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.