Afternotes Goes to Graduate School: Lectures on Advanced Numerical Analysis
By: G. W. Stewart (author)Paperback
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In this follow-up to Afternotes on Numerical Analysis (SIAM, 1996) the author continues to bring the immediacy of the classroom to the printed page. Like the original undergraduate volume, Afternotes Goes to Graduate School is the result of the author writing down his notes immediately after giving each lecture; in this case the afternotes are the result of a follow-up graduate course taught by Professor Stewart at the University of Maryland. The algorithms presented in this volume require deeper mathematical understanding than those in the undergraduate book, and their implementations are not trivial. Stewart uses a fresh presentation that is clear and intuitive as he covers topics such as discrete and continuous approximation, linear and quadratic splines, eigensystems, and Krylov sequence methods. He concludes with two lectures on classical iterative methods and nonlinear equations.
Part I. Approximation. Lecture 1. General observations; Decline and fall; The linear sine; Approximation in normed linear spaces; Significant differences; Lecture 2. The space C[0,1]; Existence of best approximations; Uniqueness of best approximations; Convergence in C[0,1]; The Weierstrass approximation theorem; Bernstein polynomials; Comments; Lecture 3. Chebyshev approximation; Uniqueness; Convergence of Chebyshev approximations; Rates of convergence; Part II. Linear and Cubic Splines. Lecture 10. Piecewise linear interpolation; The error in L(f); Approximations in the $\infty$-norm; Hat functions; Integration; Least squares approximation; Implementations issues; Lecture 11. Cubic splines; Derivation of the cubic spline; End conditions; Convergence; Locality; Part III. Eigensystems; Part III. Eigensystems; Part IV. Krylov Sequence Methods; Part V. Iterations, Linear and Nonlinear.
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