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Afternotes Goes to Graduate School: Lectures on Advanced Numerical Analysis

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by G. W. Stewart


1998 / xii + 248 pages / Softcover / ISBN: 978-0-898714-04-3 / List Price $68.00 / SIAM Member Price $47.60 / Order Code OT58

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.

Although the book is not intended as a textbook, it can be used for self study and as a reference for graduate courses in scientific computing and numerical algebra. Ask your bookstore to stock it as an optional selection.

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 VallĀŽe 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 GramÐSchmidt 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 $\infty$-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; 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; Linear and nonlinear iterations compared; Rates of convergence; Newton's method; The contractive mapping theorem; Bibliography: Introduction; Bibliography; Index.

ISBN: 9780898714043

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