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Rabu, 03 November 2010

An Introduction to Numerical Analysis

An Introduction to Numerical Analysis: Endre Suli, David F. Mayers
Cambridge University Press | ISBN: 0521810264 | 2003-09-08 | PDF / djvu (ocr) | 444 pages | 4.27 / 1.63 Mb

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.

This book has emphasis on analysis of numerical methods, including error bound, consistency, convergence, stability. In most cases, a numerical method is introduced, followed by analysis and proofs. For engineering students, who like to know more algorithms and a little bit of analysis, this book may not be the best choice.
Although this book is mainly about analysis, it does include clear presentation of many numerical methods, including topics in nonlinear equations solving, numerical linear algebra, polynomial interpolation and integration, numerical solution of ODE. In numerical linear algebra, it includes LU factorization with pivoting, Gerschgorin's theorem of eigenvalue positions, Calculating eigenvalues by Jacobi plane rotation, Householder tridiagonalization, Sturm sequence property for tridiagonal symmetric matrix. Interpolation includes Lagrange polynomial, Hermite polynomial, Newton-Cotes integration, Improved Trapezium integration through Romberg method, Oscillation theorem for minimax approximation, Chebyshev polynomial, least square polynomial approximation to a known function, Gauss quadrature using Hermite polynomial, Piecewise linear/cubic splines. Ordinary ddifferential equations section includes initial value problems with one-step and multiple steps, boundary value problems using finite difference and shooting method, Galerkin finite element method. The book gives basic definitions including norms, matrix condition numbers, real symmetric positive definite matrix, Rayleigh quotient, orthogonal polynomials, stiffness, Sobolev space.
One place that is not clear is about QR algorithm for tridiagonal matrix.
In summary, the book is written clearly. Every numerical method is presented based on mathematics. There are many proofs (there is one proof with more than 3 pages), most of them that I decided to read are pretty easy to follow. There are not much implementation details and tricks. But this book will tell you when a method will converge and when a method is better. As a non-math major reader, I wish it could present more algorithms, such as algorithms for eigenvalues of nonsymmetric matrix, more details in finite difference method, a little bit of partial differential equations etc.

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