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Book Categories |
| 1 | Introduction | 3 |
| 1.1 | Basic Concepts | 3 |
| 1.2 | Formulation of the Problem | 7 |
| 1.2.1 | Computational Methods | 8 |
| 1.2.2 | Optimal Complexity Methods | 16 |
| 1.2.3 | Asymptotic Setting | 18 |
| 1.2.4 | Exercises | 18 |
| 1.3 | Annotations | 19 |
| Bibliography | 20 | |
| 2 | Nonlinear Equations | 23 |
| 2.1 | Univariate Problems | 25 |
| 2.1.1 | Optimality of the Bisection Method | 27 |
| 2.1.2 | Root Criterion in C[superscript infinity] | 34 |
| 2.1.3 | Residual Criterion in W[superscript r subscript infinity] | 37 |
| 2.1.4 | General Error Criterion in C[superscript infinity] and W[superscript infinity subscript r] | 44 |
| 2.1.5 | Polynomial Equations | 46 |
| 2.1.6 | Asymptotic Optimality of the Bisection Method | 56 |
| 2.1.7 | Exercises | 70 |
| 2.2 | Multivariate Problems | 71 |
| 2.2.1 | Functions with Nonzero Topological Degree | 71 |
| 2.2.2 | Lipschitz Functions | 83 |
| 2.2.3 | Exercises | 97 |
| 2.3 | Annotations | 98 |
| 2.3.1 | Overview and brief history | 98 |
| 2.3.2 | Specific Comments | 106 |
| Bibliography | 108 | |
| 3 | Fixed Points-Contractive Functions | 121 |
| 3.1 | Univariate Problems | 122 |
| 3.1.1 | Relative Error Criterion | 123 |
| 3.1.2 | Absolute Error Criterion | 137 |
| 3.1.3 | Exercises | 138 |
| 3.2 | Multivariate Problems | 139 |
| 3.2.1 | A Constructive Lemma | 140 |
| 3.2.2 | Ball Iteration | 142 |
| 3.2.3 | Ellipsoid Iteration | 144 |
| 3.2.4 | Centroid Method | 155 |
| 3.2.5 | Numerical Tests | 157 |
| 3.2.6 | Exercises | 162 |
| 3.3 | Annotations | 162 |
| 3.3.1 | Specific Comments | 163 |
| Bibliography | 165 | |
| 4 | Fixed Points-Noncontractive Functions | 171 |
| 4.1 | Univariate Problems | 173 |
| 4.1.1 | Minimal Cardinality Number | 173 |
| 4.1.2 | The FPE-A Method | 175 |
| 4.1.3 | Exercises | 177 |
| 4.2 | Multivariate Problems | 178 |
| 4.2.1 | Absolute Error Criterion | 178 |
| 4.2.2 | Exercises | 187 |
| 4.3 | Annotations | 188 |
| 4.3.1 | General Comments | 188 |
| 4.3.2 | Residual Error Criterion | 188 |
| 4.3.3 | Specific Comments | 190 |
| Bibliography | 190 | |
| 5 | Topological Degree Computation | 193 |
| 5.1 | Two-Dimensional Lipschitz Functions | 194 |
| 5.1.1 | Basic Definitions | 195 |
| 5.1.2 | Lower Bound on the Minimal Cardinality Number | 196 |
| 5.1.3 | Minimal Cardinality Number | 199 |
| 5.1.4 | Complexity of the Problem | 207 |
| 5.1.5 | Numerical Experiments | 208 |
| 5.1.6 | Exercises | 211 |
| 5.2 | Lipschitz Functions in d Dimensions | 211 |
| 5.2.1 | Basic Definitions | 212 |
| 5.2.2 | Information N* | 213 |
| 5.2.3 | Algorithm [phis]* Using Information N* | 214 |
| 5.2.4 | Lower Bound on the Minimal | 217 |
| 5.2.5 | Exercises | 226 |
| 5.3 | Annotations | 226 |
| 5.3.1 | Specific Comments | 227 |
| Bibliography | 228 | |
| Index | 233 |
| Price | Condition | Delivery | Seller | Action |
| $101.60 | Digital |
| WonderClub |
Joe Richard
reviewed Optimal Solution of Nonlinear Equations on December 30, 2013
Optimal Solution of Nonlinear Equations
If you read this book as first time in order to understand Computational Complexity, then you're probably not going to like it. You need to read Sipser's textbook before you read this book and make sure that you have a strong background in discrete mathematics (if not, then see Rosen's textbook in Discrete Mathematics). This book is recommended by best people in complexity and algorithms such as Scott Aaronson, Sipser, etc. You need to solve as many exercises as you go.
The book has five parts:
Part I: Algorithms
Part II: Logic
Part III: P and NP
Part IV: Inside P
Part V: Beyond NP
The most important parts are in Part I, III.
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