2000 Solved Problems in Numerical Analysis

Description

One of the great attractions of mathematics, to its devotees, may be that a well posed problem does in fact have a unique and exact solution. This idea of absolute precision has a sort of beauty that in its own way is unsurpassable. It may also be what frightens away many who feel more comfortable where there are shades of gray. In numerical analysis we enjoy a little bit of both worlds but are much closer to the former than the latter. The perfect solution is definitely out there somewhere in thought space, and knowledge of this fact has sustained many a wearisome effort to get close to it. In what direction does it lie? How close to it are we now? In the year 1225 Leonardo of Pisa studied the equation x^3 + 2x^2 + 10x - 20 = 0 seeking its one real root, and produced x = 1.368808107. Nobody knows his method, but it is a remarkable achievement for his time. Leonardo surely knew it was not perfection and may have wondered at the true identity of the target number, but must have derived great pleasure from realizing how close he had come. Numerical analysis problems do have exact solutions, but the thrill of victory does not wait for their discovery. It is enough to come close. Error is expected. Without it we would be out of business. There is no need of approximation where the real thing is within grasp. The following problems illustrate mathematics with some controlled shades of gray. It has been a pleasure to work them through. I was almost sorry to come to the 2000th. Though error is the substance of our subject, mistakes and blunders are not. I hope there are none, but experience suggests otherwise. I will be grateful to anyone who takes the time to point them out, in the kinder and gentler way that my now advanced years can withstand. Cover CHAPTER 1 The Representation of Numbers CHAPTER 2 Algorithms and Error CHAPTER 3 Classical Numerical Analysis to Newton's Formula CHAPTER 4 Classical Numerical Analysis: Further Developments CHAPTER 5 Higher-Order Approximations CHAPTER 6 Interpolation and Prediction CHAPTER 7 Numerical Differentiation CHAPTER 8 Numerical Integration CHAPTER 9 Sums and Series CHAPTER 10 Difference Equations CHAPTER 11 Differential Equations CHAPTER 12 Least-Squares Polynomial Approximation CHAPTER 13 Min-Max and L1 Polynomial Approximation CHAPTER 14 Approximation by Rational Functions CHAPTER 15 Trigonometric Approximation CHAPTER 16 Roots of Equations CHAPTER 17 Linear Systems CHAPTER 18 Optimization CHAPTER 19 Overdetermined Systems CHAPTER 20 Boundary Value Problems CHAPTER 21 Monte Carlo Methods Index