| PREFACE 1 SOME GENERAL PRINCIPLES OF NUMERICAL CALCULATION 1.1. Introduction 1.2. Some Common Ideas and Concepts in Numerical Methods 1.3. Numerical Problems and Algorithms 1.3.1. Definitions 1.3.2. Recursive Formulas; Homer's Rule 1.3.3. An Example of Numerical Instability 2 HOW TO OBTAIN AND ESTIMATE ACCURACY IN NUMERICAL CALCULATIONS 2.1. Basic Concepts in Error Estimation 2.1.1. Introduction 2.1.2. Sources of Error 2.1.3. Absolute and Relative Errors 2.1.4. Rounding and Chopping 2.2 Propagation of Errors 2.2.1. Simple Examples of Error Analysis 2.2.2. The General Formula for Error Propagation; Maximum Error and Standard Error 2.2.3. On the Practical Application of Error Estimation 2.2.4. The Use of Experimental Perturbations 2.2.5. Automatic Control of Accuracy 2.3. Number Systems; Floating and Fixed Representation 2.3.1. The Position System 2.3.2. Floating and Fixed Representation 2.3.3. Floating Decimal Point 2.3.4. Fixed Decimal Point 2.3.5. Round-off Errors in Computation with Floating Arithmetic Operations 2.4. Backward Error Analysis; Condition Numbers 2.4.1. Backward Error Analysis 2.4.2. Condition Numbers for Problems and Algorithms 2.4.3. Geometrical Illustration of Error Analysis 3 NUMERICAL USES OF SERIES 3.1. Elementary Uses of Series 3.1.1. Simple Examples 3.1.2. Estimating the Remainder 3.1.3. Power Series 3.2. Acceleration of Convergence 3.2.1. Slowly Converging Alternating Series 3.2.2. Slowly Converging Series with Positive Terms 3.2.3. Other Simple Ways to Accelerate Convergence 3.2.4. Ill-Conditioned Series 3.2.5. Numerical Use of Divergent Series 4 APPROXIMATION OF FUNCTIONS 4.1. Basic Concepts in Approximation 4.1.1. Introduction 4.1.2. The Idea of a Function Space 4.1.3. Norms and Seminorms 4.1.4. Approximation of Functions as a Geometric Problem in Function Space 4.2. The Approximation of Functions by the Method of Least Squares 4.2.1. Statement of the Problems 4.2.2. Orthogonal Systems 4.2.3. Solution of the Approximation Problem 4.3. Polynomials 4.3.1. Basic Terminology; the Weierstrass Approximation Theorem 4.3.2. Triangle Families of Polynomials …… 5 NUMERICAL LIENEAR ALGEBRA 6 NONLIEAR EQUATIONS 7 FINTE DIFFERENCES WITH APPLICATIONS TO NUMERICAL INTEGRATION,DIFFERENTIATION,AND INTERPOLATION 8 DIFFERENTIAL EQUATIONS 9 FOURIER METHODS 10 OPTIMIZATION 11 THE MONTE CARLO METHOD AND SIMULATION 12 SOLUTIONS TO PROBLEMS 13 BIBLIOGRAPHY AND PUBLISHED ALGORITHMS |
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