An Introduction to Numerical Methods and Analysis

by James F. Epperson (Univ. of Alabama in Huntsville)

• ISBN13: 9780471316473
• ISBN10: 0471316474
• Format: Hardcover
• Copyright: 2001-08-01
• Publisher: Wiley

Summary

Emphasis on "cause and effect" in numerical mathematics. 'ˆ— Flexibility with computing languages--the book is not specific to any one computing language.

Table of Contents

Preface	vii
Acknowledgments	x
Introductory Concepts and Calculus Review	1	(36)
Basic Tools of Calculus	2	(12)
Taylor's Theorem	2	(7)
Mean Value and Extreme Value Theorems	9	(5)
Error, Approximate Equality, and Asymptotic Order Notation	14	(6)
Error	14	(1)
Notation: Approximate Equality	15	(1)
Notation: Asymptotic Order	16	(4)
A Primer on Computer Arithmetic	20	(8)
A Word on Computer Languages and Software	28	(1)
Simple Approximations	29	(4)
Application: Approximating the Natural Logarithm	33	(4)
A Survey of Simple Methods and Tools	37	(43)
Homer's Rule and Nested Multiplication	37	(3)
Difference Approximations to the Derivative	40	(8)
Application: Euler's Method for Initial Value Problems	48	(5)
Linear Interpolation	53	(7)
Application: The Trapezoid Rule	60	(9)
Solution of Tridiagonal Linear Systems	69	(6)
Application: Simple Two-Point Boundary Value Problems	75	(5)
Root Finding	80	(70)
The Bisection Method	81	(6)
Newton's Method: Derivation and Examples	87	(6)
How to Stop Newton's Method	93	(3)
Application:Division Using Newton's Method.	96	(4)
The Newton Error Formula	100	(5)
Newton's Method: Theory and Convergence	105	(4)
Application:Computation of the Square Root	109	(2)
The Secant Method: Derivation and Examples	111	(5)
Fixed-Point Iteration	116	(10)
Special Topics in Root-Finding Methods	126	(24)
Extrapolation and Acceleration	126	(5)
Variants of Newton's Method	131	(4)
The Secant Method: Derivation and Examples	135	(4)
Multiple Roots	139	(5)
In Search of Fast Global Convergence: Hybrid Algorithm	144	(5)
Literature and Software Discussion	149	(1)
Interpolation and Approximation	150	(95)
Lagrange Interpolation	151	(5)
Newton Interpolation and Divided Differences	156	(10)
Interpolation Error	166	(4)
Application: Muller's Method and Inverse Quadratic	170	(4)
Application: More Approximations to the Derivative	174	(3)
Hermite Interpolation	177	(5)
Piecewise Polynomial Interpolation	182	(7)
An Introduction to Splines	189	(15)
Definition of the Problem	189	(1)
Cubic B-Splines	190	(14)
Application: Solution of Boundary Value Problems	204	(5)
Least Squares Concepts in Approximation	209	(21)
An Introduction to Data Fitting	209	(6)
Least Squares Approximation and Orthogonal	215	(15)
Advanced Topics in Interpolation Error	230	(13)
Stability of Polynomial Interpolation	230	(4)
The Runge Example	234	(3)
The Chebyshev Nodes	237	(6)
Literature and Software Discussion	243	(2)
Numerical Integration	245	(67)
A Review of the Definite Integral	246	(2)
Improving the Trapezoid Rule	248	(5)
Simpson's Rule and Degree of Precision	253	(12)
The Midpoint Rule	265	(3)
Application: Stirling's Formula	268	(2)
Gaussian Quadrature	270	(11)
Extrapolation Methods	281	(7)
Special Topics in Numerical Integration	288	(24)
Romberg Integration	288	(5)
Quadrature with Nonsmooth Integrands	293	(6)
Adaptive Integration	299	(8)
Peano Estimates for the Trapezoid Rule	307	(4)
Literature and Software Discussion	311	(1)
Numerical Methods for Ordinary Differential Equations	312	(82)
The Initial Value Problem: Background	313	(5)
Euler's Method	318	(4)
Analysis of Euler's Method	322	(4)
Variants of Euler's Method	326	(17)
The Residual and Truncation Error	329	(2)
Implicit Methods and Predictor-Corrector Schemes	331	(6)
Starting Values and Multistep Methods	337	(2)
The Midpoint Method and Weak Stability	339	(4)
Single Step Methods: Runge-Kutta	343	(7)
Multistep Methods	350	(6)
The Adams Families	350	(4)
The BDF Family	354	(2)
Stability Issues	356	(7)
Stability Theory for Multistep Methods	356	(4)
Stability Regions	360	(3)
Application to Systems of Equations.	363	(7)
Implementation Issues and Examples	363	(3)
Stiff Equations	366	(2)
A-Stability	368	(2)
Adaptive Solvers	370	(13)
Boundary Value Problems	383	(9)
Simple Difference Methods	383	(5)
Shooting Methods	388	(4)
Literature and Software Discussion	392	(2)
Numerical Methods for the Solution of Systems	394	(62)
Linear Algebra Review	395	(2)
Linear Systems and Gaussian Elimination	397	(7)
Operation Counts	404	(2)
The LU Factorization	406	(10)
Perturbation, Conditioning, and Stability	416	(18)
Vector and Matrix Norms of Equations	417	(2)
The Condition Number and Perturbations	419	(8)
Estimating the Condition Number	427	(3)
Iterative Refinement	430	(4)
SPD Matrices and the Cholesky Decomposition	434	(3)
Iterative Methods for Linear Systems: A Brief Survey	437	(9)
Nonlinear Systems: Newton's Method and Related Ideas	446	(6)
Newton's Method	447	(3)
Fixed-Point Methods	450	(2)
Application: Numerical Solution of Nonlinear BVPs	452	(2)
Literature and Software Discussion	454	(2)
Approximate Solution of the Algebraic Eigenvalue Problem	456	(44)
Eigenvalue Review	457	(6)
Reduction to Hessenberg Form	463	(8)
Power Methods	471	(19)
An Overview of the QR Iteration	490	(10)
Literature and Software Discussion	499	(1)
A Survey of Finite Difference Methods for Partial Differential Equations	500	(35)
Difference Methods for the Diffusion Equation	501	(16)
The Basic Problem	501	(1)
The Explicit Method and Stability	502	(5)
Implicit Methods and the Crank-Nicolson Method	507	(10)
Difference Methods for Poisson Equations	517	(18)
Discretization	517	(3)
Banded Cholesky Solvers	520	(2)
Iteration and the Method of Conjugate Gradients	522	(11)
Literature and Software Discussion	533	(2)
APPENDIX A PROOFS OF SELECTED THEOREMS, AND OTHER ADDITIONAL MATERIAL	535	(7)
A.1 Proofs of the Interpolation Error Theorems	535	(2)
A.2 Proof of Stability	537	(1)
A.3 Stiff Systems of Differential Equations and Eigenvalues	538	(2)
A.4 The Matrix Perturbation Theorem	540	(2)
APPENDIX B PROOFS OF SELECTED THEOREMS, AND OTHER ADDITIONAL Material	542	(7)
Index	549