An introduction to numerical methods and analysis / 6
6
James F. Epperson
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- xv, 556 pages : illustrations 24 cm.
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Includes bibliographical references and index
Introductory Concepts and Calculus Review --;Basic Tools of Calculus --;Taylor's Theorem --;Mean Value and Extreme Value Theorems --;Error, Approximate Equality, and Asymptotic Order Notation --;Error --;Notation: Approximate Equality --;Notation: Asymptotic Order --;A Primer on Computer Arithmetic --;A Word on Computer Languages and Software --;Simple Approximations --;Application: Approximating the Natural Logarithm --;A Survey of Simple Methods and Tools --;Homer's Rule and Nested Multiplication --;Difference Approximations to the Derivative --;Application: Euler's Method for Initial Value Problems --;Linear Interpolation --;Application: The Trapezoid Rule --;Solution of Tridiagonal Linear Systems --;Application: Simple Two-Point Boundary Value Problems --;Root Finding --;The Bisection Method --;Newton's Method: Derivation and Examples --;How to Stop Newton's Method --;Application: Division Using Newton's Method --;The Newton Error Formula --;Newton's Method: Theory and Convergence --;Application: Computation of the Square Root --;The Secant Method: Derivation and Examples --;Fixed-Point Iteration --;Special Topics in Root-Finding Methods --;Extrapolation and Acceleration --;Variants of Newton's Method --;The Secant Method: Derivation and Examples --;Multiple Roots --;In Search of Fast Global Convergence: Hybrid Algorithm --;Literature and Software Discussion --;Interpolation and Approximation --;Lagrange Interpolation --;Newton Interpolation and Divided Differences --;Interpolation Error;Application: Muller's Method and Inverse Quadratic --;Application: More Approximations to the Derivative --;Hermite Interpolation --;Piecewise Polynomial Interpolation --;An Introduction to Splines --;Definition of the Problem --;Cubic B-Splines --;Application: Solution of Boundary Value Problems --;Least Squares Concepts in Approximation --;An Introduction to Data Fitting --;Least Squares Approximation and Orthogonal --;Advanced Topics in Interpolation Error --;Stability of Polynomial Interpolation --;The Runge Example --;The Chebyshev Nodes --;Literature and Software Discussion --;Numerical Integration --;A Review of the Definite Integral --;Improving the Trapezoid Rule --;Simpson's Rule and Degree of Precision --;The Midpoint Rule --;Application: Stirling's Formula --;Gaussian Quadrature --;Extrapolation Methods --;Special Topics in Numerical Integration --;Romberg Integration --;Quadrature with Nonsmooth Integrands --;Adaptive Integration --;Peano Estimates for the Trapezoid Rule --;Literature and Software Discussion --;Numerical Methods for Ordinary Differential Equations --;The Initial Value Problem: Background --;Euler's Method --;Analysis of Euler's Method --;Variants of Euler's Method --;The Residual and Truncation Error --;Implicit Methods and Predictor-Corrector Schemes --;Starting Values and Multistep Methods --;The Midpoint Method and Weak Stability --;Single Step Methods: Runge-Kutta --;Multistep Methods --;The Adams Families --;The BDF Family --;Stability Issues --;Stability Theory for Multistep Methods;Stability Regions --;Application to Systems of Equations --;Implementation Issues and Examples --;Stiff Equations --;A-Stability --;Adaptive Solvers --;Boundary Value Problems --;Simple Difference Methods --;Shooting Methods --;Literature and Software Discussion --;Numerical Methods for the Solution of Systems --;Linear Algebra Review --;Linear Systems and Gaussian Elimination --;Operation Counts --;The LU Factorization --;Perturbation, Conditioning, and Stability --;Vector and Matrix Norms of Equations --;The Condition Number and Perturbations --;Estimating the Condition Number --;Interative Refinement --;SPD Matrices and the Cholesky Decomposition --;Iterative Methods for Linear Systems: A Brief Survey --;Nonlinear Systems: Newton's Method and Related Ideas --;Newton's Method --;Fixed-Point Methods --;Application: Numerical Solution of Nonlinear BVPs --;Literature and Software Discussion --;Approximate Solution of the Algebraic Eigenvalue Problem --;Eigenvalue Review --;Reduction to Hessenberg Form --;Power Methods --;An Overview of the QR Iteration --;Literature and Software Discussion --;A Survey of Finite Difference Methods for Partial Differential Equations --;Difference Methods for the Diffusion Equation --;The Basic Problem --;The Explicit Method and Stability --;Implicit Methods and the Crank-Nicolson Method --;Difference Methods for Poisson Equations --;Discretization --;Banded Cholesky Solvers --;Iteration and the Method of Conjugate Gradients --;Literature and Software Discussion;Proofs of Selected Theorems, and Other Additional Material --;Proofs of the Interpolation Error Theorems --;Proof of Stability --;Stiff Systems of Differential Equations and Eigenvalues --;The Matrix Perturbation Theorem