An Introduction to Numerical Analysis

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An Introduction to Numerical Analysis by Endre Süli and David Mayers - 2003 Oxford Computing laboratory
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Overview: 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.



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