Numerical Analysis

Author： Scott L.

Publisher： Princeton University Press‎

Publication year： 2011

E-ISBN: 9781400838967

P-ISBN(Paperback): 9780691146867

Subject： O241 数值分析

Keyword：数值分析,应用数学

Language： ENG

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Description

Computational science is fundamentally changing how technological questions are addressed. The design of aircraft, automobiles, and even racing sailboats is now done by computational simulation. The mathematical foundation of this new approach is numerical analysis, which studies algorithms for computing expressions defined with real numbers. Emphasizing the theory behind the computation, this book provides a rigorous and self-contained introduction to numerical analysis and presents the advanced mathematics that underpin industrial software, including complete details that are missing from most textbooks.

Using an inquiry-based learning approach, Numerical Analysis is written in a narrative style, provides historical background, and includes many of the proofs and technical details in exercises. Students will be able to go beyond an elementary understanding of numerical simulation and develop deep insights into the foundations of the subject. They will no longer have to accept the mathematical gaps that exist in current textbooks. For example, both necessary and sufficient conditions for convergence of basic iterative methods are covered, and proofs are given in full generality, not just based on special cases.

The book is accessible to undergraduate mathematics majors as well as computational scientists wanting to learn the foundations of the subject.

Presents the mathematical foundations of numerical analysis

Chapter

Numerical Analysis

Contents

Preface

Chapter 1. Numerical Algorithms

1.1 Finding roots

1.2 Analyzing Heron’s algorithm

1.3 Where to start

1.4 An unstable algorithm

1.5 General roots: effects of floating-point

1.6 Exercises

1.7 Solutions

Chapter 2. Nonlinear Equations

2.1 Fixed-point iteration

2.2 Particular methods

2.3 Complex roots

2.4 Error propagation

2.5 More reading

2.6 Exercises

2.7 Solutions

Chapter 3. Linear Systems

3.1 Gaussian elimination

3.2 Factorization

3.3 Triangular matrices

3.4 Pivoting

3.5 More reading

3.6 Exercises

3.7 Solutions

Chapter 4. Direct Solvers

4.1 Direct factorization

4.2 Caution about factorization

4.3 Banded matrices

4.4 More reading

4.5 Exercises

4.6 Solutions

Chapter 5. Vector Spaces

5.1 Normed vector spaces

5.2 Proving the triangle inequality

5.3 Relations between norms

5.4 Inner-product spaces

5.5 More reading

5.6 Exercises

5.7 Solutions

Chapter 6. Operators

6.1 Operators

6.2 Schur decomposition

6.3 Convergent matrices

6.4 Powers of matrices

6.5 Exercises

6.6 Solutions

Chapter 7. Nonlinear Systems

7.1 Functional iteration for systems

7.2 Newton’s method

7.3 Limiting behavior of Newton’s method

7.4 Mixing solvers

7.5 More reading

7.6 Exercises

7.7 Solutions

Chapter 8. Iterative Methods

8.1 Stationary iterative methods

8.2 General splittings

8.3 Necessary conditions for convergence

8.4 More reading

8.5 Exercises

8.6 Solutions

Chapter 9. Conjugate Gradients

9.1 Minimization methods

9.2 Conjugate Gradient iteration

9.3 Optimal approximation of CG

9.4 Comparing iterative solvers

9.5 More reading

9.6 Exercises

9.7 Solutions

Chapter 10. Polynomial Interpolation

10.1 Local approximation: Taylor’s theorem

10.2 Distributed approximation: interpolation

10.3 Norms in infinite-dimensional spaces

10.4 More reading

10.5 Exercises

10.6 Solutions

Chapter 11. Chebyshev and Hermite Interpolation

11.1 Error term ω

11.2 Chebyshev basis functions

11.3 Lebesgue function

11.4 Generalized interpolation

11.5 More reading

11.6 Exercises

11.7 Solutions

Chapter 12. Approximation Theory

12.1 Best approximation by polynomials

12.2 Weierstrass and Bernstein

12.3 Least squares

12.4 Piecewise polynomial approximation

12.5 Adaptive approximation

12.6 More reading

12.7 Exercises

12.8 Solutions

Chapter 13. Numerical Quadrature

13.1 Interpolatory quadrature

13.2 Peano kernel theorem

13.3 Gregorie-Euler-Maclaurin formulas

13.4 Other quadrature rules

13.5 More reading

13.6 Exercises

13.7 Solutions

Chapter 14. Eigenvalue Problems

14.1 Eigenvalue examples

14.2 Gershgorin’s theorem

14.3 Solving separately

14.4 How not to eigen

14.5 Reduction to Hessenberg form

14.6 More reading

14.7 Exercises

14.8 Solutions

Chapter 15. Eigenvalue Algorithms

15.1 Power method

15.2 Inverse iteration

15.3 Singular value decomposition

15.4 Comparing factorizations

15.5 More reading

15.6 Exercises

15.7 Solutions

Chapter 16. Ordinary Differential Equations

16.1 Basic theory of ODEs

16.2 Existence and uniqueness of solutions

16.3 Basic discretization methods

16.4 Convergence of discretization methods

16.5 More reading

16.6 Exercises

16.7 Solutions

Chapter 17. Higher-order ODE Discretization Methods

17.1 Higher-order discretization

17.2 Convergence conditions

17.3 Backward differentiation formulas

17.4 More reading

17.5 Exercises

17.6 Solutions

Chapter 18. Floating Point

18.1 Floating-point arithmetic

18.2 Errors in solving systems

18.3 More reading

18.4 Exercises

18.5 Solutions

Chapter 19. Notation

Bibliography

Index

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