Numerical methods
Author(s)
Bibliographic Information
Numerical methods
AK Peters , Vieweg, c1993
- : AK Peters
- : Vieweg
Available at / 22 libraries
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
: AK PetersBOE||4||194059645
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Hokkaido University, Library, Graduate School of Science, Faculty of Science and School of Science図書
dc20:519.4/b6332070256113
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Note
Includes bibliographical references and index
Description and Table of Contents
- Volume
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: AK Peters ISBN 9781568810201
Description
This book is written for engineers and other practitioners using numerical methods in their work and serves as a textbook for courses in applied mathematics and numerical analysis.
Table of Contents
Preface -- I FUNDAMENTAL CONCEPTS -- 1 Algorithms and Error Propagation -- 1.1 Algorithms -- 1.2 The Implementation of Algorithms -- 1.3 Judging an Algorithm -- 1.4 Notes and Exercises -- 2 Matrices -- 2.1 Notations -- 2.2 Products of Matrices -- 2.3 Falk's Scheme -- 2.4 Rank and Determinant -- 2.5 Norm and Convergence -- 2.6 Notes and Exercises -- II LINEAR EQUATIONS AND INEQUALITIES -- 3 Gaussian Elimination -- 3.1 Backward Substitution -- 3.2 Gaussian Elimination -- 3.3 Pivoting -- 3.4 Notes and Exercises -- 4 The LU Factorization -- 4.1 The LU Factorization of A -- 4.2 LU Factorization with Pivoting -- 4.3 Systems of Linear Equations -- 4.4 Notes and Exercises -- 5 The Exchange Algorithm -- 5.1 Exchanging Variables -- 5.2 Scheme and Algorithm -- 5.3 Inversion -- 5.4 Linear Equations -- 5.5 Notes and Exercises -- 6 The Cholesky Factorization -- 6.1 Symmetrical Factorization -- 6.2 Existence and Uniqueness -- 6.3 Symmetric Systems of Linear Equations -- 6.4 Iterative Refinement -- 6.5 Notes and Exercises -- 7 The QU Factorization -- 7.1 The Householder Transformation -- 7.2 The Householder Algorithm -- 7.3 Systems of Linear Equations -- 7.4 Notes and Exercises -- 8 Relaxation Methods -- 8.1 Coordinate Relaxation -- 8.2 Convergence for Diagonally Dominant Matrices -- 8.3 The Minimum Problem -- 8.4 Convergence for Symmetric, Positive Definite Matrices -- 8.5 Geometric Meaning -- 8.6 Notes and Exercises -- 9 Data Fitting -- 9.1 Overdetermined Systems of Linear Equations -- 9.2 Using the QU Factorization -- 9.3 Application -- 9.4 Under determined Systems of Linear Equations -- 9.5 Application -- 9.6 Geometric Meaning and Duality -- 9.7 Notes and Exercises -- 10 Linear Optimization -- 10.1 Linear Inequalities and Linear Programming -- 10.2 Exchanging Vertices and the Simplex Method -- 10.3 Elimination -- 10.4 Data Fitting after Chebyshev -- 10.5 Notes and Exercises -- III ITERATION -- 11 Vector Iteration -- 11.1 The Eigenvalue Problem for Matrices -- 11.2 Vector Iteration after von Mises -- 11.3 Inverse Iteration -- 11.4 Improving an Approximation -- 11.5 Notes and Exercises -- 12 The LR Algorithm -- 12.1 The Algorithm of Rutishauser -- 12.2 Proving Convergence -- 12.3 Pairs of Eigenvalues with Equal Modulus -- 12.4 Notes and Exercises -- 13 One-Dimensional Iteration -- 13.1 Contractive Mappings -- 13.2 Error Bounds -- 13.3 Rate of Convergence -- 13.4 Aitken's A2-Method -- 13.5 Geometric Acceleration -- 13.6 Roots -- 13.7 Notes and Exercises -- 14 Multi-Dimensional Iteration -- 14.1 Contractive Mappings -- 14.2 Rate of Convergence -- 14.3 Accelerating the Convergence -- 14.4 Roots of Systems -- 14.5 Notes and Exercises -- 15 Roots of Polynomials -- 15.1 The Horner Scheme -- 15.2 The Extended Horner Scheme -- 15.3 Simple Roots -- 15.4 Bairstow's Method -- 15.5 The Extended Horner Scheme for Quadratic Factors -- 15.6 Notes and Exercises -- 16 Bernoulli's Method -- 16.1 Linear Difference Equations -- 16.2 Matrix Notation -- 16.3 Bernoulli's Method -- 16.4 Inverse Iteration -- 16.5 Notes and Exercises -- 17 The QD Algorithm -- 17.1 The LR Algorithm for Tridiagonal Matrices -- 17.2 The QD scheme for Polynomials -- 17.3 Pairs of Zeros with Equal Modulus -- 17.4 Notes and Exercises -- IV INTERPOLATION AND DISCRETE APPROXIMATION -- 18 Interpolation -- 18.1 Interpolation Polynomials -- 18.2 Lagrange Polynomials -- 18.3 Lagrange Form -- 18.4 Newton Form -- 18.5 Aitken's Lemma -- 18.6 Neville's Scheme -- 18.7 Hermite Interpolation -- 18.8 Piecewise Hermite Interpolation -- 18.9 The Cardinal Hermite Basis -- 18.10 More-Dimensional Interpolation -- 18.11 Surface Patches of Coons and Gordon -- 18.12 Notes and Exercises -- 19 Discrete Approximation -- 19.1 The Taylor Polynomials -- 19.2 The Interpolation Polynomial -- 19.3 Chebyshev Approximation -- 19.4 Chebyshev Polynomials -- 19.5 The Minimum Property -- 19.6 Expanding by Chebyshev Polynomials -- 19.7 Economization of Polynomials -- 19.8 Least Squares Method -- 19.9 The Orthogonality of Chebyshev Polynomials -- 19.10 Notes and Exercises -- 20 Polynomials in Bezier Form -- 20.1 Bernstein Polynomials -- 20.2 Polynomials in Bezier Form -- 20.3 The Construction of Position and Tangent -- 20.4 Bezier Surfaces -- 20.5 Notes and Exercises -- 21 Splines -- 21.1 Bezier Curves -- 21.2 Differentiability Conditions -- 21.3 Cubic Splines -- 21.4 The Minimum Property -- 21.5 B-Splines and Truncated Power Functions -- 21.6 Normalized B-Splines -- 21.7 De Boor's Algorithm -- 21.8 Notes and Exercises -- V NUMERICAL DIFFERENTIATION AND INTEGRATION -- 22 Numerical Differentiation and Integration -- 22.1 Numerical Differentiation -- 22.2 Error Estimates for the Numerical Differentiation -- 22.3 Numerical Integration -- 22.4 Composite Integration Rules -- 22.5 Error Estimation for the Numerical Integration -- 22.6 Notes and Exercises -- 23 Extrapolation -- 23.1 Sequences of Approximations -- 23.2 Richardson Extrapolation -- 23.3 Iterated Richardson Extrapolation -- 23.4 Romberg Integration -- 23.5 Notes and Exercises -- 24 One-Step Methods for Differential Equations -- 24.1 Discretization -- 24.2 Discretization Error -- 24.3 Runge-Kutta Methods -- 24.4 Error Control -- 24.5 Notes and Exercises -- 25 Linear Multi-Step Methods for Differential Equations -- 25.1 Discretization -- 25.2 Convergence of Multi-Step Methods -- 25.3 Root Condition -- 25.4 Sufficient Convergence Conditions -- 25.5 Starting Values -- 25.6 Predictor-Corrector Methods -- 25.7 Step Size Control -- 25.8 Comparing One- and Multi-Step Methods -- 25.9 Notes and Exercises -- 26 The Methods by Ritz and Galerkin -- 26.1 The Principle of Minimal Energy -- 26.2 The Ritz Method -- 26.3 Galerkin's Method -- 26.4 Relation -- 26.5 Notes and Exercises -- 27 The Finite Element Method -- 27.1 Finite Elements -- 27.2 Univariate Splines -- 27.3 Bivariate Splines -- 27.4 Numerical Examples -- 27.5 Local Coordinates -- 27.6 Notes and Exercises -- 28 Bibliography -- Index.
- Volume
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: Vieweg ISBN 9783528063504
Description
This volume addresses undergraduate students in mathematics, computing science, and all other fields of science and engineering. The student who studied advanced calculus and has a basic knowledge of linear algebra should be well equipped for the material. This book provides an introduction to methods of numerical analysis. Of principal concern in writing this book was to expose fundamental ideas of algorithms for the solution of widely varied mathematical problems. The student should be enabled to deal with related questions, and to apply principles introduced here to new problems. Difficulties arising just from the notation or the rigour of proofs should not be obstacles covering the principal ideas of a method. To encourage the programming of the methods explicit algorithms are given throughout. For better understanding, the algorithms are always presented in a pseudo code. Simple examples further illustrate the methods. This book has grown out of a one-semester upper-level undergraduate course repeatedly held by Professor Boehm at the Technical University of Braunschweig.
by "Nielsen BookData"