Numerical analysis Books

Book SynopsisLearning Data Science is the first book to cover foundational skills in both programming and statistics that encompass the entire data science lifecycle: the process of collecting, wrangling, analyzing, and drawing conclusions from data.

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Book SynopsisArising from courses taught by the authors, this largely self-contained treatment is ideal for mathematicians who are interested in applications or for students from applied fields who want to understand the mathematics behind their subject. Early chapters cover Fourier analysis, functional analysis, probability and linear algebra, all of which have been chosen to prepare the reader for the applications to come. The book includes rigorous proofs of core results in compressive sensing and wavelet convergence. Fundamental is the treatment of the linear system y=Îx in both finite and infinite dimensions. There are three possibilities: the system is determined, overdetermined or underdetermined, each with different aspects. The authors assume only basic familiarity with advanced calculus, linear algebra and matrix theory and modest familiarity with signal processing, so the book is accessible to students from the advanced undergraduate level. Many exercises are also included.Trade Review'Damelin and Miller provide a very detailed and thorough treatment of all the important mathematics related to signal processing. This includes the required background information found in elementary mathematics courses, so their book is really self-contained. The style of writing is suitable not only for mathematicians, but also for practitioners from other areas. Indeed, Damelin and Miller managed to write their text in a form that is accessible to nonspecialists, without giving up mathematical rigor.' Kai Diethelm, Computing Reviews'In the last 20 years or so, many books on wavelets have been published; most of them deal with wavelets from either the engineering or the mathematics perspective, but few try to connect the two viewpoints. The book under review falls under the last category … Overall, the book is a good addition to the literature on engineering mathematics.' Ahmed I. Zayed, Mathematical ReviewsTable of Contents1. Introduction; 2. Normed vector spaces; 3. Analytic tools; 4. Fourier series; 5. Fourier transforms; 6. Compressive sensing; 7. Discrete transforms; 8. Linear filters; 9. Windowed Fourier transforms, continuous wavelets, frames; 10. Multiresolution analysis; 11. Discrete wavelet theory; 12. Biorthogonal filters and wavelets; 13. Parsimonious representation of data; Bibliography; Index.

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Book SynopsisProvides an introduction to numerical methods for students in engineering courses. This book covers the solution of equations, interpolation and data fitting, solution of differential equations, eigenvalue problems and optimisation. The algorithms are implemented in Python 3, a high-level programming language that rivals MATLAB® in readability and ease of use.Trade Review'… a practical introduction, pushing the theory as far in the background as possible.' The European Mathematical Society (euro-math-soc.eu)'This book is nicely focused on the most frequently encountered types of numerical problems that scientists and engineers usually face and the most common and robust algorithms for solving them. The text is just the right size for a semester-long course for upper-division undergraduates or first-year graduate students … this is a well-written text that is logically organized, attractively presented, and supported with challenging problems.' Anthony J. Duben, Computing ReviewsTable of Contents1. Introduction to Python; 2. Systems of linear algebraic equations; 3. Interpolation and curve fitting; 4. Roots of equations; 5. Numerical differentiation; 6. Numerical integration; 7. Initial value problems; 8. Two-point boundary value problems; 9. Symmetric matrix eigenvalue problems; 10. Introduction to optimization.

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Book SynopsisThis comprehensive, accessible textbook covers the basics of signal processing, building up from fundamental principles to practical applications. It uses engineering notation to make mathematical concepts easy to follow, includes numerous homework problems and is accompanied by an extensive Mathematica® companion and instructor solutions manual.Trade Review'This is a major book about a serious subject - the combination of engineering and mathematics that goes into modern signal processing: discrete time, continuous time, sampling, filtering, and compression. The theory is beautiful and the applications are so important and widespread.' Gil Strang, Massachusetts Institute of Technology'A refreshing new approach to teaching the fundamentals of signal processing. Starting from basic concepts in algebra and geometry, [the authors] bring the reader to deep understandings of modern signal processing. Truly a gem!' Rico Malvar, Microsoft Research'A wonderful book that connects together all the elements of modern signal processing … it's all here and seamlessly integrated, along with a summary of history and developments in the field. A real tour-de-force, and a must-have on every signal processor's shelf!' Robert D. Nowak, University of Wisconsin, Madison'Finally a wonderful and accessible book for teaching modern signal processing to undergraduate students.' Stéphane Mallat, École Normale Supérieure'Most introductory signal processing textbooks focus on classical transforms, and study how these can be used. Instead, Foundations of Signal Processing encourages readers to think of signals first. It develops a 'signal-centric' view, one that focuses on signals, their representation and approximation, through the introduction of signal spaces. Unlike most entry-level signal processing texts, this general view, which can be applied to many different signal classes, is introduced right at the beginning. From this, starting from basic concepts, and placing an emphasis on intuition, this book develops mathematical tools that give the readers gets a fresh perspective on classical results, while providing them with the tools to understand many state of the art signal representation techniques.' Antonio Ortega, University of Southern California'Foundations of Signal Processing … is a pleasure to read. Drawing on the authors' rich experience of research and teaching of signal processing and signal representations, it provides an intellectually cohesive and modern view of the subject from the geometric point of view of vector spaces. Emphasizing Hilbert spaces, where fine technicalities can be relegated to backstage, this textbook strikes an excellent balance between intuition and mathematical rigor, that will appeal to both undergraduate and graduate engineering students. The last two chapters, on sampling and interpolation, and on localization and uncertainty, take full advantage of the machinery developed in the previous chapters to present these two very important topics of modern signal processing, that previously were only found in specialized monographs. The explanations of advanced topics are exceptionally lucid, exposing the reader to the ideas and thought processes behind the results and their derivation. Students will learn … why things work, at a deep level, which will equip them for independent further reading and research. I look forward to using this text in my own teaching.' Yoram Bresler, University of Illinois, Urbana-ChampaignTable of Contents1. On rainbows and spectra; 2. From Euclid to Hilbert: 2.1 Introduction; 2.2 Vector spaces; 2.3 Hilbert spaces; 2.4 Approximations, projections, and decompositions; 2.5 Bases and frames; 2.6 Computational aspects; 2.A Elements of analysis and topology; 2.B Elements of linear algebra; 2.C Elements of probability; 2.D Basis concepts; Exercises with solutions; Exercises; 3. Sequences and discrete-time systems: 3.1 Introduction; 3.2 Sequences; 3.3 Systems; 3.4 Discrete-time Fourier Transform; 3.5 z-Transform; 3.6 Discrete Fourier Transform; 3.7 Multirate sequences and systems; 3.8 Stochastic processes and systems; 3.9 Computational aspects; 3.A Elements of analysis; 3.B Elements of algebra; Exercises with solutions; Exercises; 4. Functions and continuous-time systems: 4.1 Introduction; 4.2 Functions; 4.3 Systems; 4.4 Fourier Transform; 4.5 Fourier series; 4.6 Stochastic processes and systems; Exercises with solutions; Exercises; 5. Sampling and interpolation: 5.1 Introduction; 5.2 Finite-dimensional vectors; 5.3 Sequences; 5.4 Functions; 5.5 Periodic functions; 5.6 Computational aspects; Exercises with solutions; Exercises; 6. Approximation and compression: 6.1 Introduction; 6.2 Approximation of functions on finite intervals by polynomials; 6.3 Approximation of functions by splines; 6.4 Approximation of functions and sequences by series truncation; 6.5 Compression; 6.6 Computational aspects; Exercises with solutions; Exercises; 7. Localization and uncertainty: 7.1 Introduction; 7.2 Localization for functions; 7.3 Localization for sequences; 7.4 Tiling the time–frequency plane; 7.5 Examples of local Fourier and wavelet bases; 7.6 Recap and a glimpse forward; Exercises with solutions; Exercises.

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Book SynopsisUseful in analyzing electromagnetic problems in a variety of engineering circumstances, the finite element method is a powerful simulation technique. This book explains the method's processes and techniques in careful, meticulous prose. It covers not only essential finite element method theory, but also its latest developments and applications.Table of ContentsPreface xix Preface to the First Edition xxiii Preface to the Second Edition xxvii 1 Basic Electromagnetic Theory 1 1.1 Brief Review of Vector Analysis 2 1.2 Maxwell's Equations 4 1.3 Scalar and Vector Potentials 6 1.4 Wave Equations 7 1.5 Boundary Conditions 8 1.6 Radiation Conditions 11 1.7 Fields in an Infinite Homogeneous Medium 11 1.8 Huygen's Principle 13 1.9 Radar Cross Sections 14 1.10 Summary 15 2 Introduction to the Finite Element Method 17 2.1 Classical Methods for Boundary-Value Problems 17 2.2 Simple Example 21 2.3 Basic Steps of the Finite Element Method 27 2.4 Alternative Presentation of the Finite Element Formulation 34 2.5 Summary 36 3 One-Dimensional Finite Element Analysis 39 3.1 Boundary-Value Problem 39 3.2 Variational Formulation 40 3.3 Finite Element Analysis 42 3.4 Plane-Wave Reflection by a Metal-Backed Dielectric Slab 53 3.5 Scattering by a Smooth, Convex Impedance Cylinder 59 3.6 Higher-Order Elements 62 3.7 Summary 74 4 Two-Dimensional Finite Element Analysis 77 4.1 Boundary-Value Problem 77 4.2 Variational Formulation 79 4.3 Finite Element Analysis 81 4.4 Application to Electrostatic Problems 98 4.5 Application to Magnetostatic Problems 103 4.6 Application to Quasistatic Problems: Analysis of Multiconductor Transmission Lines 105 4.7 Application to Time-Harmonic Problems 109 4.8 Higher-Order Elements 128 4.9 Isoparametric Elements 144 4.10 Summary 149 5 Three-Dimensional Finite Element Analysis 151 5.1 Boundary-Value Problem 151 5.2 Variational Formulation 152 5.3 Finite Element Analysis 153 5.4 Higher-Order Elements 160 5.5 Isoparametric Elements 162 5.6 Application to Electrostatic Problems 168 5.7 Application to Magnetostatic Problems 169 5.8 Application to Time-Harmonic Field Problems 176 5.9 Summary 188 6 Variational Principles for Electromagnetics 191 6.1 Standard Variational Principle 192 6.2 Modified Variational Principle 197 6.3 Generalized Variational Principle 201 6.4 Variational Principle for Anisotrpic Medium 203 6.5 Variational Principle for Resistive Sheets 207 6.6 Concluding Remarks 209 7 Eigenvalue Problems: Waveguides and Cavities 211 7.1 Scalar Formulations for Closed Waveguides 212 7.2 Vector Formulations for Closed Waveguides 225 7.3 Open Waveguides 235 7.4 Three-Dimensional Cavities 238 7.5 Summary 239 8 Vector Finite Elements 243 8.1 Two-Dimensional Edge Elements 244 8.2 Waveguide Problem Revisited 256 8.3 Three-Dimensional Edge Elements 259 8.4 Cavity Problem Revisited 270 8.5 Waveguide Discontinuities 274 8.6 Higher-Order Interpolatory Vector Elements 278 8.7 Higher-Order Hierarchical Vector Elements 293 8.8 Computational Issues 305 8.9 Summary 309 9 Absorbing Boundary Conditions 315 9.1 Two-Dimensional Absorbing Boundary Conditions 316 9.2 Three-Dimensional Absorbing Boundary Conditions 323 9.3 Scattering Analysis Using Absorbing Boundary Conditons 328 9.4 Adaptive Absorbing Boundary Conditons 339 9.5 Fictitious Absorbers 348 9.6 Perfectly Matched Layers 350 9.7 Application of PML to Body-of-Revolutions Problems 368 9.8 Summary 371 10 Finite Element-Boundary Integral Methods 379 10.1 Scattering by Two-Dimensional Cavity-Backed Apertures 381 10.2 Scattering by Two-Dimensional Cylindrical Structures 399 10.3 Scattering by Three-Dimensional Cavity-Backed Apertures 411 10.4 Radiation by Microstrip Patch Antennas in a Cavity 425 10.5 Scattering by General Three-Dimensional Bodies 430 10.6 Solution of the Finite Element-Boundary Integral System 436 10.7 Symmetric Finite Element-Boundary Integral Formulations 447 10.8 Summary 462 11 Finite Element-Eigenfunction Expansion Methods 469 11.1 Waveguide Port Boundary Conditions 470 11.2 Open-Region Scattering 487 11.3 Coupled Basis Functions: The Unimoment Method 494 11.4 Finite Element-Extended Boundary Condition Method 502 11.5 Summary 509 12 Finite Element Analysis in the Time Domain 513 12.1 Finite Element Formulation and Temporal Excitation 514 12.2 Time-Domain Discretization 518 12.3 Stability Analysis 523 12.4 Modeling of Dispersive Media 529 12.5 Truncation via Absorbing Boundary Conditions 538 12.6 Truncation via Perfectly Matched Layers 541 12.7 Truncation via Boundary Integral Equations 551 12.8 Time-Domain Wqaveguide Port Boundary Conditions 562 12.9 Hybrid Field-Circuit Analysis 569 12.10 Dual-Field Domain Decomposition and Element-Level Methods 587 12.11 Discontinuous Galerkin Time-Domain Methods 605 12.12 Summary 625 13 Finite Element Analysis of Periodic Structures 637 13.1 Finite Element Formulation for a Unit Cell 638 13.2 Scattering by One-Dimensional Periodic Structures: Frequency-Domain Analysis 651 13.3 Scattering by One-Dimensional Periodic Structures: Time-Domain Analysis 656 13.4 Scattering by Two-Dimensional Periodic Structures: Frequency-Domain Analysis 663 13.5 Scattering by Two-Dimensonal Periodic Structures: Time-Domain Analysis 670 13.6 Analysis of Angular Periodic Strctures 678 13.7 Summary 682 14 Domain Decompsition for Large-Scale Analysis 687 14.1 Schwarz Methods 688 14.2 Schur Complement Methods 693 14.3 FETI-DP Method for Low-Frequency Problems 705 14.4 FETI-DP Method for High-Frequency Problems 728 14.5 Noncomformal FETI-DP Method Based on Cement Elements 743 14.6 Application of Second-Order Transmission Conditions 753 14.7 Summary 760 15 Solution of Finite Element Equations 767 15.1 Decomposition Methods 769 15.2 Conjugate Gradient Methods 778 15.3 Solution of Eigenvalue Problems 791 15.4 Fast Frequency-Sweep Computation 797 15.5 Summary 803 Appendix A: Basic Vector Identities and Integral Theorems 809 Appendix B: The Ritz Procedure for Complex-Valued Problems 813 Appendix C: Green's Functions 817 Appendix D: Singular Integral Evaluation 825 Appendix E: Some Special Functions 829 Index 837

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Book SynopsisThis book presents the first comprehensive account of fast sequential Monte Carlo (SMC) methods for counting and optimization at an exceptionally accessible level. Written by authorities in the field, it places great emphasis on cross-entropy, minimum cross-entropy, splitting, and stochastic enumeration.Table of ContentsPreface xi 1. Introduction to Monte Carlo Methods 1 2. Cross-Entropy Method 6 2.1. Introduction 6 2.2. Estimation of Rare-Event Probabilities 7 2.3. Cross-Entrophy Method for Optimization 18 2.3.1. The Multidimensional 0/1 Knapsack Problem 21 2.3.2. Mastermind Game 23 2.3.3. Markov Decision Process and Reinforcement Learning 25 2.4. Continuous Optimization 31 2.5. Noisy Optimization 33 2.5.1. Stopping Criterion 35 3. Minimum Cross-Entropy Method 37 3.1. Introduction 37 3.2. Classic MinxEnt Method 39 3.3. Rare Events and MinxEnt 43 3.4. Indicator MinxEnt Method 47 3.4.1. Connection between CE and IME 51 3.5. IME Method for Combinatorial Optimization 52 3.5.1. Unconstrained Combinatorial Optimization 52 3.5.2. Constrained Combinatorial Optimization: The Penalty Function Approach 54 4. Splitting Method for Counting and Optimization 56 4.1. Background 56 4.2. Quick Glance at the Splitting Method 58 4.3. Splitting Algorithm with Fixed Levels 64 4.4. Adaptive Splitting Algorithm 68 4.5. Sampling Uniformly on Discrete Regions 74 4.6. Splitting Algorithm for Combinatorial Optimization 75 4.7. Enhanced Splitting Method for Counting 76 4.7.1. Counting with the Direct Estimator 76 4.7.2. Counting with the Capture–Recapture Method 77 4.8. Application of Splitting to Reliability Models 79 4.8.1. Introduction 79 4.8.2. Static Graph Reliability Problem 82 4.8.3. BMC Algorithm for Computing S(Y) 84 4.8.4. Gibbs Sampler 85 4.9. Numerical Results with the Splitting Algorithms 86 4.9.1. Counting 87 4.9.2. Combinatorial Optimization 101 4.9.3. Reliability Models 102 4.10. Appendix: Gibbs Sampler 104 5. Stochastic Enumeration Method 106 5.1. Introduction 106 5.2. OSLA Method and Its Extensions 110 5.2.1. Extension of OSLA: nSLA Method 112 5.2.2. Extension of OSLA for SAW: Multiple Trajectories 115 5.3. SE Method 120 5.3.1. SE Algorithm 120 5.4. Applications of SE 127 5.4.1. Counting the Number of Trajectories in a Network 127 5.4.2. SE for Probabilities Estimation 131 5.4.3. Counting the Number of Perfect Matchings in a Graph 132 5.4.4. Counting SAT 135 5.5. Numerical Results 136 5.5.1. Counting SAW 137 5.5.2. Counting the Number of Trajectories in a Network 137 5.5.3. Counting the Number of Perfect Matchings in a Graph 140 5.5.4. Counting SAT 143 5.5.5. Comparison of SE with Splitting and SampleSearch 146 A. Additional Topics 148 A.1. Combinatorial Problems 148 A.1.1. Counting 149 A.1.2. Combinatorial Optimization 154 A.2. Information 162 A.2.1. Shannon Entropy 162 A.2.2. Kullback–Leibler Cross-Entropy 163 A.3. Efficiency of Estimators 164 A.3.1. Complexity 165 A.3.2. Complexity of Randomized Algorithms 166 Bibliography 169 Abbreviations and Acronyms 177 List of Symbols 178 Index 181

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Book SynopsisAn informative look at the theory, computer implementation, and application of the scaled boundary finite element method This reliable resource, complete with MATLAB, is an easy-to-understand introduction to the fundamental principles of the scaled boundary finite element method. It establishes the theory of the scaled boundary finite element method systematically as a general numerical procedure, providing the reader with a sound knowledge to expand the applications of this method to a broader scope. The book also presents the applications of the scaled boundary finite element to illustrate its salient features and potentials. The Scaled Boundary Finite Element Method: Introduction to Theory and Implementation covers the static and dynamic stress analysis of solids in two and three dimensions. The relevant concepts, theory and modelling issues of the scaled boundary finite element method are discussed and the unique features of the method are highlightedTable of ContentsPreface xv Acknowledgements xix About the Companion Website xxi 1 Introduction 1 1.1 Numerical Modelling 1 1.2 Overview of the Scaled Boundary Finite Element Method 6 1.3 Features and Example Applications of the Scaled Boundary Finite Element Method 10 1.3.1 Linear Elastic Fracture Mechanics: Crack Terminating at Material Interface 11 1.3.2 Automatic Mesh Generation Based on Quadtree/Octree 13 1.3.3 Treatment of Non-matching Meshes 14 1.3.4 Crack Propagation 17 1.3.5 Adaptive Analysis 17 1.3.6 TransientWave Scattering in an Alluvial Basin 19 1.3.7 Automatic Image-based Analysis 19 1.3.7.1 Two-dimensional Elastoplastic Analysis of Cast Iron 20 1.3.7.2 Three-dimensional Concrete Specimen 22 1.3.8 Automatic Analysis of STL Models 24 1.4 Summary 26 Part I Basic Concepts and MATLAB Implementation of the Scaled Boundary Finite Element Method in Two Dimensions 27 2 Basic Formulations of the Scaled Boundary Finite Element Method 31 2.1 Introduction 31 2.2 Modelling of Geometry in Scaled Boundary Coordinates 31 2.2.1 S-domains: Scaling Requirement on Geometry, Scaling Centre and Scaling of Boundary 31 2.2.2 S-elements: Boundary Discretization of S-domains 37 2.2.3 Scaled Boundary Transformation 40 2.2.3.1 Scaled Boundary Coordinates 40 2.2.3.2 Coordinate Transformation of Partial Derivatives 42 2.2.3.3 Geometrical Properties in Scaled Boundary Coordinates 44 2.3 Governing Equations of Linear Elasticity in Scaled Boundary Coordinates 50 2.4 Semi-analytical Representation of Displacement and Strain Fields 51 2.5 Derivation of the Scaled Boundary Finite Element Equation by the Virtual Work Principle 53 2.5.1 Virtual Displacement and Strain Fields in Scaled Boundary Coordinates 54 2.5.2 Nodal Force Functions 54 2.5.3 The Scaled Boundary Finite Element Equation 55 2.6 Computer Program Platypus: Coefficient Matrices of an S-element 63 2.6.1 Element Coefficient Matrices of a 2-node Line Element 63 2.6.2 Assembly of Coefficient Matrices of an S-element 67 3 Solution of the Scaled Boundary Finite Element Equation by Eigenvalue Decomposition 73 3.1 Solution Procedure for the Scaled Boundary Finite Element Equations in Displacement 73 3.2 Pre-conditioning of Eigenvalue Problems 77 3.3 Computer Program Platypus: Solution of the Scaled Boundary Finite Element Equation of a Bounded S-element by the Eigenvalue Method 78 3.4 Assembly of S-elements and Solution of Global System of Equations 84 3.4.1 Assembly of S-elements 84 3.4.2 Surface Tractions 85 3.4.3 Enforcing Displacement Boundary Conditions 87 3.5 Computer Program Platypus: Assembly and Solution 87 3.5.1 Assembly of Global Stiffness Matrix 87 3.5.2 Assembly of Load Vector 95 3.5.3 Solution of Global System of Equations 96 3.5.4 Utility Functions 97 3.6 Examples of Static Analysis Using Platypus 102 3.7 Evaluation of Internal Displacements and Stresses of an S-element 111 3.7.1 Integration Constants and Internal Displacements 111 3.7.2 Strain/Stress Modes and Strain/Stress Fields 112 3.7.3 Shape Functions of Polygon Elements Modelled as S-elements 114 3.8 Computer Program Platypus: Internal Displacements and Strains 114 3.9 Body Loads 132 3.10 Dynamics and Vibration Analysis 135 3.10.1 Mass Matrix and Equation of Motion 135 3.10.2 Natural Frequencies and Mode Shapes 140 3.10.3 Response History Analysis Using the Newmark Method 143 4 Automatic Polygon Mesh Generation for Scaled Boundary Finite Element Analysis 149 4.1 Introduction 149 4.2 Basics of Geometrical Representation by Signed Distance Functions 150 4.3 Computer Program Platypus: Generation of Polygon S-elementMesh 154 4.3.1 Mesh Data Structure 157 4.3.2 Centroid of a Polygon 165 4.3.3 Converting a TriangularMesh to an S-elementMesh 166 4.3.4 Use of Polygon Meshes Generated by PolyMesher in a Scaled Boundary Finite Element Analysis 171 4.3.5 Dividing Edges of Polygons into Multiple Elements 172 4.4 Examples of Scaled Boundary Finite Element Analysis Using Platypus 175 4.4.1 A Deep Beam 178 4.4.1.1 Static Analysis 186 4.4.1.2 Modal Analysis 189 4.4.1.3 Response History Analysis 190 4.4.1.4 Pure Bending of a Beam: 2 Line Elements on an Edge of Polygons 190 4.4.2 A Circular Hole in an Infinite Plane Under Remote Uniaxial Tension 193 4.4.3 An L-shaped Panel 197 4.4.3.1 Static Analysis 203 4.4.3.2 Modal Analysis 204 4.4.3.3 Response History Analysis 207 5 Modelling Considerations in the Scaled Boundary Finite Element Analysis 209 5.1 Effect of Location of Scaling Centre on Accuracy 209 5.2 Mesh Transition 212 5.2.1 Local Mesh Refinement 212 5.2.2 Rapid Mesh Transition 214 5.2.3 Effect of Nonuniformity of Line Element Length on the Boundary of S-elements 216 5.3 Connecting Non-matching Meshes of Multiple Domains 218 5.3.1 Computer Program Platypus: Combining Two Non-matching Meshes 220 5.3.2 Computer Program Platypus: Modelling of a Problem by Multiple Domains with Non-matching Meshes 223 5.3.3 Examples 225 5.4 Modelling of Stress Singularities 234 Part II Theory and Applications of the Scaled Boundary Finite Element Method 237 6 Derivation of the Scaled Boundary Finite Element Equation in Three Dimensions 239 6.1 Introduction 239 6.2 Scaling of Boundary 239 6.3 Boundary Discretization of an S-domain 242 6.3.1 Isoparametric Quadrilateral Elements 243 6.3.1.1 Four-node Quadrilateral Element 243 6.3.1.2 Quadrilateral Element of Variable Number of Nodes 245 6.3.2 Isoparametric Triangular Elements 246 6.3.2.1 Three-node Triangular Elements 247 6.3.2.2 Six-node Triangular Elements 248 6.4 Scaled Boundary Transformation of Geometry 249 6.5 Geometrical Properties in Scaled Boundary Coordinates 253 6.6 Governing Equations of Elastodynamics with Geometry in Scaled Boundary Coordinates 257 6.7 Derivation of the Scaled Boundary Finite Element Equation by the Galerkin’s Weighted Residual Technique 259 6.7.1 Displacement, Strain Fields and Nodal Force Functions in Scaled Boundary Coordinates 259 6.7.2 The Scaled Boundary Finite Element Equation 262 6.8 Unified Formulations in Two andThree Dimensions 267 6.9 Formulation of the Scaled Boundary Finite Element Equation as a System of First-order Differential Equations 268 6.10 Properties of Coefficient Matrices 269 6.10.1 Coefficient Matrices [E0] and [M0] 270 6.10.2 Coefficient Matrix [E2] 270 6.10.3 Matrix [Zp] 271 6.11 Linear Completeness of the Scaled Boundary Finite Element Solution 272 6.11.1 Constant Displacement Field 272 6.11.2 Linear Displacement Field 273 6.12 Scaled Boundary Finite Element Equation in Stiffness 278 7 Solution of the Scaled Boundary Finite Element Equation in Statics by Schur Decomposition 281 7.1 Introduction 281 7.2 Basics of Matrix Exponential Function 283 7.3 Schur Decomposition 287 7.3.1 Introduction 287 7.3.2 Treatment of the Diagonal Block of Eigenvalues of 0 288 7.4 Solution Procedure for a Bounded S-element by Schur Decomposition 291 7.4.1 Transformation of the Scaled Boundary Finite Element Equation 291 7.4.2 Enforcing the Boundary Condition at the Scaling Centre 292 7.4.3 Determining the Solution for Displacement and Nodal Force Functions 294 7.4.4 Determining the Static Stiffness Matrix 295 7.5 Solution of Displacement and Stress Fields of an S-element 295 7.5.1 Integration Constants 295 7.5.2 Stress Modes and Stresses on the Boundary 296 7.6 Block-diagonal Schur Decomposition 297 7.7 Solution Procedure by Block-diagonal Schur Decomposition 303 7.7.1 General Solution of the Scaled Boundary Finite Element Equation 303 7.7.1.1 [Zp] Having No Eigenvalues of Zero 304 7.7.1.2 [Zp] Having Eigenvalues of Zero 304 7.7.2 Solution for Bounded S-elements 305 7.7.3 Solution for Unbounded S-elements 307 7.7.3.1 [Zp] Having No Eigenvalues of Zero 307 7.7.3.2 [Zp] Having Eigenvalues of Zero 308 7.8 Displacements and Stresses of an S-element by Block-diagonal Schur Decomposition 310 7.8.1 Integration Constants and Displacement Fields 310 7.8.2 Stress Modes and Stress Fields 311 7.8.3 Shape Functions of Polytope Elements 312 7.9 Body Loads 313 7.10 Mass Matrix 315 7.11 Remarks 317 7.12 Examples 319 7.12.1 Circular Cavity in Full-plane 319 7.12.2 Bi-materialWedge 322 7.12.3 Interface Crack in Anisotropic Bi-material Full-plane 325 7.13 Summary 327 8 High-order Elements 329 8.1 Lagrange Interpolation 330 8.2 One-dimensional Spectral Elements 333 8.2.1 Shape Functions 334 8.2.2 Numerical Integration of Element Coefficient Matrices 337 8.2.2.1 Gauss-Legendre Quadrature 337 8.2.2.2 Gauss-Lobatto-Legendre Quadrature 338 8.3 Two-dimensional Quadrilateral Spectral Elements 341 8.3.1 Shape Functions 341 8.3.2 Integration of Element Coefficient Matrices by Gauss-Lobatto-Legendre Quadrature 342 8.4 Examples 344 8.4.1 A Cantilever Beam Subject to End Loading 345 8.4.2 A Circular Hole in an Infinite Plate 347 8.4.3 An L-shaped Panel 349 8.4.4 A 3D Cantilever Beam Subject to End-shear Loading 351 8.4.5 A Pressurized Hollow Sphere 352 9 Quadtree/Octree Algorithm of Mesh Generation for Scaled Boundary Finite Element Analysis 355 9.1 Introduction 355 9.1.1 Mesh Generation 355 9.1.2 The Quadtree/Octree Algorithm 357 9.2 Data Structure of S-element Meshes 360 9.3 Quadtree/Octree Mesh Generation of Digital Images 361 9.3.1 Illustration of Quadtree Decomposition of Two-dimensional Images by an Example 361 9.3.2 Octree Decomposition 366 9.4 Solutions of S-elements with the Same Pattern of Node Configuration 370 9.4.1 Two-dimensional S-elements 370 9.4.2 Three-dimensional S-elements 372 9.5 Examples of Image-based Analysis 374 9.5.1 A 2D Concrete Specimen 374 9.5.2 A 3D Concrete Specimen 376 9.6 Quadtree/Octree Mesh Generation for CAD Models 378 9.6.1 Quadtree/Octree Grid 380 9.6.2 Trimming of Boundary Cells 381 9.7 Examples Using Quadtree/Octree Meshes of CAD Models 383 9.7.1 Square Body with Multiple Holes 384 9.7.2 An Evolving Void in a Square Body 385 9.7.3 Adaptive Analysis of an L-shaped Panel 386 9.7.4 A Mechanical Part 387 9.7.5 STL Models 389 9.8 Remarks 394 10 Linear Elastic Fracture Mechanics 395 10.1 Introduction 395 10.2 Basics of Fracture Analysis: Asymptotic Solutions, Stress Intensity Factors, and the T-stress 397 10.2.1 Crack in Homogeneous Isotropic Material 397 10.2.2 Interfacial Cracks between Two Isotropic Materials 401 10.2.3 Interfacial Cracks between Two AnisotropicMaterials 402 10.2.4 Multi-materialWedges 405 10.3 Modelling of Singular Stress Fields by the Scaled Boundary Finite Element Method 406 10.4 Stress Intensity Factors and the T-stress of a Cracked Homogeneous Body 407 10.5 Definition and Evaluation of Generalized Stress Intensity Factors 416 10.6 Examples of Highly Accurate Stress Intensity Factors and T-stress 432 10.6.1 A Single Edge-cracked Rectangular Body Under Tension 433 10.6.2 A Single Edge-cracked Rectangular Body Under Bending 435 10.6.3 A Centre-cracked Rectangular Body Under Tension 437 10.6.4 A Double Edge-cracked Rectangular Body Under Tension 438 10.6.5 A Single Edge-cracked Rectangular Body Under End Shearing 439 10.7 Modelling of Crack Propagation 440 10.7.1 Modelling of Crack Paths by Polygon Meshes 442 10.7.2 Modelling of Crack Paths by Quadtree Meshes 443 10.7.3 Examples of Crack PropagationModelling 444 10.7.3.1 Fatigue Crack Propagation Using Polygon Mesh 444 10.7.3.2 Crack Propagation in a Beam with Three Holes 447 Appendix A Governing Equations of Linear Elasticity 449 A.1 Three-dimensional Problems 449 A.1.1 Strain 449 A.1.2 Stress and Equilibrium Equation 450 A.1.3 Stress-strain Relationship and Material Elasticity Matrix 451 A.1.4 Boundary Conditions 453 A.2 Two-dimensional Problems 454 A.2.1 Elasticity Matrix in Plane Stress 455 A.2.2 Elasticity Matrix in Plane Strain 456 A.3 Unified Expressions of Governing Equations 457 Appendix B Matrix Power Function 459 B.1 Definition of Matrix Power Function 459 B.2 Application to Solution of System of Ordinary Differential Equations 460 B.3 Computation of Matrix Power Function by Eigenvalue Method 461 Bibliography 463 Index 475

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Book SynopsisApproaches computational engineering sciences from the perspective of engineering applications Uniting theory with hands-on computer practice, this book gives readers a firm appreciation of the error mechanisms and control that underlie discrete approximation implementations in the engineering sciences. Key features: Illustrative examples include heat conduction, structural mechanics, mechanical vibrations, heat transfer with convection and radiation, fluid mechanics and heat and mass transport Takes a cross-discipline continuum mechanics viewpoint Includes Matlab toolbox and .m data files on a companion website, immediately enabling hands-on computing in all covered disciplines Website also features eight topical lectures from the author's own academic courses It provides a holistic view of the topic from covering the different engineering problems that can be solved using finite element to how each pTable of ContentsPreface viii Notation xi 1 COMPUTATIONAL ENGINEERING SCIENCE 1 1.1 Engineering simulation 1 1.2 A problem solving environment 2 1.3 Problem statements in engineering 4 1.4 Decisions on forming WSN 6 1.5 Discrete approximate WSh implementation 8 1.6 Chapter summary 9 1.7 Chapter references 10 2 PROBLEM STATEMENTS 11 2.1 Engineering simulation 11 2.2 Continuum mechanics viewpoint 12 2.3 Continuum conservation law forms 12 2.4 Constitutive closure for conservation law PDEs 14 2.5 Engineering science continuum mechanics 18 2.6 Chapter references 20 3 SOME INTRODUCTORY MATERIAL 21 3.1 Introduction 21 3.2 Multi-dimensional PDEs, separation of variables 22 3.3 Theoretical foundations, GWSh 27 3.4 A legacy FD construction 28 3.5 An FD approximate solution 30 3.6 Lagrange interpolation polynomials 31 3.7 Chapter summary 32 3.8 Exercises 34 3.9 Chapter references 34 4 HEAT CONDUCTION35 4.1 A steady heat conduction example 35 4.2 Weak form approximation, error minimization 37 4.3 GWSN discrete implementation, FE basis38 4.4 Finite element matrix statement 41 4.5 Assembly of {WS}e to form algebraic GWSh 43 4.6 Solution accuracy, error distribution 45 4.7 Convergence, boundary heat flux 47 4.8 Chapter summary 47 4.9 Exercises 48 4.10 Chapter reference 48 5 STEADY HEAT TRANSFER, n =149 5.1 Introduction 49 5.2 Steady heat transfer, n = 1 50 5.3 FE k = 1 trial space basis matrix library 52 5.4 Object-oriented GWSh programming 55 5.5 Higher completeness degree trial space bases58 5.6 Global theory, asymptotic error estimate 62 5.7 Non-smooth data, theory generalization 66 5.8 Temperature dependent conductivity, non-linearity 69 5.9 Static condensation, p-elements 72 5.10 Chapter summary 75 5.11 Exercises 76 5.12 Computer labs 77 5.13 Chapter references 78 6 ENGINEERING SCIENCES, n =1 79 6.1 Introduction 79 6.2 The Euler-Bernoulli beam equation 80 6.3 Euler-Bernoulli beam theory GWSh reformulation 85 6.4 The Timoshenko beam theory 92 6.5 Mechanical vibrations of a beam 99 6.6 Fluid mechanics, potential flow 106 6.7 Electromagnetic plane wave propagation110 6.8 Convective-radiative finned cylinder heat transfer 112 6.9 Chapter summary 120 6.10 Exercises122 6.10 Computer labs 123 6.11 Chapter references 124 7 STEADY HEAT TRANSFER, n > 1 125 7.1 Introduction 125 7.2 Multi-dimensional FE bases and DOF 126 7.3 Multi-dimensional FE operations 129 7.4 The NC k = 1,2 basis FE matrix library 132 7.5 NC basis {WS}e template, accuracy, convergence 136 7.6 The tensor product basis element family 139 7.7 Gauss numerical quadrature, k = 1 TP basis library 141 7.8 Convection-radiation BC GWSh implementation 146 7.9 Linear basis GWSh template unification 150 7.10 Accuracy, convergence revisited 152 7.11 Chapter summary 153 7.12 Exercises155 7.13 Computer labs 155 7.14 Chapter references 156 8 FINITE DIFFERENCES OF OPINION 159 8.1 The FD-FE correlation159 8.2 The FV-FE correlation162 8.3 Chapter summary 167 8.4 Exercises168 9 CONVECTION-DIFFUSION, n = 1 169 9.1 Introduction169 9.2 The Galerkin weak statement 170 9.3 GWSh completion for time dependence172 9.4 GWSh + qTS algorithm templates 173 9.5 GWSh + qTS algorithm asymptotic error estimates 175 9.6 Performance verification test cases 177 9.7 Dispersive error characterization 180 9.8 A modified Galerkin weak statement 184 9.9 Verification problem statements revisited 187 9.10 Unsteady heat conduction 190 9.11 Chapter summary 193 9.12 Exercises 193 9.13 Computer labs 194 9.14 Chapter references 195 10 CONVECTION-DIFFUSION, n > 1 197 10.1 The problem statement 197 10.2 GWSh + qTS formulation reprise 198 10.3 Matrix library additions, templates 200 10.4 mPDE Galerkin weak forms, theoretical analyses 202 10.5 Verification, benchmarking and validation 207 10.6 Mass transport, the rotating cone verification 208 10.7 The gaussian plume benchmark 211 10.8 The steady n-D Peclet problem verification 213 10.9 Mass transport, a validated n = 3 experiment 215 10.10 Numerical linear algebra, matrix iteration 222 10.11 Newton and AF TP jacobian templates 227 10.12 Chapter summary 229 10.13 Exercises231 10.14 Computer labs 231 10.15 Chapter references232 11 ENGINEERING SCIENCES, n > 1 235 11.1 Introduction 235 11.2 Structural mechanics236 11.3 Structural mechanics, virtual work FE form 240 11.4 Plane stress/strain, GWSh implementation 242 11.5 Elasticity computer lab 246 11.6 Fluid mechanics, incompressible-thermal flow 251 11.7 Vorticity-streamfunction GWSh + qTS algorithm 254 11.8 An isothermal INS validation experiment 258 11.9 Multi-mode convection heat transfer262 11.10 Mechanical vibrations, normal mode GWSh 267 11.11 Normal modes of a vibrating membrane270 11.12 Multi-physics solid-fluid mass transport 276 11.13 Chapter summary 280 11.14 Exercises 282 11.15 Computer labs283 11.14 Chapter references 284 12 CONCLUSION 287 Index 289

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Book SynopsisBasic R Programming.- Random Variable Generation.- Monte Carlo Integration.- Controlling and Accelerating Convergence.- Monte Carlo Optimization.- Metropolis#x2013;Hastings Algorithms.- Gibbs Samplers.- Convergence Monitoring and Adaptation for MCMC Algorithms.Trade ReviewFrom the reviews:“Robert and Casella’s new book uses the programming language R, a favorite amongst (Bayesian) statisticians to introduce in eight chapters both basic and advanced Monte Carlo techniques … . The book could be used as the basic textbook for a semester long course on computational statistics with emphasis on Monte Carlo tools … . useful for (and should be next to the computer of) a large body of hands on graduate students, researchers, instructors and practitioners … .” (Hedibert Freitas Lopes, Journal of the American Statistical Association, Vol. 106 (493), March, 2011)“Chapters focuses on MCMC methods the Metropolis–Hastings algorithm, Gibbs sampling, and monitoring and adaptation for MCMC algorithms. … There are exercises within and at the end of all chapters … . Overall, the level of the book makes it suitable for graduate students and researchers. Others who wish to implement Monte Carlo methods, particularly MCMC methods for Bayesian analysis will also find it useful.” (David Scott, International Statistical Review, Vol. 78 (3), 2010)“The primary audience is graduate students in statistics, biostatistics, engineering, etc. who need to know how to utilize Monte Carlo simulation methods to analyze their experiments and/or datasets. … this text does an effective job of including a selection of Monte Carlo methods and their application to a broad array of simulation problems. … Anyone who is an avid R user and has need to integrate and/or optimize complex functions will find this text to be a necessary addition to his or her personal library.” (Dean V. Neubauer, Technometrics, Vol. 53 (2), May, 2011)Table of ContentsBasic R Programming.- Random Variable Generation.- Monte Carlo Integration.- Controlling and Accelerating Convergence.- Monte Carlo Optimization.- Metropolis#x2013;Hastings Algorithms.- Gibbs Samplers.- Convergence Monitoring and Adaptation for MCMC Algorithms.

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Book SynopsisAs well as providing an overview of the current state of science in the analysis and synthesis of non-rigid shapes, the authors include everyday examples to explain concepts. Practice problems follow at the end of each chapter, along with detailed solutions.Trade ReviewFrom the reviews: "This book provides an introduction to this geometry. … Overall, the book … does explain relevant mathematical notions, such as Gromov’s metric geometry ideas, in a very understandable and entertaining way, with numerous images and exercises. … I highly recommend it to both computer scientists interested in learning more about the latest advances in computational geometry and to geometers looking for applications. This unique book can serve as an excellent textbook for many related courses, for self-study, or as a reference." (V. Kreinovich, ACM Computing Reviews, May, 2009) “Numerical geometry of non-rigid shapes by A. Bronstein, M. Bronstein, and R. Kimmel combines the beauty of modern mathematics … with the interesting field of computer vision and pattern recognition. … The book is developed at an intermediate-advanced level. Students will find the material clear and easy to understand, and will benefit from its good presentation.” (Stefan Henn, Mathematical Reviews, Issue 2010 b)Table of ContentsA Taste of Geometry.- Discrete Geometry.- Shortest Paths and Fast Marching Methods.- Numerical Optimization.- In the Rigid Kingdom.- Multidimensional Scaling.- Spectral Embedding.- Non Euclidean Embedding.- Isometry Invariant Similarity.- Partial Similarity.- Non rigid Correspondence and Calculus of Shapes.- Three dimensional Face Recognition.- Epilogue.

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Book SynopsisHistory of Experimental Structural Mechanics.- Sensors .- Instrumentation.- Applied Digital Signal Processing.- Basic Measurements.- Structural Measurements.- Environmental Measurements.- Design of Tests.- Modal Parameter Estimation.- Modal Analysis of Rotating Systems.- Operating Modal Analysis.- Computational Methods in Structural Dynamics.- Finite/Boundary Element Modeling and Model Reduction.- FE Model Correlation.- Model Updating.- Damping of Materials and Stuctures.- Model Validation/Verification/Calibration.- Uncertainty Quantification and Statistical Issues.- Nonlinear System Analysis.- Rotating System Analysis.- Structural Health Monitoring and Damage Detection.- System Modeling.- Modal Modeling.- Impedance Modeling.- Acoustics of Structural Systems-VibroAcoustics.- Automotive Structural Testing.- Civil Structural Testing.- Aerospace Structural Testing.- Sports Equipment Testing.Table of ContentsHistory of Experimental Structural Mechanics.- Sensors .- Instrumentation.- Applied Digital Signal Processing.- Basic Measurements.- Structural Measurements.- Environmental Measurements.- Design of Tests.- Modal Parameter Estimation.- Modal Analysis of Rotating Systems.- Operating Modal Analysis.- Computational Methods in Structural Dynamics.- Finite/Boundary Element Modeling and Model Reduction.- FE Model Correlation.- Model Updating.- Damping of Materials and Stuctures.- Model Validation/Verification/Calibration.- Uncertainty Quantification and Statistical Issues.- Nonlinear System Analysis.- Rotating System Analysis.- Structural Health Monitoring and Damage Detection.- System Modeling.- Modal Modeling.- Impedance Modeling.- Acoustics of Structural Systems-VibroAcoustics.- Automotive Structural Testing.- Civil Structural Testing.- Aerospace Structural Testing.- Sports Equipment Testing.

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Book SynopsisProceedings volume contains carefully selected papers presented during the 17th IFIP Conference on System Modelling and Optimization.Table of Contents1 On the convergence of a trust region SQP algorithm for nonlinearly constrained optimization problems.- 2 Decomposition and suboptimal control in dynamical systems.- 3 Network flow — theory and applications with practical impact.- 4 The mathematical theory of evidence — a short introduction.- 5 Algebraic methods in control, theory and applications.- 6 One method for robust control of uncertain systems — theory and practice.- 7 Stochastic optimization methods in engineering.- 8 Robust stabilization of nonlinear systems by optimal controllers.- 9 Weighted H2 approximation of transfer functions.- 10 On design of H? optimal controls for uncertain nonlinear systems.- 11 Constrained optimization algorithms and automatic differentiation for parameter estimation with application to granulocytics models.- 12 Expert system for diagnosis of womens’ menstrual cycle using natural family planning method.- 13 Metabolic flux determination by 13-C tracer experiments: analysis of sensitivity, identifiability and redundancy.- 14 Binding-time analysis applied to mathematical algorithms.- 15 Invariant state progress and relation modelling of DEDS.- 16 Remarks on the observability of nonlinear discrete time systems.- 17 Risk-sensitive control and dynamic games: the discrete-time case.- 18 Dynamic portfolio optimization based on reference trajectories.- 19 Stability analysis of time-varying discrete interval systems.- 20 The relaxation theory applied to optimal control problems of semilinear elliptic equations.- 21 On the use of space invariant imbedding to solve optimal control problems for second order elliptic equations.- 22 Semismoothness in parametrized quasi-variational inequalities.- 23 Optimal control problem governed by a semilinear parabolic equation.- 24 Shape optimization of hyperelastic rod.- 25 Dynamic modelling and optimal hierarchical control of a multiple-effect evaporator — superconcentrator plant.- 26 On the use of consistent approximations for the optimal design of beams.- 27 A game-theoretical model for a controlled process of heat transfer.- 28 Constrained predictive control of a counter-current extractor.- 29 Optimal policies under different assumptions about target values: an optimal control analysis for Austria.- 30 Optimal usage of saline and non saline irrigation water; a policy tool.- 31 Fuzzy integer sharing problem with fuzzy capacity constraints.- 32 A fuzzy-PID-concept with minimal rule set.- 33 A numerical procedure for minimizing the maximum cost.- 34 Game of pursuit with zero stop probability.- 35 Solution concepts in multicriteria cooperative games without side payments.- 36 Computer models for maximising tumor cell kill and for minimizing side effects in radiation therapy.- 37 Decision makin& problems: AIDS prevention and energy development.- 38 A mathematical model of HIV infection: the role of CD8+ lymphocytes.- 39 Mathematical modelling of conjugate formation by cytotoxic lymphocytes and tumour cells.- 40 Reliability optimization of complex systems using sharp lower bounds.- 41 Knowledge retrieval for autonomous agents.- 42 Simulation and optimization of complex systems reliability characteristics in grouped data structure.- 43 A modular system of software tools for multicriteria model analysis.- 44 Methodology and modular tool for aspiration-led analysis of LP models.- 45 Interactive multiobjective optimization system NIMBUS applied to nonsmooth structural design problems.- 46 Preliminary computational experience with a descent level method for convex nondifferentiable optimization.- 47 Bundle methods applied to the unit-commitment problem.- 48 Nondifferentiable optimization solver: basic theoretical assumptions.- 49 Discrete approximation of nonlinear control problems.- 50 Convergence of Lagrange—Newton method for control-state and pure state constrained optimal control problems.- 51 Descent methods for optimal periodic hereditary control problems.- 52 Aircraft trajectory optimization using nonlinear programming.- 53 Feedback control of state constrained optimal control problems.- 54 Primal-dual interior point method for multicommodity network flows with side constraints and comparison with alternative methods.- 55 Dual Bregman proximal methods for large-scale 0–1 problems.- 56 On long-step surrogate projection methods for solving convex feasibility problems.- 57 Theoretical and experimental analyis of random linkage algorithms for global optimization.- 58 A dynamic list heuristic for 2D-cutting.- 59 About solving linear integer programs through hermite normal form decomposition.- 60 Software system for solving multi-scale optimization problems.- 61 Dual barrier-projection and barrier-Newton methods in linear programming.- 62 Flow and release optimization in manufacturing systems represented as timed event graphs.- 63 A control model for assembly manufacturing systems.- 64 Numerical experiment on the 2D cutting-stock algorithms based on local optimization.- 65 An algorithm for the transportation problem with given frequencies.- 66 The traveling salesman problem with precedence constraints and binary costs.- 67 Cost oriented competing processes — a new handling of assignment problems.- 68 Modelling and solving of the allocation problem of non-convex polygons with rotations.- 69 Parameters identification of a time-varying stochastic dynamic systems using Viterbi algorithm.- 70 Management of bond portfolios via stochastic programming — postoptimality and sensitivity analysis.- 71 A note on objective functions in multistage stochastic nonlinear programming problems.- 72 Dynamic search for shortest multimodal paths in a transportation network.- 73 Arc routing for rural Irish networks.- 74 Arc routing vehicle routing problems with vehicle/site dependencies.- Index of contributors.- Keyword index.

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Book SynopsisEmploy essential tools and functions of the MATLAB and Simulink packages, which are explained and demonstrated via interactive examples and case studies. This revised edition covers features from the latest MATLAB 2022b release, as well as other features that have been released since the first edition published.  This book contains dozens of simulation models and solved problems via m-files/scripts and Simulink models which will help you to learn programming and modelling essentials. You''ll become efficient with many of the built-in tools and functions of MATLAB/Simulink while solving engineering and scientific computing problems. Beginning MATLAB and Simulink, Second Edition explains various practical issues of programming and modelling in parallel by comparing MATLAB and Simulink. After studying and using this book, you''ll be proficient at using MATLAB and Simulink and applying the source code and models from the book''s examples as templTable of Contents1. Introduction to MATLAB.- 2. Programming Essentials.- 3. Graphical User Interface Model Development.- 4. MEX files, C/C++ and Standalone Applications.- 5. Simulink Modeling Essentials.- 6. Plots.- 7. Matrix Algebra.- 8. Ordinary Differential Equations.

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Book SynopsisPreliminary concepts.- Nonlinear Finite Element Analysis Procedure.- Finite Element Analysis for Nonlinear Elastic Systems.- Finite Element Analysis for Elastoplastic Problems.- Finite Element Analysis for Contact Problems. Table of ContentsPreliminary concepts.- Nonlinear Finite Element Analysis Procedure.- Finite Element Analysis for Nonlinear Elastic Systems.- Finite Element Analysis for Elastoplastic Problems.- Finite Element Analysis for Contact Problems.

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Book SynopsisImplicit Functions and Solution MappingsTrade Review“The book represents the state of the art of the modern theory of inverse and implicit functions and provides an important source for studies of numerical methods and applications in this area. It can be warmly recommended to all specialists and advanced students working in optimization, analysis, numerical mathematics, and other mathematical fields, as well as to all those who apply variational analysis in engineering, physics, operations research, economics, finance, and more.” (Diethard Klatte, SIAM Review, Vol. 57 (2), June, 2015)“The book commences with a helpful context-setting preface followed by six chapters. Each chapter starts with a useful preamble and concludes with a careful and instructive commentary, while a good set of references, a notation guide and a somewhat brief index complete this study. … I unreservedly recommended this book to all practitioners and graduate students interested in modern optimization theory or control theory or to those just engaged by beautiful analysis cleanly described.” (Jonathan Michael Browein, IEEE Control Systems Magazine, February, 2012).“This book is devoted to the theory of inverse and implicit functions and some of its modifications for solution mappings in variational problems. … The book is targeted to a broad audience of researchers, teachers and graduate students. It can be used as well as a textbook as a reference book on the topic. Undoubtedly, it will be used by mathematicians dealing with functional and numerical analysis, optimization, adjacent branches and also by specialists in mechanics, physics, engineering, economics and so on.” (Peter Zabreiko, Zentralblatt MATH, Vol. 1178, 2010).“The present monograph will be a most welcome and valuable addition. … This book will save much time and effort, both for those doing research in variational analysis and for students learning the field. This important contribution fills a gap in the existing literature.” (Stephen M. Robinson, Mathematical Reviews, Issue 2010).Table of ContentsIntroduction and equation-solving background.- Solution mappings for variational problems.- Set-valued analysis of solution mappings.- Regularity properties through generalized derivatives.- Metric regularity in infinite dimensions.- Applications in numerical variational analysis.

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Book SynopsisRandom Number Generators, Principles and Practices has been written for programmers, hardware engineers, and sophisticated hobbyists interested in understanding random numbers generators and gaining the tools necessary to work with random number generators with confidence and knowledge. Using an approach that employs clear diagrams and running code examples rather than excessive mathematics, random number related topics such as entropy estimation, entropy extraction, entropy sources, PRNGs, randomness testing, distribution generation, and many others are exposed and demystified. If you have ever Wondered how to test if data is really random Needed to measure the randomness of data in real time as it is generated Wondered how to get randomness into your programs Wondered whether or not a random number generator is trustworthy Wanted to be able to choose between random number generator solutions Needed to turn uniform random data into a different distribution NeededTable of Contents1 Introduction 1.1 Tools 1.2 Terminology 1.3 The Many Types of Random Numbers 1.3.1 Uniform Random Numbers 2 Random Number Generators 2.1 Classes of Random Number Generators 2.2 Names for RNGs 3 Making Random Numbers 3.1 A Quick Overview of the RNG Types 3.2 The Structure of Full RNG Implementations 3.3 Pool Extractor Structures 3.4 Multiple Input Extractors 4 Physically Uncloneable Functions 21 4.1 The other kind âAS Static vs. Dynamic Random Number Generators . 5 Testing Random Numbers 5.1 Known Answer Tests 5.2 Distinguishing From Random 5.3 PRNG Test Suites 5.4 Entropy Measurements 5.5 Min Entropy Estimation 5.6 Model Equivalence Testing 5.7 Statistical Prerequisite Testing 5.8 The problem Distinguishing Entropy and Pseudo-randomness 5.9 PRNG Tests: DieHarder, NIST SP800-22,TestU01, China ICS 35.040 5.10 Entropy Measurements 5.11 Min Entropy Measurements 5.12 Modeling to Test a Source 5.13 Statistical Prerequisites 5.14 Testing for bias . 5.15 results that are âAŸtoo goodâAZ (E.G. Chi-square == 0.5) 5.16 Distinguishing Correlation from Bias 5.17 Testing for Stationary properties 5.18 FFT analysis 5.19 Online Testing 5.20 Working From the Source RNG 5.21 Tools 5.22 Summary 6 Entropy Extraction or Distillation 6.1 A simple extractor, the XOR gate 6.2 A simple way of improving the distribution of random numbers that have known missing values using XOR 7 Quantifying Entropy 7.1 Rényi Entropy 7.2 Distance From Uniform Topics to put somewhere in the book- in existing chapters and new chapters 8.1 XOR as a 2 bit extractor 8.2 Properties of real random numbers 8.3 Binomial distributions 8.4 Normal distributions 8.4.1 Dice, more dice 8.4.2 Central limit theorem 8.5 Seeing patterns 8.6 Regression to the mean 8.7 Lack of correlation, bias, algorithmic connections, predictability 8.8 What’s a True random number? 8.9 Random numbers in cryptography 8.10 Things they help with liveness, unpredictability, resistance to attacks 8.11 Examples of use 8.11.1 Salting Passwords . 8.11.2 802.11i exchange 8.11.3 PKMv2 exchange 8.11.4 Making Keys 8.12 Examples of RNG crypto failures 8.12.1 Sony PS3 attack 8.12.2 MiFare Classic 8.12.3 Online Poker 8.12.4 Debian OpenSSL Fiasco 8.12.5 Linux Boot Time Entropy 8.13 Humans and random numbers 8.14 Result of asking people for a random number 8.14.1 Normal People 8.14.2 Crypto People 8.15 Mental Random Number Tricks 8.15.1 How to think of a really random number 8.16 PRNGs 8.17 extractors 8.17.1 CBC MAC 8.17.2 BIW 8.17.3 Von Neumann 8.18 Extractor Theory 8.19 Random Number Standards 8.19.1 SP800-90A B C . 8.19.2 Ansi X9.82 8.20 PRNG Algorithms 8.20.1 SP800-90A CTR DRBG 8.20.2 SP800-90A SHA DRBG 8.20.3 XOR Construction 8.20.4 Oversampling Construction 8.21 Yarrow 8.22 Whirlpool 8.23 Linux Kernel random service 8.24 Appendices 8.25 Resources 8.25.1 SW Sources 8.25.2 Online random number sources 8.26 Example Algorithm Vectors 8.26.1 SP800-90A CTR DRBG 128 & 256 8.26.2 SP800-90A Hash DRBG SHA-1 & SHA 256 8.26.3 AES-CBC-MAC Conditioner 128 8.26.4 AES-CBC-MAC Conditioner 8.27 SP800-90 LZ Tests Issues

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Book SynopsisThe recent development of the fuzzy set theory has given scientists the opportunity to model under conditions which are vague or not precisely defined, thus succeeding to solve mathematically problems whose statements are expressed in our natural language. Since Zadeh introduced the concept of fuzzy set in 1965, many efforts have been made by specialists for improving its effectiveness to deal with uncertain, ambiguous and vague situations. As a result a series of extensions and generalizations of the ordinary fuzzy set followed and several theories have been proposed as alternatives to the fuzzy set theory. The spectre of applications of those theories has been rapidly expanded during the last years covering physical sciences, economics and management, expert systems like financial planners, diagnostic, meteorological, information-retrieval, control systems, etc, industry, robotics, decision making, programming, medicine, biology, humanities, education and almost all the other sectors of the human activity, including human reasoning itself. The target of the present book is to become an essential guide to fuzzy sets and systems and to related theories. The whole book consists of ten chapters and a shorter commentary. It starts from the history and an introduction to fuzzy sets and logic and from a brief exposition of related theories. The management of the uncertainty in fuzzy environment as well as the evaluation of fuzzy data, frequently appearing nowadays in science and technology, are also studied. Assessment methods are presented using tools such as triangular fuzzy numbers, fuzzy relation equations and the grey system theory. An introduction to the theory of fuzzy graphs, a review of the hybrids of neural networks and fuzzy logic and an introduction to single valued neutrosophic numbers and the granular calculus of single valued neutrosophic functions are also contained among the topics of the book. More specialized topics include the controllability of non linear fuzzy fractional differential systems, the use of fuzzy probability and fuzzy possibility theory for integrating the voltage sag type detection of electrical networks, the presentation of an algorithm to highlight the importance of using statistical methods in pattern recognition, the study of the known from Physics Goursat problem for a fuzzy hyperbolic equation under the fractional Caputo g-derivative for fuzzy-valued multivariable functions an a hybrid fuzzy potential field method for the navigation of Sumo robots. It is hoped that all the above information can provide a framework to the readers of the book that enable them to proceed to a deeper study of fuzzy systems and the related to them theories.

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Book SynopsisNumerous problems from diverse disciplines can be converted using mathematical modelling to an equation defined on suitable abstract spaces usually involving the n-dimensional Euclidean space or Hilbert space or Banach Space or even more general spaces. The solution of these equations is sought in closed form. But this is possible only in special cases. That is why researchers and practitioners use algorithms which seems to be the only alternative. Due to the explosion of technology, scientific and parallel computing, faster and faster computers become available. This development simply means that new optimized algorithms should be developed to take advantage of these improvements. There is exactly where we come in with our book containing such algorithms with application especially in problems from Economics but also from other areas such as Mathematical: Biology, Chemistry, Physics, Scientific, Parallel Computing, and also Engineering. The book can be used by senior undergraduate students, graduate students, researchers and practitioners in the aforementioned area in the classroom or as a reference material. Readers should know the fundamentals of numerical functional analysis, economic theory, and Newtonian physics. Some knowledge of computers and contemporary programming shall be very helpful to the readers.Table of ContentsPreface; Definition, Existence and Uniqueness of Equilibrium in Oligopoly Markets; Numerical Methodology for Solving Oligopoly Problems; Global Convergence of Iterative Methods with Inverses; Ball Convergence of Third and Fourth Order Methods for Multiple Zeros; Local Convergence of Two Methods for Multiple Roots Eight Order; Choosing the Initial Points for Newtons Method; Extending the Applicability of an Ulm-Like Method under Weak Conditions; Projection Methods for Solving Equations with a Non-differentiable Term; Efficient Seventh Order of Convergence Solver; An Extended Result of Rall-Type for Newtons Method; Extension of Newtons Method for Cone Valued Operators; Inexact Newtons Method under Robinsons Condition; Newtons Method for Generalized Equations with Monotone Operators; Convergence of Newtons method and uniqueness of the solution for Banach Space Valued Equations; Convergence of Newtons method and uniqueness of the solution for Banach Space Valued Equations II; Extended Gauss-Newton Method: Convergence and Uniqueness Results; Newtons Method for Variational Problems: Wangs g-condition and Smales a-theory; Extending the Applicability of Newtons Method; On the Convergence of a Derivative Free Method using Recurrent Functions; Inexact Newton-like Method under Weak Lipschitz Conditions; Ball Convergence Theorem for Inexact Newton Methods in Banach Space; Extending the Semi-Local Convergence of a Stirling-Type Method; Newtons Method for Systems of Equations with Constant Rank Derivatives; Extended Super-Halley Method; Chebyshev-Type Method of Order Three; Extended Semi-Local Convergence of the Chebyshev-Halley Method; Gauss-Newton-Type Schemes for Undetermined Least Squares Problems; Glossary of Symbols.

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Book SynopsisSystems of linear equations are ubiquitous in numerical analysis and scientific computing. and iterative methods are indispensable for the numerical treatment of such systems. This book offers a rigorous introduction to fundamental iterative methods for systems of linear algebraic equations. The book distinguishes itself from other texts on the topic by providing a straightforward yet comprehensive analysis of the Krylov subspace methods, approaching the development and analysis of algorithms from various perspectives, and going beyond the standard description of iterative methods by connecting them in a natural way to the idea of preconditioning. The book supplements standard texts on numerical mathematics for first-year graduate and advanced undergraduate courses and is suitable for advanced graduate classes covering numerical linear algebra and Krylov subspace and multigrid iterative methods. It will be useful to researchers interested in numerical linear algebra and engineers who use iterative methods for solving large algebraic systems.Table of ContentsList of figures; List of algorithms; Preface; 1. Krylov subspace methods; 2. Toeplitz matrices and preconditioners; 3. Multigrid preconditioners; 4. Preconditioners by space decomposition; 5. Some applications; Bibliography; Index.

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Book SynopsisInverse problems need to be solved in order to properly interpret indirect measurements. Often, inverse problems are ill-posed and sensitive to data errors. Therefore one has to incorporate some sort of regularization to reconstruct significant information from the given data.This book presents the main achievements that have emerged in regularization theory over the past 50 years, focusing on linear ill-posed problems and the development of methods that can be applied to them. Some of this material has previously appeared only in journal articles.A Taste of Inverse Problems: Basic Theory and Examples rigorously discusses state-of-the-art inverse problems theory, focusing on numerically relevant aspects and omitting subordinate generalizations; presents diverse real-world applications, important test cases, and possible pitfalls; and treats these applications with the same rigor and depth as the theory.

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Book Synopsis“If mathematical modeling is the process of turning real phenomena into mathematical abstractions, then numerical computation is largely about the transformation from abstract mathematics to concrete reality. Many science and engineering disciplines have long benefited from the tremendous value of the correspondence between quantitative information and mathematical manipulation.” -from the PrefaceFundamentals of Numerical Computation is an advanced undergraduate-level introduction to the mathematics and use of algorithms for the fundamental problems of numerical computation: linear algebra, finding roots, approximating data and functions, and solving differential equations. The book is organized with simpler methods in the first half and more advanced methods in the second half, allowing use for either a single course or a sequence of two courses. The authors take readers from basic to advanced methods, illustrating them with over 200 self-contained MATLAB functions and examples designed for those with no prior MATLAB experience. Although the text provides many examples, exercises, and illustrations, the aim of the authors is not to provide a cookbook per se, but rather an exploration of the principles of cooking.Professors Driscoll and Braun have developed an online resource that includes well-tested materials related to every chapter. Among these materials are lecture-related slides and videos, ideas for student projects, laboratory exercises, computational examples and scripts, and all the functions presented in the book.

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Book SynopsisIn the intervening years since this book was published in 1981, the field of optimization has been exceptionally lively. This fertility has involved not only progress in theory, but also faster numerical algorithms and extensions into unexpected or previously unknown areas such as semidefinite programming. Despite these changes, many of the important principles and much of the intuition can be found in this Classics version of Practical Optimization.This book provides model algorithms and pseudocode, useful tools for users who prefer to write their own code as well as for those who want to understand externally provided code. It presents algorithms in a step-by-step format, revealing the overall structure of the underlying procedures and thereby allowing a high-level perspective on the fundamental differences. And it contains a wealth of techniques and strategies that are well suited for optimization in the twenty-first century, and particularly in the now-flourishing fields of data science, “big data,” and machine learning. Practical Optimization is appropriate for advanced undergraduates, graduate students, and researchers interested in methods for solving optimization problems.

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Book SynopsisThis textbook develops the fundamental skills of numerical analysis: designing numerical methods, implementing them in computer code, and analyzing their accuracy and efficiency. A number of mathematical problems—interpolation, integration, linear systems, zero finding, and differential equations—are considered, and some of the most important methods for their solution are demonstrated and analyzed. Notable features of this book include the development of Chebyshev methods alongside more classical ones; a dual emphasis on theory and experimentation; the use of linear algebra to solve problems from analysis, which enables students to gain a greater appreciation for both subjects; and many examples and exercises.Numerical Analysis: Theory and Experiments is designed to be the primary text for a junior- or senior-level undergraduate course in numerical analysis for mathematics majors. Scientists and engineers interested in numerical methods, particularly those seeking an accessible introduction to Chebyshev methods, will also be interested in this book.

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Book SynopsisBased on first principle quantum mechanics, electronic structure theory is widely used in physics, chemistry, materials science, and related fields and has recently received increasing research attention in applied and computational mathematics. This book provides a self-contained, mathematically oriented introduction to the subject and its associated algorithms and analysis. It will help applied mathematics students and researchers with minimal background in physics understand the basics of electronic structure theory and prepare them to conduct research in this area.A Mathematical Introduction to Electronic Structure Theory begins with an elementary introduction of quantum mechanics, including the uncertainty principle and the Hartree–Fock theory, which is considered the starting point of modern electronic structure theory. The authors then provide an in-depth discussion of two carefully selected topics that are directly related to several aspects of modern electronic structure calculations: density matrix based algorithms and linear response theory. Chapter 2 introduces the Kohn–Sham density functional theory with a focus on the density matrix based numerical algorithms, and Chapter 3 introduces linear response theory, which provides a unified viewpoint of several important phenomena in physics and numerics. An understanding of these topics will prepare readers for more advanced topics in this field. The book concludes with the random phase approximation to the correlation energy.The book is written for advanced undergraduate and beginning graduate students, specifically those with mathematical backgrounds but without a priori knowledge of quantum mechanics, and can be used for self-study by researchers, instructors, and other scientists. The book can also serve as a starting point to learn about many-body perturbation theory, a topic at the frontier of the study of interacting electrons.

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Book SynopsisDynamical systems are a principal tool in the modeling, prediction, and control of a wide range of complex phenomena. As the need for improved accuracy leads to larger and more complex dynamical systems, direct simulation often becomes the only available strategy for accurate prediction or control, inevitably creating a considerable burden on computational resources. This is the main context where one considers model reduction, seeking to replace large systems of coupled differential and algebraic equations that constitute high fidelity system models with substantially fewer equations that are crafted to control the loss of fidelity that order reduction may induce in the system response. Interpolatory methods are among the most widely used model reduction techniques, and Interpolatory Methods for Model Reduction is the first comprehensive analysis of this approach available in a single, extensive resource. It introduces state-of-the-art methods reflecting significant developments over the past two decades, covering both classical projection frameworks for model reduction and data-driven, nonintrusive frameworks.This textbook is appropriate for a wide audience of engineers and other scientists working in the general areas of large-scale dynamical systems and data-driven modeling of dynamics.

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Book SynopsisThis companion piece to the author’s 2018 book, A Software Repository for Orthogonal Polynomials, focuses on Gaussian quadrature and the related Christoffel function. The book makes Gauss quadrature rules of any order easily accessible for a large variety of weight functions and for arbitrary precision. It also documents and illustrates known as well as original approximations for Gauss quadrature weights and Christoffel functions.The repository contains 60 datasets, each dealing with a particular weight function. Included are classical, quasi-classical, and, most of all, nonclassical weight functions and associated orthogonal polynomials.

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Book SynopsisNumerical Linear Algebra and Optimization covers the fundamentals of closely related topics: linear systems (linear equations and least-squares) and linear programming (optimizing a linear function subject to linear constraints). For each problem class, stable and efficient numerical algorithms intended for a finite-precision environment are derived and analyzed. In 1991, when the book first appeared, these topics were rarely taught with a unified perspective, and, somewhat surprisingly, this remains true almost 30 years later. As a result, some of the material in this book can be difficult to find elsewhere—in particular, techniques for updating the LU factorization, descriptions of the simplex method applied to all-inequality form, and the analysis of what happens when using an approximate inverse to solve Ax=b.This book is appropriate for students who want to learn about numerical techniques for solving linear systems and/or linear programming using the simplex method.

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Book SynopsisOver seventy years ago, Richard Bellman coined the term "the curse of dimensionality" to describe phenomena and computational challenges that arise in high dimensions. These challenges, in tandem with the ubiquity of high-dimensional functions in real-world applications, have led to a lengthy, focused research effort on high-dimensional approximation—that is, the development of methods for approximating functions of many variables accurately and efficiently from data. This book provides an in-depth treatment of one of the latest installments in this long and ongoing story: sparse polynomial approximation methods. These methods have emerged as useful tools for various high-dimensional approximation tasks arising in a range of applications in computational science and engineering. It begins with a comprehensive overview of best s-term polynomial approximation theory for holomorphic, high-dimensional functions, as well as a detailed survey of applications to parametric differential equations. It then describes methods for computing sparse polynomial approximations, focusing on least squares and compressed sensing techniques.Sparse Polynomial Approximation of High-Dimensional Functions presents the first comprehensive and unified treatment of polynomial approximation techniques that can mitigate the curse of dimensionality in high-dimensional approximation, including least squares and compressed sensing. It develops main concepts in a mathematically rigorous manner, with full proofs given wherever possible, and it contains many numerical examples, each accompanied by downloadable code. The authors provide an extensive bibliography of over 350 relevant references, with an additional annotated bibliography available on the book's companion website (www.sparse-hd-book.com).This text is aimed at graduate students, postdoctoral fellows, and researchers in mathematics, computer science, and engineering who are interested in high-dimensional polynomial approximation techniques.

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Book SynopsisIterative methods use successive approximations to obtain more accurate solutions. Iterative Methods and Preconditioners for Systems of Linear Equations presents historical background, derives complete convergence estimates for all methods, illustrates and provides Matlab codes for all methods, and studies and tests all preconditioners first as stationary iterative solvers. This textbook is appropriate for undergraduate and graduate students in need of an overview or of deeper knowledge about iterative methods. It can be used in courses on Advanced Numerical Analysis, Special Topics on Numerical Analysis, Topics on Data Science, Topics on Numerical Optimization, and Topics on Approximation Theory. Scientists and engineers interested in new topics and applications will also find the text useful.

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Book SynopsisFit for students just starting to build a background in mathematics, this textbook provides an introduction to numerical methods for linear algebra problems.Introduction to Numerical Linear Algebra is ideal for a flipped classroom, as it provides detailed explanations that allow students to read on their own and instructors to go beyond lecturing, assumes that the reader has taken a course on linear algebra, but reviews background as needed, and covers several topics not commonly addressed in related introductory books, including diffusion, a toy model of computed tomography, global positioning systems, the use of eigenvalues in analyzing stability of equilibria, a detailed derivation and careful motivation of the QR method for eigenvalues starting from power iteration, a discussion of the use of the SVD for assigning grades, and multigrid methods. This textbook is appropriate for undergraduate and beginning graduate students in mathematics and related fields. It can be used in the following courses: Advanced Numerical Analysis, Special Topics on Numerical Analysis, Topics on Data Science, Topics on Numerical Optimization, and Topics on Approximation Theory

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Book SynopsisThis expansive volume describes the history of numerical methods proposed to solve linear algebra problems,, from antiquity to the present day. The authors focus on methods for solving linear systems of equations and eigenvalue problems and describe the interplay between numerical methods and the computing tools available at the time. The second part of the book consists of 78 biographies of the main important contributors to the field.A Journey through the History of Numerical Linear Algebra will be of special interest to applied mathematicians, especially researchers in numerical linear algebra, and to applied mathematiciansas well as to and historians of mathematics as well.

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Book SynopsisReduced order modeling is an important, growing field in computational science and engineering, and this is the first book to address the subject in relation to computational fluid dynamics. It focuses on complex parametrization of shapes for their optimization and includes recent developments in advanced topics such as turbulence, stability of flows, inverse problems, optimization, and flow control, as well as applications.This book will be of interest to researchers and graduate students in the field of reduced order modeling.

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Book SynopsisRounding Errors in Algebraic Processes was the first book to give systematic analyses of the effects of rounding errors on a variety of key computations involving polynomials and matrices.A detailed analysis is given of the rounding errors made in the elementary arithmetic operations and inner products, for both floating-point arithmetic and fixed-point arithmetic. The results are then applied in the error analyses of a variety of computations involving polynomials as well as the solution of linear systems, matrix inversion, and eigenvalue computations.The conditioning of these problems is investigated. The aim was to provide a unified method of treatment, and emphasis is placed on the underlying concepts.This book is intended for mathematicians, computer scientists, those interested in the historical development of numerical analysis, and students in numerical analysis and numerical linear algebra.Trade Review[This book] combines a rigorous mathematical analysis with a practicality that stems from an obvious first-hand contact with the actual numerical computation. The well-chosen examples alone show vividly both the importance of the study of rounding errors and the perils of its neglect. A. A. Grau, SIAM Review (1966)

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Book SynopsisDivided into two self-contained parts, this textbook is an introduction to modern real analysis. More than 350 exercises and 100 examples are integrated into the text to help clarify the theoretical considerations and the practical applications to differential geometry, Fourier series, differential equations, and other subjects. The first section of Classical Analysis of Real-Valued Functions covers the theorems of existence of supremum and infimum of bounded sets on the real line and the Lagrange formula for differentiable functions. Applications of these results are crucial for classical mathematical analysis, andmany are threaded through the text. In the second part of the book, the implicit function theorem plays a central role, while the Gauss–Ostrogradskii formula, surface integration, Heine–Borel lemma, the Ascoli–Arzelà theorem, and the one-dimensional indefinite Lebesgue integral are also covered. This book is intended for students in the first and second years of classical universities majoring in pure and applied mathematics, but students of engineering disciplines will also gain important and helpful insights. It is appropriate for courses in mathematical analysis, functional analysis, real analysis, and calculus and can be used for self-study as well.

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