Introduction To Numerical Computation: Analysis And MATLAB Illustrations

by Elden, Lars; Wittmeyer-Koch, Linde; Neilsen, Hans Bruun

• ISBN13: 9789144037271
• ISBN10: 9144037279
• Format: Paperback
• Copyright: 2004-08-31
• Publisher: Lightning Source Inc

Summary

This text, designed for an introductory course in scientific computation at an advanced undergraduate level, is a revision and translation of a Swedish work which focuses on simplicity and readability in order to help students prepare for complex mathematical software. (Computer Books - Languages/Programming)

Table of Contents

PREFACE	v
1. INTRODUCTION
1.1. Mathematical Models and Numerical Approximations	1	(5)
1.2. Numerical Computation	6	(2)
1.3. References	8	(1)
2. ERROR ANALYSIS AND COMPUTER ARITHMETIC
2.1. Sources of Error	9	(1)
2.2. Basic Concepts	10	(3)
2.3. Error Propagation	13	(7)
2.4. Number Representation	20	(4)
2.5. Rounding Errors in Floating Point	24	(2)
2.6. Arithmetic Operations in Floating Point	26	(4)
2.7. Accumulated Errors	30	(4)
2.8. IEEE Standard	34	(3)
Exercises	37	(4)
Computer Exercises	41	(1)
References	42	(3)
3. FUNCTION EVALUATION
3.1. Introduction	45	(1)
3.2. Remainder Term Estimates	46	(3)
3.3. Standard Functions	49	(4)
3.4. Range Reduction	53	(3)
3.5. Trigonometric Functions	56	(6)
Exercises	62	(1)
References	63	(2)
4. NONLINEAR EQUATIONS
4.1. Introduction	65	(1)
4.2. Crude Localization	66	(2)
4.3. Iteration Methods	68	(6)
4.4. Convergence Analysis	74	(4)
4.5. Error Estimation and Stopping Criteria	78	(6)
4.6. Algebraic Equations	84	(3)
4.7. Square Root	87	(3)
4.8. Nonlinear Systems	90	(6)
Exercises	96	(1)
Computer Exercises	96	(1)
References	97	(2)
5. INTERPOLATION
5.1. Introduction	99	(1)
5.2. Interpolation by Polynomials	100	(4)
5.3. Linear Interpolation	104	(3)
5.4. Newton's Interpolation Formula	107	(6)
5.5. Neville's Method	113	(2)
5.6. Lagrange Interpolation	115	(1)
5.7. Hermite Interpolation	116	(2)
5.8. Runge's Phenomenon	118	(1)
5.9. Splines	119	(2)
5.10. Linear Spline Functions	121	(3)
5.11. Cubic Splines	124	(10)
5.12. Cubic B-Splines	134	(3)
Exercises	137	(2)
Computer Exercises	139	(2)
References	141	(2)
6. DIFFERENTIATION AND RICHARDSON EXTRAPOLATION
6.1. Introduction	143	(1)
6.2. Difference Approximations	144	(1)
6.3. Difference Approximation Errors	145	(5)
6.4. Richardson Extrapolation	150	(10)
Exercises	160	(1)
Computer Exercises	161	(1)
References	162	(1)
7. INTEGRATION
7.1. Introduction	163	(1)
7.2. Trapezoidal Rule	163	(4)
7.3. Newton-Cotes Formulas	167	(4)
7.4. Romberg's Method	171	(5)
7.5. Difficulties with Numerical Integration	176	(4)
7.6. Adaptive Quadrature	180	(3)
Exercises	183	(1)
Computer Exercises	184	(1)
References	185	(2)
8. LINEAR SYSTEMS OF EQUATIONS
8.1. Introduction	187	(3)
8.2. Triangular Systems	190	(2)
8.3. Gaussian Elimination	192	(6)
8.4. Pivoting	198	(5)
8.5. Permutations, Gauss Transformations	203	(5)
8.6. LU Factorization	208	(6)
8.7. Symmetric, Positive Definite Matrices	214	(5)
8.8. Band Matrices	219	(4)
8.9. Inverse Matrix	223	(2)
8.10. Vector and Matrix Norms	225	(6)
8.11. Sensitivity Analysis	231	(5)
8.12. Rounding Errors	236	(5)
8.13. Estimation of Condition Number	241	(3)
8.14. Overdetermined Systems	244	(6)
8.15. QR Factorization	250	(7)
Exercises	257	(1)
Computer Exercises	258	(1)
References	259	(2)
9. APPROXIMATION
9.1. Introduction	261	(3)
9.2. Important Concepts	264	(5)
9.3. Least Squares Method	269	(5)
9.4. Orthogonal Functions	274	(2)
9.5. Orthogonal Polynomials	276	(5)
9.6. Legendre Polynomials	281	(3)
9.7. Chebyshev Polynomials	284	(4)
9.8. Discrete Cosine Transform	288	(8)
9.9. Minimax Approximation	296	(7)
Exercises	303	(2)
Computer Exercises	305	(2)
References	307	(2)
10. ORDINARY DIFFERENTIAL EQUATIONS
10.1. Introduction	309	(3)
10.2. Initial Value Problems	312	(4)
10.3. Local and Global Error	316	(3)
10.4. Runge-Kutta Methods	319	(4)
10.5. An Implicit Method	323	(2)
10.6. Stability	325	(9)
10.7. Adaptive Step Length Control	334	(3)
10.8. Boundary Value Problems	337	(2)
10.9. A Difference Method	339	(5)
10.10. A Finite Element Method	344	(8)
10.11. The Shooting Method	352	(3)
Exercises	355	(1)
Computer Exercises	356	(1)
References	357	(2)
SHORT BIOGRAPHIES	359	(5)
ANSWERS TO EXERCISES	364	(6)
INDEX	370