X 1 − X d1 0 ⎢ ⎥ or F 0 (X) = ⎢ ⎣ . √ dn (X dn−1 0 X n − X dn 0 ⎡ ⎢ ˜F 0 (X) = F 1 (X) − F 1 (e 0 ) ⎣ X d1 0 . X dn 0 ⎥ ) ⎦ ⎤ ⎥ ⎦ . (10.20) Then, one has to integrate the expected length of the path. None of those linear paths is known to be polynomially bounded length in average. Another possibility is to look for more insight. metric on V \ Σ ′ is 〈·, ·〉 ′ f,x = µ 2 (f, x)〈·, ·〉 f,x The condition This reduces complexity to lengths. This new Riemannian metric is akin to the hyperbolic metric in Poincaré plane y > 2, 〈·, ·〉 Poincaré x,y = y −2 〈·, ·〉 2 . A new difficulty arises. All geometry books seem to be written under differentiability assumptions for the metric. Here, µ is not differentiable at all points. (See fig. 10.3) The differential equation defining geodesics has to be replaced by a differential inequality [21]. In [8,9] it was proved in the linear case that the condition number is self-convex. This means that log µ is a convex function along geodesics in the condition metric.
[SEC. 10.4: THE GEOMETRIC VERSION... 155 B A Figure 10.3: The condition metric for diagonal, real matrices is min(|x|, |y|) −2 〈·, ·〉. Geodesics in the smooth part are easy to construct. But what is the shortest path from A to B? In particular, the maximum of µ along a geodesic arc is attained at the extremities. The non-linear case is still open. Starting the homotopy at a global minimum of µ (such as (10.20)), one would have a guarantee that the condition number along the path is bounded above by the condition number of the target F 1 . Moreover, a ‘short’ geodesic between F 1 and a global minimum is known to exist [14]. There is nothing very particular about geodesics, except that they minimize distance. One can settle for a short path, that is a piecewise linear path with condition length bounded by a uniform polynomial in the input size. This book finishes with a question. Question 10.22. Given a random f 1 , is it possible to deterministically find a starting pair (f 0 , z 0 ) and a short path to (f 1 , z 1 ) in polynomial time?
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Nonlinear Equations
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Copyright © 2011 by Gregorio Malaj
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Foreword I added together the ratio
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ix another book with a systematic p
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Contents Foreword vii 1 Counting so
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CONTENTS xiii 10 Homotopy 135 10.1
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2 [CH. 1: COUNTING SOLUTIONS if and
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4 [CH. 1: COUNTING SOLUTIONS connec
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6 [CH. 1: COUNTING SOLUTIONS 1.2 Sh
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8 [CH. 1: COUNTING SOLUTIONS has ex
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10 [CH. 1: COUNTING SOLUTIONS equat
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Chapter 2 The Nullstellensatz The s
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14 [CH. 2: THE NULLSTELLENSATZ prin
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16 [CH. 2: THE NULLSTELLENSATZ or a
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18 [CH. 2: THE NULLSTELLENSATZ 3. T
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20 [CH. 2: THE NULLSTELLENSATZ and
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22 [CH. 2: THE NULLSTELLENSATZ 1. T
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24 [CH. 2: THE NULLSTELLENSATZ Proo
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26 [CH. 2: THE NULLSTELLENSATZ To e
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28 [CH. 2: THE NULLSTELLENSATZ for
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30 [CH. 2: THE NULLSTELLENSATZ 2.7
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32 [CH. 2: THE NULLSTELLENSATZ Coro
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34 [CH. 3: TOPOLOGY AND ZERO COUNTI
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Chapter 4 Differential forms Throug
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44 [CH. 4: DIFFERENTIAL FORMS Prope
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46 [CH. 4: DIFFERENTIAL FORMS Lemma
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48 [CH. 4: DIFFERENTIAL FORMS 2. cl
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50 [CH. 4: DIFFERENTIAL FORMS be me
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52 [CH. 4: DIFFERENTIAL FORMS Expan
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54 [CH. 4: DIFFERENTIAL FORMS We co
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56 [CH. 5: REPRODUCING KERNEL SPACE
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Chapter 6 Exponential sums and spar
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74 [CH. 6: EXPONENTIAL SUMS AND SPA
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Chapter 7 Newton Iteration and Alph
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84 [CH. 7: NEWTON ITERATION As long
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86 [CH. 7: NEWTON ITERATION Proposi
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88 [CH. 7: NEWTON ITERATION 1 y =
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90 [CH. 7: NEWTON ITERATION t 1 t 2
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92 [CH. 7: NEWTON ITERATION The Tay
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94 [CH. 7: NEWTON ITERATION 2 63 2
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96 [CH. 7: NEWTON ITERATION Exercis
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98 [CH. 7: NEWTON ITERATION The con
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100 [CH. 7: NEWTON ITERATION Let us
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102 [CH. 7: NEWTON ITERATION Passin
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Page 162 and 163: 146 [CH. 10: HOMOTOPY Lemma 10.13.
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Page 166 and 167: 150 [CH. 10: HOMOTOPY Corollary 10.
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Page 189 and 190: Glossary of notations As a general
Page 191 and 192: Index algorithm discrete, x Homotop
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Nonlinear Equations - UFRJ

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