Solutions Manual to Accompany an Introduction to Numerical Methods and Analysis

ISBN-13: 9781119604532 / Angielski / Miękka / 2021 / 304 str.

James F. Epperson

Spis treści

1 Introductory Concepts and Calculus Review 11.1 Basic Tools of Calculus 11.2 Error, Approximate Equality, and Asymptotic Order Notation 101.3 A Primer on Computer Arithmetic 131.4 A Word on Computer Languages and Software 171.5 A Brief History of Scientific Computing 182 A Survey of Simple Methods and Tools 192.1 Horner's Rule and Nested Multiplication 192.2 Difference Approximations to the Derivative 222.3 Application: Euler's Method for Initial Value Problems 302.4 Linear Interpolation 342.5 Application--The Trapezoid Rule 382.6 Solution of Tridiagonal Linear Systems 462.7 Application: Simple Two-Point Boundary Value Problems 503 Root-Finding 553.1 The Bisection Method 553.2 Newton's Method: Derivation and Examples 593.3 How to Stop Newton's Method 633.4 Application: Division Using Newton's Method 663.5 The Newton Error Formula 693.6 Newton's Method: Theory and Convergence 723.7 Application: Computation of the Square Root 763.8 The Secant Method: Derivation and Examples 793.9 Fixed Point Iteration 833.10 Roots of Polynomials (Part 1) 853.11 Special Topics in Root-finding Methods 883.12 Very High-order Methods and the Efficiency Index 984 Interpolation and Approximation 1014.1 Lagrange Interpolation 1014.2 Newton Interpolation and Divided Differences 1044.3 Interpolation Error 1144.4 Application: Muller's Method and Inverse Quadratic Interpolation 1194.5 Application: More Approximations to the Derivative 1214.6 Hermite Interpolation 1224.7 Piecewise Polynomial Interpolation 1254.8 An Introduction to Splines 1294.9 Tension Splines 1354.10 Least Squares Concepts in Approximation 1374.11 Advanced Topics in Interpolation Error 1425 Numerical Integration 1495.1 A Review of the Definite Integral 1495.2 Improving the Trapezoid Rule 1515.3 Simpson's Rule and Degree of Precision 1545.4 The Midpoint Rule 1625.5 Application: Stirling's Formula 1665.6 Gaussian Quadrature 1675.7 Extrapolation Methods 1735.8 Special Topics in Numerical Integration 1776 Numerical Methods for Ordinary Differential Equations 1856.1 The Initial Value Problem--Background 1856.2 Euler's Method 1876.3 Analysis of Euler's Method 1896.4 Variants of Euler's Method 1906.5 Single Step Methods--Runge-Kutta 1976.6 Multistep Methods 2006.7 Stability Issues 2046.8 Application to Systems of Equations 2066.9 Adaptive Solvers 2106.10 Boundary Value Problems 2127 Numerical Methods for the Solution of Systems of Equations 2177.1 Linear Algebra Review 2177.2 Linear Systems and Gaussian Elimination 2187.3 Operation Counts 2237.4 The LU Factorization 2247.5 Perturbation, Conditioning and Stability 2297.6 SPD Matrices and the Cholesky Decomposition 2357.7 Application: Numerical Solution of Linear Least Squares Problems 2367.8 Sparse and Structured Matrices 2407.9 Iterative Methods for Linear Systems - A Brief Survey 2417.10 Nonlinear Systems: Newton's Method and Related Ideas 2427.11 Application: Numerical Solution of Nonlinear BVP's 2448 Approximate Solution of the Algebraic Eigenvalue Problem 2478.1 Eigenvalue Review 2478.2 Reduction to Hessenberg Form 2498.3 Power Methods 2508.4 Bisection and Inertia to Compute Eigenvalues of Symmetric Matrices 2538.5 An Overview of the QR Iteration 2578.6 Application: Roots of Polynomials, II 2608.7 Application: Computation of Gaussian Quadrature Rules 2619 A Survey of Numerical Methods for Partial Differential Equations 2659.1 Difference Methods for the Diffusion Equation 2659.2 Finite Element Methods for the Diffusion Equation 2709.3 Difference Methods for Poisson Equations 27110 An Introduction to Spectral Methods 27710.1 Spectral Methods for Two-Point Boundary Value Problems 27710.2 Spectral Methods in Two Dimensions 27910.3 Spectral Methods for Time-Dependent Problems 28210.4 Clenshaw-Curtis Quadrature 283