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.
Preface Part 1: 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: Jackson's theorem Lecture 4: A theorem of de la Vallee Poussin A general approximation strategy Chebyshev polynomials Economization of power series Farewell to C[a,b] Lecture 5: Discrete, continuous, and weighted least squares Inner-product space Quasi-matrices Positive definite matrices The Cauchy and triangle inequalities Orthogonality The QR factorization Lecture 6: Existence and uniqueness of the QR factorization The Gram-Schmidt algorithm Projections Best approximation on inner-product spaces Lecture 7: Expansions in orthogonal functions Orthogonal polynomials Discrete least squares and the QR decomposition Lecture 8: Householder transformations Orthogonal triangularization Implementation Comments on the algorithm Solving least squares problems Lecture 9: Operation counts The Frobenius and spectral norms Stability of orthogonal triangularization Error analysis of the normal equations Perturbation of inverses and linear systems Perturbation of pseudoinverses and least squares solutions Summary Part 2: Linear and Cubic Splines. Lecture 10: Piecewise linear interpolation The error in L(f) Approximations in the Y-norm Hat functions Integration Least squares approximation Implementations issues Lecture 11: Cubic splines Derivation of the cubic spline End conditions Convergence Locality Part 3: Eigensystems. Lecture 12: A system of differential equations Complex vectors and matrices Eigenvalues and eigenvectors Existence and uniqueness Left eigenvectors Real matrices Multiplicity and defective matrices Functions of matrices Similarity transformations and diagonalization The Schur decomposition Lecture 13: Real Schur form Block diagonalization Diagonalization Jordan canonical form Hermitian matrices Perturbation of a simple eigenvalue Lecture 14: A backward perturbation result The Rayleigh quotient Powers of matrices The power method Lecture 15: The inverse power method Derivation of the QR algorithm Local convergence analysis Practical considerations Hessenberg matrices Lecture 16: Reduction to Hessenberg form The Hessenberg QR algorithm Return to Upper Hessenberg Lecture 17: The implicit double shift Some implementation details The singular value decomposition Lecture 18: Rank and Schmidt's theorem Computational considerations Reduction to bidiagonal form Plane rotations The implicit QR algorithm for singular values Part 4: Krylov Sequence Methods. Lecture 19: Introduction Invariant subspaces Krylov subspaces Arnoldi decompositions Implicit restarting Deflation Lecture 20: The Lanczos algorithm Relation to orthogonal polynomials Golub-Kahan-Lanczos bidiagonalization Lecture 21: Linear systems, errors, and residuals Descending to a solution Conjugate directions The method of conjugate gradients Termination Lecture 22: Operation counts and storage requirements Conjugate gradients as an iterative method Convergence in the A-norm Monotone convergence in the 2-norm Lecture 23: Preconditioned conjugate gradients Preconditioners Incomplete LU preconditioners Lecture 24: Diagonally dominant matrices Return to incomplete factorization Part 5: Iterations, Linear and Nonlinear. Lecture 25: Some classical iterations Splittings and iterative methods Convergence Irreducibility Splittings of irreducibly diagonally dominant matrices M-matrices and positive definite matrices Lecture 26: Linear and nonlinear Continuity and derivatives The fixed-point iteration
Afternotes on Numerical Analysis is the result of the author writing down his notes immediately after giving each lecture.
