Numerical Mathematics and Computing

by Cheney, E. Ward; Kincaid, David R.

• ISBN13: 9780534389932
• ISBN10: 0534389937
• Edition: 5th
• Format: Hardcover
• Copyright: 2003-07-25
• Publisher: Brooks Cole

Summary

1. INTRODUCTION. Preliminary Remarks. Review of Taylor Series. 2. NUMBER REPRESENTATION AND ERRORS. Representation of Numbers in Different Bases. Floating-Point Representation. Loss of Significance. 3. LOCATING ROOTS OF EQUATIONS. Bisection Method. Newton's Method. Secant Method. 4. INTERPOLATION AND NUMERICAL DIFFERENTIATION. Polynomial Interpolation. Errors in Polynomial Interpolation. Estimating Derivatives and Richardson Extrapolation. 5. NUMERICAL INTEGRATION. Definite Integral. Trapezoid Rule. Romberg Algorithm. 6. MORE ON NUMERICAL INTEGRATION. An Adaptive Simpson's Scheme. Gaussian Quadrature Formulas. 7. SYSTEMS OF LINEAR EQUATIONS. Naive Gaussian Elimination. Gaussian Elimination with Scaled Partial Pivoting. Tridiagonal and Banded Systems. 8. MORE ON SYSTEMS OF LINEAR EQUATIONS. Factorizations. Iterative Solution of Linear Systems. Eigenvalues and Eigenvectors. Power Methods. 9. APPROXIMATION BY SPLINE FUNCTIONS. First-Degree and Second-Degree Splines. Natural Cubic Splines. B splines: Interpolation and Approximation by B Splines. 10. ORDINARY DIFFERENTIAL EQUATIONS. Initial-Value Problem: Analytical vs. Numerical Solution. Taylor Series Methods. Runge-Kutta Methods. Stability and Adaptive Runge-Kutta and Multi-Step Methods. 11. SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS. Methods for First-Order Systems. Higher-Order Equations and Systems. Adams-Moulton Methods. 12. SMOOTHING OF DATA AND THE METHOD OF LEAST SQUARES. The Method of Least Squares. Orthogonal Systems and Chebyshev Polynomials. Other Examples of the Least-Squares Principle. 13. MONTE CARLO METHODS AND SIMULATION. Random Numbers. Estimation of Areas and Volumes by\hfill\break Monte Carlo Techniques. Simulation. 14. BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS. Shooting Method. A Discretization Method. 15. PARTIAL DIFFERENTIAL EQUATIONS. Some Partial Differential Equations from Applied Problems. Parabolic Problems. Hyperbolic Problems. Elliptic Problems. 16. MINIMIZATION OF MULTIVARIATE FUNCTIONS. One-Variable Case. Multivariate Case. 17. LINEAR PROGRAMMING. Standard Forms and Duality. Simplex Method. Approximate Solution of Inconsistent Linear Systems. Appendices. Advice on Good Programming Practices. An Overview of Mathematical Software on the Web. Additional Details on IEEE Floating-Point Arithmetic. Linear Algebra Concepts and Notation. Sir Isaac Newton: Never at Rest. Answers for Selected Problems. Bibliography.

Table of Contents

1 Introduction	1	(42)
1.1 Preliminary Remarks	1	(16)
Nested Multiplication	2	(2)
Errors: Absolute and Relative	4	(1)
Rounding and Chopping	4	(2)
First Programming Experiment	6	(1)
Mathematical Software	7	(10)
1.2 Review of Taylor Series	17	(26)
Taylor Series	17	(5)
Taylor's Theorem in Terms of (x-c)	22	(2)
Mean-Value Theorem	24	(1)
Taylor's Theorem in Terms of h	25	(2)
Alternating Series	27	(16)
2 Number Representation and Errors	43	(48)
2.1 Representation of Numbers in Different Bases	43	(11)
Base ß Numbers	44	(1)
Conversion of Integer Parts	45	(2)
Conversion of Fractional Parts	47	(1)