This textbook provides an accessible and concise introduction to numerical analysis for upper undergraduate and beginning graduate students from various backgrounds. It was developed from the lecture notes of four successful courses on numerical analysis taught within the MPhil of Scientific Computing at the University of Cambridge. The book is easily accessible, even to those with limited knowledge of mathematics.
Students will get a concise, but thorough introduction to numerical analysis. In addition the algorithmic principles are emphasized to encourage a deeper understanding of why an algorithm is suitable, and sometimes unsuitable, for a particular problem.
A Concise Introduction to Numerical Analysis strikes a balance between being mathematically comprehensive, but not overwhelming with mathematical detail. In some places where further detail was felt to be out of scope of the book, the reader is referred to further reading.
The book uses MATLAB® implementations to demonstrate the workings of the method and thus MATLAB's own implementations are avoided, unless they are used as building blocks of an algorithm. In some cases the listings are printed in the book, but all are available online on the book’s page at www.crcpress.com.
Most implementations are in the form of functions returning the outcome of the algorithm. Also, examples for the use of the functions are given. Exercises are included in line with the text where appropriate, and each chapter ends with a selection of revision exercises. Solutions to odd-numbered exercises are also provided on the book’s page at www.crcpress.com.
This textbook is also an ideal resource for graduate students coming from other subjects who will use numerical techniques extensively in their graduate studies.
Fundamentals
Floating Point Arithmetic
Overflow and Underflow
Absolute, Relative Error, Machine Epsilon
Forward and Backward Error Analysis
Loss of Significance
Robustness
Error Testing and Order of Convergence
Computational Complexity
Condition
Revision Exercises
Linear Systems
Simultaneous Linear Equations
Gaussian Elimination and Pivoting
LU Factorization
Cholesky Factorization
QR Factorization
The Gram–Schmidt Algorithm
Givens Rotations
Householder Reflections
Linear Least Squares
Singular Value Decomposition
Iterative Schemes and Splitting
Jacobi and Gauss–Seidel Iterations
Relaxation
Steepest Descent Method
Conjugate Gradients
Krylov Subspaces and Pre-Conditioning
Eigenvalues and Eigenvectors
The Power Method
Inverse Iteration
Deflation
Revision Exercises
Interpolation and Approximation Theory
Lagrange Form of Polynomial Interpolation
Newton Form of Polynomial Interpolation
Polynomial Best Approximations
Orthogonal polynomials
Least-Squares Polynomial Fitting
The Peano Kernel Theorem
Splines
B-Spline
Revision Exercises
Non-Linear Systems
Bisection, Regula Falsi, and Secant Method
Newton’s Method
Broyden’s Method
Householder Methods
Müller’s Method
Inverse Quadratic Interpolation
Fixed Point Iteration Theory
Mixed Methods
Revision Exercises
Numerical Integration
Mid-Point and Trapezium Rule
The Peano Kernel Theorem
Simpson’s Rule
Newton–Cotes Rules
Gaussian Quadrature
Composite Rules
Multi-Dimensional Integration
Monte Carlo Methods
Revision Exercises
ODEs
One-Step Methods
Multistep Methods, Order, and Consistency
Order Conditions
Stiffness and A-Stability
Adams Methods
Backward Differentiation Formulae
The Milne and Zadunaisky Device
Rational Methods
Runge–Kutta Methods
Revision Exercises
Numerical Differentiation
Finite Differences
Differentiation of Incomplete or Inexact Data
PDEs
Classification of PDEs
Parabolic PDEs
Elliptic PDEs
Parabolic PDEs in Two Dimensions
Hyperbolic PDEs
Spectral Methods
Finite Element Method
Revision Exercises
Biography
A. C. Faul
"This is a thorough and detailed introduction to the important topic of numerical analysis. It is based upon an existing successful postgraduate course and assumes a significant level of mathematical sophistication. The material is very clearly presented, with lots of end-of-topic questions and examples in MATLAB that really illustrate the key issues involved, such as error propagation. The range of topics is conventional, but the level of detail is not. This is definitely a textbook that should be on the shelf of anyone working in the field who wants a deep understanding of the methods, limitations, and practical issues."
—Dr. Matt Probert, Department of Physics, University of York, UK
