Nonlinear Equations - UFRJ

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116 [CH. 8: CONDITION NUMBER THEORY Lemma 8.15. Let L = L ix : F i → Tx(M/H) ∗ be defined by 〉 L ix (f) : u ↦→ √ 〈f(·), 1 1 √ P x D¯x K(·, x)ū di K(x, x) F i . Then L is onto, and L | ker L ⊥ satisfies: √ ei d i ‖f‖ ≤ ‖L | ker L ⊥f‖ T ∗ x (M/H) ≤ ‖f‖ Again, L x : F 1 × · · · × F s → L(T ⎡ x M, C ⎤ s ), L 1x (f 1 ) (f 1 , . . . , f s ) ↦→ ⎢ ⎣ . L sx (f s ) ⎥ ⎦ . As before, Lemma 8.16. L x is onto, and (min √ e i /d i ) ‖h‖ ≤ ‖L | ker L ⊥h‖ ≤ ‖h‖ The condition number of f at x is defined by µ(f, x) = ‖f‖ ( σ min(n,s) (L x (f)) ) −1 . By construction and the implicit (inverse) function theorem, Proposition 8.17. Let f t ∈ F 1 × · · · × F s a one-parameter family, with f 0 (x 0 ) = 0. If s ≤ n, then there is locally a solution x t , f t (x t ) with 1 ‖ẋ t ‖ ≤ min √ µ(f 0 , x t )‖ f ˙ t ‖ d i Moreover, we have: Theorem 8.18 (Condition number theorem). Let f ∈ F x = (F 1 × · · · × F s ) x .
[SEC. 8.5: CONDITION NUMBERS IN GENERAL 117 Let r = min(n, s). Then ( √ ) ei min min d i g∈Fx Proof. rank(D(f+h)(x))
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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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Page 82 and 83: 66 [CH. 5: REPRODUCING KERNEL SPACE
Page 88 and 89: Chapter 6 Exponential sums and spar
Page 90 and 91: 74 [CH. 6: EXPONENTIAL SUMS AND SPA
Page 98 and 99: Chapter 7 Newton Iteration and Alph
Page 100 and 101: 84 [CH. 7: NEWTON ITERATION As long
Page 102 and 103: 86 [CH. 7: NEWTON ITERATION Proposi
Page 104 and 105: 88 [CH. 7: NEWTON ITERATION 1 y =
Page 106 and 107: 90 [CH. 7: NEWTON ITERATION t 1 t 2
Page 108 and 109: 92 [CH. 7: NEWTON ITERATION The Tay
Page 110 and 111: 94 [CH. 7: NEWTON ITERATION 2 63 2
Page 112 and 113: 96 [CH. 7: NEWTON ITERATION Exercis
Page 114 and 115: 98 [CH. 7: NEWTON ITERATION The con
Page 116 and 117: 100 [CH. 7: NEWTON ITERATION Let us
Page 118 and 119: 102 [CH. 7: NEWTON ITERATION Passin
Page 120 and 121: 104 [CH. 7: NEWTON ITERATION 13−3
Page 122 and 123: 106 [CH. 7: NEWTON ITERATION Proof.
Page 124 and 125: 108 [CH. 8: CONDITION NUMBER THEORY
Page 138 and 139: 122 [CH. 9: THE PSEUDO-NEWTON OPERA
Page 152 and 153: 136 [CH. 10: HOMOTOPY form for x i+
Page 154 and 155: 138 [CH. 10: HOMOTOPY Again, f t is
Page 156 and 157: 140 [CH. 10: HOMOTOPY We will need
Page 158 and 159: 142 [CH. 10: HOMOTOPY P n+1 x i [N(
Page 160 and 161: 144 [CH. 10: HOMOTOPY Let d Riem de
Page 162 and 163: 146 [CH. 10: HOMOTOPY Lemma 10.13.
Page 164 and 165: 148 [CH. 10: HOMOTOPY above, the se
Page 166 and 167: 150 [CH. 10: HOMOTOPY Corollary 10.
Page 168 and 169: 152 [CH. 10: HOMOTOPY by the geomet
Page 170 and 171: 154 [CH. 10: HOMOTOPY 2. The condit
Page 173 and 174: Appendix A Open Problems, by Carlos
Page 175 and 176: [SEC. A.3: EQUIDISTRIBUTION OF ROOT
Page 177 and 178: [SEC. A.5: EXTENSION OF THE ALGORIT
Page 179: [SEC. A.7: INTEGER ZEROS OF A POLYN
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166 BIBLIOGRAPHY [12] , Smale’s 1
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168 BIBLIOGRAPHY [42] Michael R. Ga
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170 BIBLIOGRAPHY [69] , Complexity
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Glossary of notations As a general
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Index algorithm discrete, x Homotop
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INDEX 177 complexity of homotopy, 1
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Nonlinear Equations - UFRJ

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