Nonlinear Equations - UFRJ

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112 [CH. 8: CONDITION NUMBER THEORY Theorem 8.6 (Condition number theorem, unmixed). Let f ∈ F s . Let r = min(n, s). Then µ(f, x) −1 = ‖g‖ min g∈Fx ‖f‖ . rank(D(f+g)(x))
[SEC. 8.4: CONDITION NUMBERS FOR HOMOGENEOUS SYSTEMS 113 8.4 Condition numbers for homogeneous systems We consider now a possibly unmixed situation. Let f ∈ H d1 × · · · × H dn , where each f i is homogeneous in n + 1 variables. Let M = C n+1 \ {0}, H = C × and thus M/H = P n . Projective space is endowed with the Fubini-Study metric 〈·, ·〉. Each of the H di has reproducing kernel K i (x, y) = (x 0 ȳ 0 + · · · + x n ȳ n ) di and therefore (Exercise 5.5) induces a metric 〈·, ·〉 P n ,i = d i 〈·, ·〉. Lemma 8.8. Let L = L ix : H di → Tx(P ∗ n ) be defined by 〉 L ix (f) : u ↦→ √ 〈f(·), 1 1 √ P x D¯x K(·, x)ū di K(x, x) Then L is onto, and L | ker L ⊥ is an isometry. H di . Proof. If we assume the 〈·, ·〉 P n ,i norm on Tx(P ∗ n ), Lemma 8.4 implies that the operator above is onto and L | ker L ⊥ is d −1/2 i times an isometry. For vectors, the relation between Fubini-Study and H di -induced norm is ‖u‖ = √ 1 ‖u‖ i . di For covectors, it is therefore ‖ω‖ = √ d i ‖ω‖ i . Hence, we deduce that L | ker L ⊥ is an isometry, when Fubini-Study metric is assumed on P n . Now we define As before, L x : F s → L(T x M, C s ), ⎡ ⎤ L 1x (f 1 ) (f 1 , . . . , f s ) ↦→ ⎢ ⎣ . L sx (f s ) ⎥ ⎦ .
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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 88 and 89: Chapter 6 Exponential sums and spar
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Page 98 and 99: Chapter 7 Newton Iteration and Alph
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Page 108 and 109: 92 [CH. 7: NEWTON ITERATION The Tay
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Page 112 and 113: 96 [CH. 7: NEWTON ITERATION Exercis
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Page 122 and 123: 106 [CH. 7: NEWTON ITERATION Proof.
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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.
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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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[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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