by David R. Kincaid and E. Ward Cheney
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This highly successful and scholarly book introduces students with diverse backgrounds to the various types of mathematical analysis that are commonly needed in scientific computing. The subject of numerical analysis is treated from a mathematical point of view, offering a complete analysis of methods for scientific computing with careful proofs and scientific background. An in-depth treatment of the topics of numerical analysis, a more scholarly approach, and a different menu of topics sets this book apart from the authors' well-respected and best-selling text: NUMERICAL MATHEMATICS AND COMPUTING, FOURTH EDITION.
Benefits:
Author Bio
Kincaid, David R. : University of Texas at Austin
Cheney, E. Ward : University of Texas at Austin
Preface. Numerical Analysis: What Is It?
1. Mathematical Preliminaries.
Basic Concepts and Taylor's Theorem. Orders of Convergence and Additional Basic Concepts. Difference Equations.
2. Computer Arithmetic.
Floating-Point Numbers and Roundoff Errors. Absolute and Relative Errors: Loss of Significance. Stable and Unstable Computations: Conditioning.
3. Solution of Nonlinear Equations.
Bisection (Interval Halving) Method. Newton's Method. Secant Method. Fixed Points and Functional Iteration. Computing Zeros of Polynomials. Hornotopy and Continuation Methods.
4. Solving Systems of Linear Equations.
Matrix Algebra. The LU and Cholesky Factorizations. Pivoting and Constructing an Algorithm. Norms and the Analysis of Errors. Nuemann Series and Interactive Refinement. Solution of Equations by Iterative Methods. Steepest Descent and Conjugate Gradient Methods. Analysis of Roundoff Error in the Gaussian Algorithm.
5. Selected Topics in Numerical Linear Algebra.
Matrix Eigenvalue Problem: Power Method. Schur's and Gershgorin's Theorems. Orthogonal Factorizations and Least-Squares Problems. Singular-Value Decomposition and Pseudoinverses. The QR-Algorithm of Francis for the Eigenvalue Problem.
6. Approximation Functions.
Polynomial Interpolation. Divided Differences. Hermite Interpolation. Spline Interpolation. The B-Splines: Basic Theory. The B-Splines: Applications. Taylor Series. Best Approximation: Least-Squares Theory. Best Approximation: Chebyshev Theory. Interpolation in Higher Dimension. Continued Fractions. Trigonometric Interpolation. Fast Fourier Transform. Adaptive Approximation.
7. Numerical Differentiation and Integration.
Numerical Differentiation and Richardson Extrapolation. Numerical Integration Based on Interpolation. Gaussian Quadrature. Romberg Integration. Adaptive Quadrature. Sard's Theory of Approximating Functionals. Bernoulli Polynomials and the Euler-Maclaurin Formula.
8. Numerical Solution of Ordinary Differential Equations.
The Existence and Uniqueness of Solutions. Taylor-Series Methods. Runge-Kutta Methods. Multistep Methods. Local and Global Errors: Stability. Systems and Higher-Order Ordinary Differential Equations. Boundary-Value Problems. Boundary-Value Problems: Shooting Methods. Boundary-Value Problems: Finite-Difference Methods. Boundary-Value Problems: Collocation. Linear Differential Equations. Stiff Equations.
9. Numerical Solution of Partial Differential Equations.
Parabolic Equations: Explicit Methods. Parabolic Equations: Implicit Methods. Problems Without Time Dependence: Finite-Difference Methods. Problems Without Time Dependence: Galerkin and Ritz Methods. First-Order Partial Differential Equations: Characteristic Curves. Quasilinear Second-Order Equations: Characteristics. Other Methods for Hyperbolic Problems. Multigrid Method. Fast Methods for Poisson's Equation.
10. Linear Programming and Related Topics.
Convexity and Linear Inequalities. Linear Inequalities. Linear Programming. The Simplex Algorithm.
11. Optimization.
One-Variable Case. Descent Methods. Analysis of Quadratic Objective Functions. Nelder-Meade Algorithm. Simulated Annealing. Genetic Algorithms. Convex Programming. Constrained Minimization. Pareto Optimization.
Overview of Mathematical Software.
Bibliography.
Index.
David R. Kincaid and E. Ward Cheney
ISBN13: 978-0821847886This highly successful and scholarly book introduces students with diverse backgrounds to the various types of mathematical analysis that are commonly needed in scientific computing. The subject of numerical analysis is treated from a mathematical point of view, offering a complete analysis of methods for scientific computing with careful proofs and scientific background. An in-depth treatment of the topics of numerical analysis, a more scholarly approach, and a different menu of topics sets this book apart from the authors' well-respected and best-selling text: NUMERICAL MATHEMATICS AND COMPUTING, FOURTH EDITION.
Benefits:
Author Bio
Kincaid, David R. : University of Texas at Austin
Cheney, E. Ward : University of Texas at Austin
Table of Contents
Preface. Numerical Analysis: What Is It?
1. Mathematical Preliminaries.
Basic Concepts and Taylor's Theorem. Orders of Convergence and Additional Basic Concepts. Difference Equations.
2. Computer Arithmetic.
Floating-Point Numbers and Roundoff Errors. Absolute and Relative Errors: Loss of Significance. Stable and Unstable Computations: Conditioning.
3. Solution of Nonlinear Equations.
Bisection (Interval Halving) Method. Newton's Method. Secant Method. Fixed Points and Functional Iteration. Computing Zeros of Polynomials. Hornotopy and Continuation Methods.
4. Solving Systems of Linear Equations.
Matrix Algebra. The LU and Cholesky Factorizations. Pivoting and Constructing an Algorithm. Norms and the Analysis of Errors. Nuemann Series and Interactive Refinement. Solution of Equations by Iterative Methods. Steepest Descent and Conjugate Gradient Methods. Analysis of Roundoff Error in the Gaussian Algorithm.
5. Selected Topics in Numerical Linear Algebra.
Matrix Eigenvalue Problem: Power Method. Schur's and Gershgorin's Theorems. Orthogonal Factorizations and Least-Squares Problems. Singular-Value Decomposition and Pseudoinverses. The QR-Algorithm of Francis for the Eigenvalue Problem.
6. Approximation Functions.
Polynomial Interpolation. Divided Differences. Hermite Interpolation. Spline Interpolation. The B-Splines: Basic Theory. The B-Splines: Applications. Taylor Series. Best Approximation: Least-Squares Theory. Best Approximation: Chebyshev Theory. Interpolation in Higher Dimension. Continued Fractions. Trigonometric Interpolation. Fast Fourier Transform. Adaptive Approximation.
7. Numerical Differentiation and Integration.
Numerical Differentiation and Richardson Extrapolation. Numerical Integration Based on Interpolation. Gaussian Quadrature. Romberg Integration. Adaptive Quadrature. Sard's Theory of Approximating Functionals. Bernoulli Polynomials and the Euler-Maclaurin Formula.
8. Numerical Solution of Ordinary Differential Equations.
The Existence and Uniqueness of Solutions. Taylor-Series Methods. Runge-Kutta Methods. Multistep Methods. Local and Global Errors: Stability. Systems and Higher-Order Ordinary Differential Equations. Boundary-Value Problems. Boundary-Value Problems: Shooting Methods. Boundary-Value Problems: Finite-Difference Methods. Boundary-Value Problems: Collocation. Linear Differential Equations. Stiff Equations.
9. Numerical Solution of Partial Differential Equations.
Parabolic Equations: Explicit Methods. Parabolic Equations: Implicit Methods. Problems Without Time Dependence: Finite-Difference Methods. Problems Without Time Dependence: Galerkin and Ritz Methods. First-Order Partial Differential Equations: Characteristic Curves. Quasilinear Second-Order Equations: Characteristics. Other Methods for Hyperbolic Problems. Multigrid Method. Fast Methods for Poisson's Equation.
10. Linear Programming and Related Topics.
Convexity and Linear Inequalities. Linear Inequalities. Linear Programming. The Simplex Algorithm.
11. Optimization.
One-Variable Case. Descent Methods. Analysis of Quadratic Objective Functions. Nelder-Meade Algorithm. Simulated Annealing. Genetic Algorithms. Convex Programming. Constrained Minimization. Pareto Optimization.
Overview of Mathematical Software.
Bibliography.
Index.