"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"--Provided by publisher.

• Numerical Analysis
  • Copyright Page
  • 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