Introduction to Numerical Analysis

ISBN-10: 0521007941
ISBN-13: 9780521007948
Edition: 2002
List price: $74.99 Buy it from $56.00
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Description: This textbook is written primarily for undergraduate mathematicians and also appeals to students working at an advanced level in other disciplines. The text begins with a clear motivation for the study of numerical analysis based on real-world  More...

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Book details

List price: $74.99
Copyright year: 2002
Publisher: Cambridge University Press
Publication date: 8/28/2003
Binding: Paperback
Pages: 444
Size: 6.00" wide x 9.00" long x 0.75" tall
Weight: 1.628
Language: English

This textbook is written primarily for undergraduate mathematicians and also appeals to students working at an advanced level in other disciplines. The text begins with a clear motivation for the study of numerical analysis based on real-world problems. The authors then develop the necessary machinery including iteration, interpolation, boundary-value problems and finite elements. Throughout, the authors keep an eye on the analytical basis for the work and add historical notes on the development of the subject. There are numerous exercises for students.

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Preface
Solution of equations by iteration
Introduction
Simple iteration
Iterative solution of equations
Relaxation and Newton's method
The secant method
The bisection method
Global behaviour
Notes
Exercises
Solution of systems of linear equations
Introduction
Gaussian elimination
LU factorisation
Pivoting
Solution of systems of equations
Computational work
Norms and condition numbers
Hilbert matrix
Least squares method
Notes
Exercises
Special matrices
Introduction
Symmetric positive definite matrices
Tridiagonal and band matrices
Monotone matrices
Notes
Exercises
Simultaneous nonlinear equations
Introduction
Simultaneous iteration
Relaxation and Newton's method
Global convergence
Notes
Exercises
Eigenvalues and eigenvectors of a symmetric matrix
Introduction
The characteristic polynomial
Jacobi's method
The Gerschgorin theorems
Householder's method
Eigenvalues of a tridiagonal matrix
The QR algorithm
The QR factorisation revisited
The definition of the QR algorithm
Inverse iteration for the eigenvectors
The Rayleigh quotient
Perturbation analysis
Notes
Exercises
Polynomial interpolation
Introduction
Lagrange interpolation
Convergence
Hermite interpolation
Differentiation
Notes
Exercises
Numerical integration-I
Introduction
Newton-Cotes formulae
Error estimates
The Runge phenomenon revisited
Composite formulae
The Euler-Maclaurin expansion
Extrapolation methods
Notes
Exercises
Polynomial approximation in the [infinity]-norm
Introduction
Normed linear spaces
Best approximation in the [infinity]-norm
Chebyshev polynomials
Interpolation
Notes
Exercises
Approximation in the 2-norm
Introduction
Inner product spaces
Best approximation in the 2-norm
Orthogonal polynomials
Comparisons
Notes
Exercises
Numerical integration - II
Introduction
Construction of Gauss quadrature rules
Direct construction
Error estimation for Gauss quadrature
Composite Gauss formulae
Radau and Lobatto quadrature
Note
Exercises
Piecewise polynomial approximation
Introduction
Linear interpolating splines
Basis functions for the linear spline
Cubic splines
Hermite cubic splines
Basis functions for cubic splines
Notes
Exercises
Initial value problems for ODEs
Introduction
One-step methods
Consistency and convergence
An implicit one-step method
Runge-Kutta methods
Linear multistep methods
Zero-stability
Consistency
Dahlquist's theorems
Systems of equations
Stiff systems
Implicit Runge-Kutta methods
Notes
Exercises
Boundary value problems for ODEs
Introduction
A model problem
Error analysis
Boundary conditions involving a derivative
The general self-adjoint problem
The Sturm-Liouville eigenvalue problem
The shooting method
Notes
Exercises
The finite element method
Introduction: the model problem
Rayleigh-Ritz and Galerkin principles
Formulation of the finite element method
Error analysis of the finite element method
A posteriori error analysis by duality
Notes
Exercises
An overview of results from real analysis
WWW-resources
Bibliography
Index

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