Introduction to Numerical Analysis

Author： Arnold Neumaier

Publisher： Cambridge University Press‎

Publication year： 2001

E-ISBN: 9780511890222

P-ISBN(Paperback): 9780521333238

Subject： O241 数值分析

Keyword：数学理论

Language： ENG

Access to resources Favorite

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Introduction to Numerical Analysis

Description

Numerical analysis is an increasingly important link between pure mathematics and its application in science and technology. This textbook provides an introduction to the justification and development of constructive methods that provide sufficiently accurate approximations to the solution of numerical problems, and the analysis of the influence that errors in data, finite-precision calculations, and approximation formulas have on results, problem formulation and the choice of method. It also serves as an introduction to scientific programming in MATLAB, including many simple and difficult, theoretical and computational exercises. A unique feature of this book is the consequent development of interval analysis as a tool for rigorous computation and computer assisted proofs, along with the traditional material.

Chapter

1. The Numerical Evaluation of Expressions

1.1 Arithmetic Expressions and Automatic Differentiation

1.2 Numbers, Operations, and Elementary Functions

1.3 Numerical Stability

1.4 Error Propagation and Condition

1.5 Interval Arithmetic

1.6 Exercises

2. Linear Systems of Equations

2.1 Gaussian Elimination

2.2 Variations on a Theme

2.3 Rounding Errors, Equilibration, and Pivot Search

2.4 Vector and Matrix Norms

2.5 Condition Numbers and Data Perturbations

2.6 Iterative Refinement

2.7 Error Bounds for Solutions of Linear Systems

2.8 Exercises

3. Interpolation and Numerical Differentiation

3.1 Interpolation by Polynomials

3.2 Extrapolation and Numerical Differentiation

3.3 Cubic Splines

3.4 Approximation by Splines

3.5 Radial Basis Functions

3.6 Exercises

4. Numerical Integration

4.1 The Accuracy of Quadrature Formulas

4.2 Gaussian Quadrature Formulas

4.3 The Trapezoidal Rule

4.4 Adaptive Integration

4.5 Solving Ordinary Differential Equations

4.6 Step Size and Order Control

4.7 Exercises

5. Univariate Nonlinear Equations

5.1 The Secant Method

5.2 Bisection Methods

5.3 Spectral Bisection Methods for Eigenvalues

5.4 Convergence Order

5.5 Error Analysis

5.6 Complex Zeros

5.7 Methods Using Derivative Information

5.8 Exercises

6. Systems of Nonlinear Equations

6.1 Preliminaries

6.2 Newton's Method and Its Variants

6.3 Error Analysis

6.4 Further Techniques for Nonlinear Systems

6.5 Exercises

References

Index

The users who browse this book also browse