A Survey of Numerical Mathematics, Volume I

by Young, David M.; Gregory, Robert Todd

• ISBN13: 9780486656915
• ISBN10: 0486656918
• Format: Paperback
• Copyright: 2011-11-30
• Publisher: Dover Publications

Summary

Volume 2 of 2-volume set. Broad self-contained coverage of computer-oriented numerical algorithms for solving various types of mathematical problems in linear algebra, ordinary and partial, differential equations, much more. Includes exercises.

Table of Contents

Chapter 1 Numerical Analysis as a Subject Area	1	(18)
1.1 Introduction	1	(1)
1.2 Some pitfalls in computation	2	(3)
1.3 Mathematical and computer aspects of an algorithm	5	(2)
1.4 Numerical instability of algorithms and ill-conditioned problems	7	(3)
1.5 Typical problems of interest to the numerical analyst	10	(3)
1.6 Iterative methods	13	(6)
Chapter 2 Elementary Operations with Automatic Digital Computers	19	(51)
2.1 Introduction	19	(1)
2.2 Binary arithmetic	19	(3)
2.3 Conversion from base D to base B representation	22	(6)
2.4 Representation of integers on a binary computer	28	(7)
2.5 Floating-point representations	35	(5)
2.6 Computer-representable numbers	40	(5)
2.7 Floating-point arithmetic operations	45	(11)
2.8 Fortran analysis of a floating-point number	56	(5)
2.9 Calculation of elementary functions	61	(9)
Chapter 3 Surveillance of Number Ranges	70	(23)
3.1 Introduction	70	(1)
3.2 Allowable number ranges	71	(3)
3.3 Basic real arithmetic operations	74	(3)
3.4 The quadratic equation	77	(6)
3.5 Complex arithmetic operations	83	(10)
Chapter 4 Solution of Equations	93	(83)
4.1 Introduction	93	(3)
4.2 Attainable accuracy	96	(7)
4.3 Graphical methods	103	(2)
4.4 The method of bisection	105	(7)
4.5 The method of false position	112	(6)
4.6 The secant method	118	(1)
4.7 General properties of iterative methods	119	(8)
4.8 Generation of iterative methods	127	(5)
4.9 The Newton method	132	(4)
4.10 Muller's method	136	(9)
4.11 Orders of convergence of iterative methods	145	(14)
4.12 Acceleration of the convergence	159	(1)
4.13 Systems of nonlinear equations	160	(16)
Chapter 5 Roots of Polynomial Equations	176	(70)
5.1 Introduction	176	(2)
5.2 General properties of polynomials	178	(11)
5.3 The Newton method and related methods	189	(6)
5.4 Muller's method and Cauchy's method	195	(1)
5.5 Location of the roots	196	(16)
5.6 Root acceptance and refinement	212	(7)
5.7 Matrix related methods: the modified Bernoulli method	219	(8)
5.8 Matrix related methods: the IP method	227	(8)
5.9 Polyalgorithms	235	(6)
5.10 Other methods	241	(5)
Chapter 6 Interpolation and Approximation	246	(98)
6.1 Introduction	246	(2)
6.2 Linear interpolation	248	(3)
6.3 Convergence and accuracy of linear interpolation	251	(7)
6.4 Lagrangian interpolation	258	(8)
6.5 Convergence and accuracy of Lagrangian interpolation	266	(4)
6.6 Interpolation with equal intervals	270	(18)
6.7 Hermite interpolation	288	(7)
6.8 Limitations on polynomial interpolation: smooth interpolation	295	(6)
6.9 Inverse interpolation	301	(7)
6.10 Approximation by polynomials	308	(10)
6.11 Least squares approximation by polynomials	318	(8)
6.12 Rational approximation	326	(3)
6.13 Trigonometric interpolation and approximation	329	(11)
6.14 Interpolation in two variables	340	(4)
Chapter 7 Numerical Differentiation and Quadrature	344	(78)
7.1 Introduction	344	(1)
7.2 The method of undetermined weights	345	(5)
7.3 Numerical differentiation	350	(11)
7.4 Numerical quadrature--equal intervals	361	(13)
7.5 The Euler-MacLaurin formula	374	(7)
7.6 Romberg integration	381	(4)
7.7 Error determination	385	(16)
7.8 Numerical quadrature--unequal intervals	401	(21)
Chapter 8 Ordinary Differential Equations	422
8.1 Introduction	422	(3)
8.2 Existence and uniqueness	425	(6)
8.3 Analytic methods	431	(5)
8.4 Integral equation formulation--the Picard method of successive approximations	436	(5)
8.5 The Euler method	441	(8)
8.6 Methods based on numerical quadrature	449	(11)
8.7 Error estimation for predictor-corrector methods	460	(9)
8.8 A numerical example	469	(4)
8.9 Runge-Kutta methods	473	(5)
8.10 Methods based on numerical differentiation	478	(4)
8.11 Higher-order equations and systems of first-order equations	482	(6)
8.12 The use of high-speed computers	488
Appendix A	A-1	(18)
Appendix B	A-19	(2)
Appendix C	A-21
Bibliography	B-1
Index	I-1