An Introduction to Numerical Methods and Analysis Book

	Preface
	Acknowledgments
Ch. 1	Introductory Concepts and Calculus Review	1
1.1	Basic Tools of Calculus	2
1.2	Error, Approximate Equality, and Asymptotic Order Notation	14
1.3	A Primer on Computer Arithmetic	20
1.4	A Word on Computer Languages and Software	28
1.5	Simple Approximations	29
1.6	Application: Approximating the Natural Logarithm	33
Ch. 2	A Survey of Simple Methods and Tools	37
2.1	Homer's Rule and Nested Multiplication	37
2.2	Difference Approximations to the Derivative	40
2.3	Application: Euler's Method for Initial Value Problems	48
2.4	Linear Interpolation	53
2.5	Application: The Trapezoid Rule	60
2.6	Solution of Tridiagonal Linear Systems	69
2.7	Application: Simple Two-Point Boundary Value Problems	75
Ch. 3	Root Finding	80
3.1	The Bisection Method	81
3.2	Newton's Method: Derivation and Examples	87
3.3	How to Stop Newton's Method	93
3.4	Application: Division Using Newton's Method	96
3.5	The Newton Error Formula	100
3.6	Newton's Method: Theory and Convergence	105
3.7	Application: Computation of the Square Root	109
3.8	The Secant Method: Derivation and Example	111
3.9	Fixed-Point Iteration	116
3.10	Special Topics in Root-Finding Methods	126
Ch. 4	Interpolation and Approximation	150
4.1	Lagrange Interpolation	151
4.2	Newton Interpolation and Divided Differences	156
4.3	Interpolation Error	166
4.4	Application: Muller's Method and Inverse Quadratic	170
4.5	Application: More Approximation to the Derivative	174
4.6	Hermite Interpolation	177
4.7	Piecewise Polynomial Interpolation	182
4.8	An Introduction to Splines	189
4.9	Application: Solution of Boundary Value Problems	204
4.10	Least Squares Concepts in Approximation	209
4.11	Advanced Topics in Interpolation Error	230
4.12	Literature and Software Discussion	243
Ch. 5	Numerical Integration	245
5.1	A Review of the Definite Integral	246
5.2	Improving the Trapezoid Rule	248
5.3	Simpson's Rule and Degree of Precision	253
5.4	The Midpoint Rule	265
5.5	Application: Stirling's Formula	268
5.6	Gaussian Quadrature	270
5.7	Extrapolation Methods	281
5.8	Special Topics in Numerical Integration	288
Ch. 6	Numerical Methods for Ordinary Differential Equations	312
6.1	The Initial Value Problem: Background	313
6.2	Euler's Method	318
6.3	Analysis of Euler's Method	322
6.4	Variants of Euler's Method	326
6.5	Single Step Methods: Runge-Kutta	343
6.6	Multistep Methods	350
6.7	Stability Issues	356
6.8	Application to Systems of Equations	363
6.9	Adaptive Solvers	370
6.10	Boundary Value Problems	383
6.11	Literature and Software Discussion	392
Ch. 7	Numerical Methods for the Solution of Systems	394
7.1	Linear Algebra Review	395
7.2	Linear Systems and Gaussian Elimination	397
7.3	Operation Counts	404
7.4	The LU Factorization	406
7.5	Perturbation, Conditioning, and Stability	416
7.6	SPD Matrices and the Cholesky Decomposition	434
7.7	Iterative Methods for Linear Systems: A Brief Survey	437
7.8	Nonlinear Systems: Newton's Method and Related Ideas	446
7.9	Application: Numerical Solution of Nonlinear BVPs	452
7.10	Literature and Software Discussion	454
Ch. 8	Approximate Solution of the Algebraic Eigenvalue Problem	456
8.1	Eigenvalue Review	457
8.2	Reduction to Hessenberg Form	463
8.3	Power Methods	471
8.4	An Overview of the QR Iteration	490
Ch. 9	A Survey of Finite Difference Methods for Partial Differential Equations	500
9.1	Difference Methods for the Diffusion Equation	501
9.2	Difference Methods for Poisson Equations	517
App. A	Proofs of Selected Theorems, and other Additional Material	535
App. B	Proofs of Selected Theorems, and other Additional Material	542
	Index	549