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Trim Size: 6in x 9inEpperson - 06/09/2021 10:50pm Page iii �
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� James F. Epperson Solutions Manual For An Introduction to Numerical Methods and Analysis Third Edition 1 / 4
CONTENTS
- Introductory Concepts and Calculus Review 1
- A Survey of Simple Methods and Tools 19
1.1 Basic Tools of Calculus 1 1.2 Error, Approximate Equality, and Asymptotic Order Notation 10 1.3 A Primer on Computer Arithmetic 13 1.4 A Word on Computer Languages and Software 17 1.5 A Brief History of Scientific Computing 18
2.1 Horner’s Rule and Nested Multiplication19 2.2 Difference Approximations to the Derivative 22
2.3 Application: Euler’s Method for Initial Value Problems 30
2.4 Linear Interpolation 34 2.5 Application — The Trapezoid Rule 38 2.6 Solution of Tridiagonal Linear Systems 46
2.7 Application: Simple Two-Point Boundary Value Problems 50
- Root-Finding 55
3.1 The Bisection Method 55
3.2 Newton’s Method: Derivation and Examples 59
3.3 How to Stop Newton’s Method 63 v Preface to the Solutions Manual for the Third Edition ix 2 / 4
vi CONTENTS
3.4 Application: Division Using Newton’s Method66
3.5 The Newton Error Formula69
3.6 Newton’s Method: Theory and Convergence72
3.7 Application: Computation of the Square Root76
3.8 The Secant Method: Derivation and Examples79
3.9 Fixed Point Iteration83 3.10 Roots of Polynomials (Part 1)85 3.11 Special Topics in Root-finding Methods88 3.12 Very High-order Methods and the Efficiency Index 98
- Interpolation and Approximation101
4.1 Lagrange Interpolation101 4.2 Newton Interpolation and Divided Differences104 4.3 Interpolation Error114
4.4 Application: Muller’s Method and Inverse Quadratic Interpolation 119
4.5 Application: More Approximations to the Derivative 121
4.6 Hermite Interpolation122 4.7 Piecewise Polynomial Interpolation125 4.8 An Introduction to Splines129 4.9 Tension Splines135 4.10 Least Squares Concepts in Approximation137 4.11 Advanced Topics in Interpolation Error142
- Numerical Integration149
5.1 A Review of the Definite Integral149 5.2 Improving the Trapezoid Rule151 5.3 Simpson’s Rule and Degree of Precision154 5.4 The Midpoint Rule162
5.5 Application: Stirling’s Formula166
5.6 Gaussian Quadrature167 5.7 Extrapolation Methods173 5.8 Special Topics in Numerical Integration177
- Numerical Methods for Ordinary Differential Equations 185
6.1 The Initial Value Problem—Background185 6.2 Euler’s Method187 6.3 Analysis of Euler’s Method189 6.4 Variants of Euler’s Method190 6.5 Single Step Methods—Runge-Kutta197 6.6 Multistep Methods200 6.7 Stability Issues204 3 / 4
CONTENTS vii 6.8 Application to Systems of Equations206 6.9 Adaptive Solvers210 6.10 Boundary Value Problems212
- Numerical Methods for the Solution of Systems of Equations 217
7.1 Linear Algebra Review217 7.2 Linear Systems and Gaussian Elimination218 7.3 Operation Counts223 7.4 TheLUFactorization224 7.5 Perturbation, Conditioning and Stability229 7.6 SPD Matrices and the Cholesky Decomposition235
7.7 Application: Numerical Solution of Linear Least Squares Problems 236
7.8 Sparse and Structured Matrices240 7.9 Iterative Methods for Linear Systems – A Brief Survey 241
7.10 Nonlinear Systems: Newton’s Method and Related Ideas 242
7.11 Application: Numerical Solution of Nonlinear BVP’s 244
- Approximate Solution of the Algebraic Eigenvalue Problem 247
8.1 Eigenvalue Review247 8.2 Reduction to Hessenberg Form249 8.3 Power Methods250 8.4 Bisection and Inertia to Compute Eigenvalues of Symmetric Matrices 253 8.5 An Overview of theQRIteration257
8.6 Application: Roots of Polynomials, II260
8.7 Application: Computation of Gaussian Quadrature Rules 261
- A Survey of Numerical Methods
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for Partial Differential Equations265 9.1 Difference Methods for the Diffusion Equation265 9.2 Finite Element Methods for the Diffusion Equation 270 9.3 Difference Methods for Poisson Equations271 10 An Introduction to Spectral Methods277 10.1 Spectral Methods for Two-Point Boundary Value Problems 277 10.2 Spectral Methods in Two Dimensions279 10.3 Spectral Methods for Time-Dependent Problems282 10.4 Clenshaw-Curtis Quadrature283
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