Nonlinear Equations - UFRJ

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170 BIBLIOGRAPHY [69] , Complexity of Bezout’s theorem. VI. Geodesics in the condition (number) metric, Found. Comput. Math. 9 (2009), no. 2, 171–178, DOI 10.1007/s10208-007-9017-6. [70] Michael Shub and Steve Smale, Complexity of Bézout’s theorem. I. Geometric aspects, J. Amer. Math. Soc. 6 (1993), no. 2, 459–501, DOI 10.2307/2152805. [71] M. Shub and S. Smale, Complexity of Bezout’s theorem. II. Volumes and probabilities, Computational algebraic geometry (Nice, 1992), Progr. Math., vol. 109, Birkhäuser Boston, Boston, MA, 1993, pp. 267–285. [72] Michael Shub and Steve Smale, Complexity of Bezout’s theorem. III. Condition number and packing, J. Complexity 9 (1993), no. 1, 4–14, DOI 10.1006/jcom.1993.1002. Festschrift for Joseph F. Traub, Part I. [73] , Complexity of Bezout’s theorem. IV. Probability of success; extensions, SIAM J. Numer. Anal. 33 (1996), no. 1, 128–148, DOI 10.1137/0733008. [74] M. Shub and S. Smale, Complexity of Bezout’s theorem. V. Polynomial time, Theoret. Comput. Sci. 133 (1994), no. 1, 141–164, DOI 10.1016/0304- 3975(94)90122-8. Selected papers of the Workshop on Continuous Algorithms and Complexity (Barcelona, 1993). [75] S. Smale, Topology and mechanics. I, Invent. Math. 10 (1970), 305–331. [76] Steve Smale, On the efficiency of algorithms of analysis, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 87–121, DOI 10.1090/S0273-0979-1985-15391-1. [77] , Newton’s method estimates from data at one point, computational mathematics (Laramie, Wyo., 1985), Springer, New York, 1986, pp. 185–196. [78] , Mathematical problems for the next century, Math. Intelligencer 20 (1998), no. 2, 7–15, DOI 10.1007/BF03025291. [79] , Mathematical problems for the next century, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 271–294. [80] Andrew J. Sommese and Charles W. Wampler II, The numerical solution of systems of polynomials, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. Arising in engineering and science. [81] A. M. Turing, Rounding-off errors in matrix processes, Quart. J. Mech. Appl. Math. 1 (1948), 287–308. [82] Constantin Udrişte, Convex functions and optimization methods on Riemannian manifolds, Mathematics and its Applications, vol. 297, Kluwer Academic Publishers Group, Dordrecht, 1994. [83] Jan Verschelde, Polyhedral methods in numerical algebraic geometry, Interactions of classical and numerical algebraic geometry, Contemp. Math., vol. 496, Amer. Math. Soc., Providence, RI, 2009, pp. 243–263. [84] Wang Xinghua, Some result relevant to Smale’s reports, in: M.Hirsch, J. Marsden and S. Shub(eds): From Topolgy to Computation: Proceedings of Smalefest, Springer, new-york, 1993, pp. 456-465.
BIBLIOGRAPHY 171 [85] Hermann Weyl, The theory of groups and quantum mechanics, Dover Publications, New York, 1949. XVII+422 pp. [86] J. H. Wilkinson, Rounding errors in algebraic processes, Dover Publications Inc., New York, 1994. Reprint of the 1963 original [Prentice-Hall, Englewood Cliffs, NJ; MR0161456 (28 #4661)].
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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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104 [CH. 7: NEWTON ITERATION 13−3
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106 [CH. 7: NEWTON ITERATION Proof.
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108 [CH. 8: CONDITION NUMBER THEORY
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Page 136 and 137: 120 [CH. 8: CONDITION NUMBER THEORY
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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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Page 182 and 183: 166 BIBLIOGRAPHY [12] , Smale’s 1
Page 184 and 185: 168 BIBLIOGRAPHY [42] Michael R. Ga
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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