Nonlinear Equations - UFRJ

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142 [CH. 10: HOMOTOPY P n+1 x i [N(F t, X i)] x i+1 z t R t i t i+1 Figure 10.1: The homotopy step. This picture is in projective space. For the picture in linear space, the reader can imagine that he stands at the origin. The points X i+1 , N(F ti+1 , X i ) and the origin are in the same complex line. Lemma 10.8. Assume the conditions of Theorem 10.5. For short, write β = β(F ti , X i ) and µ = µ(F ti , X i ). If D 3/2 2 βµ ≤ a 0, (10.5) ‖F t − F s ‖ ≤ ɛ 1 , and (10.6) µ 2ɛ 2 Φ t,s (X) ≤ ∀s ∈ [t D 3/2 i , t i+1 ], (10.7) µ then the following estimates hold for all s ∈ [t i , t i+1 ]: D 3/2 µ(f s , x i ) β(F s , X i ) β(F s , X i ) ≤ ≤ ≥ µ 1 − ɛ 1 (10.8) 2 (1 − ɛ 1 )α D 3/2 µ (10.9) 2 (ɛ 2 − a 0 ) √ 1 − ɛ 2 1 D 3/2 (1 + ɛ 1 )µ (10.10) 2 β(F s, X i )µ(f s , x i ) ≤ α (10.11)
[SEC. 10.2: PROOF OF THEOREM 10.5 143 Proof. Because of (10.1), ‖F ti ‖, ‖F s ‖ ≥ 1 and F ti ∥‖F ti ‖ − F s ‖F s ‖ ∥ ≤ ‖F t i − F s ‖ ≤ ɛ 1 µ Then Lemma 8.22 with u = 0, v = ɛ 1 implies (10.8). For (10.9) and (10.10), we write β(F s , X i ) = DF s (X i ) † DF ti (X i ) ( DF ti (X i ) † F ti (X i )+ +DF ti (X i ) † (F s (X i ) − F ti (X i )) ) . Let v = ‖Fs−Ft i ‖ ‖F ti ‖ µ. By (10.2) ‖F ti ‖ > 1 so that v ≤ ɛ 1 . From Lemma 9.5, we deduce that √ 1 − ɛ 2 1 1 + ɛ 1 ( 2ɛ2 D 3/2 µ − β ) ≤ β(F s , X i ) ≤ β + (10.11) is ob- Now equation (10.3) implies (10.9) and (10.10). tained by multiplying (10.8) and (10.9). Lemma 10.9. Under the conditions of Lemma 10.8, and D 3/2 µ(f s , [N(F s , X i )]) β(F s , N(F s , X i )) ≤ ≤ 2ɛ2 D 3/2 µ 1 − ɛ 1 µ 1 − ɛ 1 − πa 0 / √ D (10.12) 2 1 − ɛ 1 1 − α D 3/2 µ ψ(α) α2 (10.13) 2 β(F s, N(F s , X i ))µ(f s , [N(F s , X i )]) ≤ (1 − (1 − ɛ 1 )α/2) a 0 (10.14) Proof. The proof of (10.12) is similar to the one of (10.8). We need to keep in mind that X ti is scaled but N(F s , X ti ) is not assumed scaled. Anyway, we know that ‖X ti − N(F s , X ti )‖ = β.
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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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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 173 and 174: Appendix A Open Problems, by Carlos
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Page 182 and 183: 166 BIBLIOGRAPHY [12] , Smale’s 1
Page 184 and 185: 168 BIBLIOGRAPHY [42] Michael R. Ga
Page 186 and 187: 170 BIBLIOGRAPHY [69] , Complexity
Page 189 and 190: Glossary of notations As a general
Page 191 and 192: Index algorithm discrete, x Homotop
Page 193: INDEX 177 complexity of homotopy, 1
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Nonlinear Equations - UFRJ

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