Numerical analysis : a graduate course / / David E. Stewart

• Record Nr.: UNISA996499866403316
• Autore: Stewart David E.
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2023]; ©2023
• ISBN: 9783031081217; 9783031081200
• Descrizione fisica: 1 online resource (645 pages)
• Collana: CMS/CAIMS Books in Mathematics ; ; v. 4
• Disciplina: 620.00151535
• Soggetti: Differential equations; Numerical analysis; Equacions diferencials; Anàlisi numèrica; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- Preface -- Contents -- 1 Basics of Numerical Computation -- 1.1 How Computers Work -- 1.1.1 The Central Processing Unit -- 1.1.2 Code and Data -- 1.1.3 On Being Correct -- 1.1.4 On Being Efficient -- 1.1.5 Recursive Algorithms and Induction -- 1.1.6 Working in Groups: Parallel Computing -- 1.1.7 BLAS and LAPACK -- Exercises -- 1.2 Programming Languages -- 1.2.1 MATLABTM -- 1.2.2 Julia -- 1.2.3 Python -- 1.2.4 C/C++ and Java -- 1.2.5 Fortran -- Exercises -- 1.3 Floating Point Arithmetic -- 1.3.1 The IEEE Standards -- 1.3.2 Correctly Rounded Arithmetic -- 1.3.3 Future of Floating Point Arithmetic -- Exercises -- 1.4 When Things Go Wrong -- 1.4.1 Underflow and Overflow -- 1.4.2 Subtracting Nearly Equal Quantities -- 1.4.3 Numerical Instability -- 1.4.4 Adding Many Numbers -- Exercises -- 1.5 Measuring: Norms -- 1.5.1 What Is a Norm? -- 1.5.2 Norms of Functions -- Exercises -- 1.6 Taylor Series and Taylor Polynomials -- 1.6.1 Taylor Series in One Variable -- 1.6.2 Taylor Series and Polynomials in More than One Variable -- 1.6.3 Vector-Valued Functions -- Exercises -- Project -- 2 Computing with Matrices and Vectors -- 2.1 Solving Linear Systems -- 2.1.1 Gaussian Elimination -- ; 2.1.2 LU Factorization -- 2.1.3 Errors in Solving Linear Systems -- 2.1.4 Pivoting and PA=LU -- 2.1.5 Variants of LU Factorization -- Exercises -- 2.2 Least Squares Problems -- 2.2.1 The Normal Equations -- 2.2.2 QR Factorization -- Exercises -- 2.3 Sparse Matrices -- 2.3.1 Tridiagonal Matrices -- 2.3.2 Data Structures for Sparse Matrices -- 2.3.3 Graph Models of Sparse Factorization -- 2.3.4 Unsymmetric Factorizations -- Exercises -- 2.4 Iterations -- 2.4.1 Classical Iterations -- 2.4.2 Conjugate Gradients and Krylov Subspaces -- 2.4.3 Non-symmetric Krylov Subspace Methods -- Exercises -- 2.5 Eigenvalues and Eigenvectors -- 2.5.1 The Power Method &amp -- Google.; 2.5.2 Schur Decomposition -- 2.5.3 The QR Algorithm -- 2.5.4 Singular Value Decomposition -- 2.5.5 The Lanczos and Arnoldi Methods -- Exercises -- 3 Solving nonlinear equations -- 3.1 Bisection method -- 3.1.1 Convergence -- 3.1.2 Robustness and reliability -- Exercises -- 3.2 Fixed-point iteration -- 3.2.1 Convergence -- 3.2.2 Robustness and reliability -- 3.2.3 Multivariate fixed-point iterations -- Exercises -- 3.3 Newton's method -- 3.3.1 Convergence of Newton's method -- 3.3.2 Reliability of Newton's method -- 3.3.3 Variant: Guarded Newton method -- 3.3.4 Variant: Multivariate Newton method -- Exercises -- 3.4 Secant and hybrid methods -- 3.4.1 Convenience: Secant method -- 3.4.2 Regula Falsi -- 3.4.3 Hybrid methods: Dekker's and Brent's methods -- Exercises -- 3.5 Continuation methods -- 3.5.1 Following paths -- 3.5.2 Numerical methods to follow paths -- Exercises -- Project -- 4 Approximations and Interpolation -- 4.1 Interpolation-Polynomials -- 4.1.1 Polynomial Interpolation in One Variable -- 4.1.2 Lebesgue Numbers and Reliability -- Exercises -- 4.2 Interpolation-Splines -- 4.2.1 Cubic Splines -- 4.2.2 Higher Order Splines in One Variable -- Exercises -- 4.3 Interpolation-Triangles and Triangulations -- 4.3.1 Interpolation over Triangles -- 4.3.2 Interpolation over Triangulations -- 4.3.3 5021671En4FigdPrint.eps Approximation Error over Triangulations -- 4.3.4 Creating Triangulations -- Exercises -- 4.4 Interpolation-Radial Basis Functions -- Exercises -- 4.5 Approximating Functions by Polynomials -- 4.5.1 Weierstrass' Theorem -- 4.5.2 Jackson's Theorem -- 4.5.3 Approximating Functions on Rectangles and Cubes -- 4.6 Seeking the Best-Minimax Approximation -- 4.6.1 Chebyshev's Equi-oscillation Theorem -- 4.6.2 Chebyshev Polynomials and Interpolation -- 4.6.3 Remez Algorithm -- 4.6.4 Minimax Approximation in Higher Dimensions -- Exercises.; 4.7 Seeking the Best-Least Squares -- 4.7.1 Solving Least Squares -- 4.7.2 Orthogonal Polynomials -- 4.7.3 Trigonometric Polynomials and Fourier Series -- 4.7.4 Chebyshev Expansions -- Exercises -- Project -- 5 Integration and Differentiation -- 5.1 Integration via Interpolation -- 5.1.1 Rectangle, Trapezoidal and Simpson's Rules -- 5.1.2 Newton-Cotes Methods -- 5.1.3 Product Integration Methods -- 5.1.4 Extrapolation -- 5.2 Gaussian Quadrature -- 5.2.1 Orthogonal Polynomials Reprise -- 5.2.2 Orthogonal Polynomials and Integration -- 5.2.3 Why the Weights are Positive -- 5.3 Multidimensional Integration -- 5.3.1 Tensor Product Methods -- 5.3.2 Lagrange Integration Methods -- 5.3.3 Symmetries and Integration -- 5.3.4 Triangles and Tetrahedra -- 5.4 High-Dimensional Integration -- 5.4.1 Monte Carlo Integration -- 5.4.2 Quasi-Monte Carlo Methods -- 5.5 Numerical Differentiation -- 5.5.1 Discrete Derivative Approximations -- 5.5.2 Automatic Differentiation -- 6 Differential Equations -- 6.1 Ordinary Differential Equations - Initial Value Problems -- 6.1.1 Basic Theory -- 6.1.2 Euler's Method and Its Analysis -- 6.1.3 Improving on Euler: Trapezoidal, Midpoint, and Heun -- 6.1.4 Runge-Kutta Methods ; -- 6.1.5 Multistep Methods -- 6.1.6 Stability and Implicit Methods -- 6.1.7 Practical Aspects of Implicit Methods -- 6.1.8 Error Estimates and Adaptive Methods -- 6.1.9 Differential Algebraic Equations (DAEs) -- Exercises -- 6.2 Ordinary Differential Equations-Boundary Value Problems -- 6.2.1 Shooting Methods -- 6.2.2 Multiple Shooting -- 6.2.3 Finite Difference Approximations -- Exercises -- 6.3 Partial Differential Equations-Elliptic Problems -- 6.3.1 Finite Difference Approximations -- 6.3.2 Galerkin Method -- 6.3.3 Handling Boundary Conditions -- 6.3.4 Convection-Going with the Flow -- 6.3.5 Higher Order Problems -- Exercises.; 6.4 Partial Differential Equations-Diffusion and Waves -- 6.4.1 Method of Lines -- Exercises -- Projects -- 7 Randomness -- 7.1 Probabilities and Expectations -- 7.1.1 Random Events and Random Variables -- 7.1.2 Expectation and Variance -- 7.1.3 Averages -- Exercises -- 7.2 Pseudo-Random Number Generators -- 7.2.1 The Arithmetical Generation of Random Digits -- 7.2.2 Modern Pseudo-Random Number Generators -- 7.2.3 Generating Samples from Other Distributions -- 7.2.4 Parallel Generators -- Exercises -- 7.3 Statistics -- 7.3.1 Averages and Variances -- 7.3.2 Regression and Curve Fitting -- 7.3.3 Hypothesis Testing -- Exercises -- 7.4 Random Algorithms -- 7.4.1 Random Choices -- 7.4.2 Monte Carlo Algorithms and Markov Chains -- Exercises -- 7.5 Stochastic Differential Equations -- 7.5.1 Wiener Processes -- 7.5.2 Itô Stochastic Differential Equations -- 7.5.3 Stratonovich Integrals and Differential Equations -- 7.5.4 Euler-Maruyama Method -- 7.5.5 Higher Order Methods for Stochastic Differential Equations -- Exercises -- Project -- 8 Optimization -- 8.1 Basics of Optimization -- 8.1.1 Existence of Minimizers -- 8.1.2 Necessary Conditions for Local Minimizers -- 8.1.3 Lagrange Multipliers and Equality-Constrained Optimization -- Exercises -- 8.2 Convex and Non-convex -- 8.2.1 Convex Functions -- 8.2.2 Convex Sets -- Exercises -- 8.3 Gradient Descent and Variants -- 8.3.1 Gradient Descent -- 8.3.2 Line Searches -- 8.3.3 Convergence -- 8.3.4 Stochastic Gradient Method -- 8.3.5 Simulated Annealing -- Exercises -- 8.4 Second Derivatives and Newton's Method -- Exercises -- 8.5 Conjugate Gradient and Quasi-Newton Methods -- 8.5.1 Conjugate Gradients for Optimization -- 8.5.2 Variants on the Conjugate Gradient Method -- 8.5.3 Quasi-Newton Methods -- Exercises -- 8.6 Constrained Optimization -- 8.6.1 Equality Constrained Optimization.; 8.6.2 Inequality Constrained Optimization -- Exercises -- Project -- Appendix A What You Need from Analysis -- A.1 Banach and Hilbert Spaces -- A.1.1 Normed Spaces and Completeness -- A.1.2 Inner Products -- A.1.3 Dual Spaces and Weak Convergence -- A.2 Distributions and Fourier Transforms -- A.2.1 Distributions and Measures -- A.2.2 Fourier Transforms -- A.3 Sobolev Spaces -- Appendix References -- -- Index.

Beijing Urban Memory : Historic Buildings and Historic Areas, Central Axes and City Walls / / by Fang Wang

• Record Nr.: UNINA9910254877203321
• Autore: Wang Fang
• Pubbl/distr/stampa: Singapore : , : Springer Singapore : , : Imprint : Springer, , 2016
• ISBN: 981-10-0678-4
• Edizione: [1st ed. 2016.]
• Descrizione fisica: 1 online resource (267 p.)
• Disciplina: 330
• Soggetti: Culture—Economic aspects; Regional planning; City planning; Cultural Economics; Landscape/Regional and Urban Planning; Urbanism
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references.
• Nota di contenuto: 1 About the Development of Urban Memory -- 2 Point: Historic Buildings in Beijing’s Measurement of Urban Memory -- 3 Line: Central Axis of Beijing’s Measurement of Urban Memory -- 4 Line: City Wall of Beijing’s Measurement of Urban Memory -- 5 Place: Historic Sites of Beijing’s Measurement of Urban Memory -- 6 Conclusion and Discussion -- References. .
• Sommario/riassunto: From the cross-disciplinary perspective of urban management and planning, geography and architecture, this book explores the theory and methods of urban memory, selecting Beijing's historic buildings, historic areas, central areas and city walls as research cases. It is divided into three parts: factors analysis, modeling and practical application. It lays a scientific foundation and provides practical methods for the management of historical spaces, residents’ and commercial activities, optimizing the layout and structure of the historic spaces, updating the protection of old buildings, promoting the organic growth of historic sites and the sustainable development of urbanization with new concepts.