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E-raamat: Computational Methods for Numerical Analysis with R

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(Johns Hopkins Applied Physics Laboratory, USA)
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Computational Methods for Numerical Analysis with R is an overview of traditional numerical analysis topics presented using R. This guide shows how common functions from linear algebra, interpolation, numerical integration, optimization, and differential equations can be implemented in pure R code. Every algorithm described is given with a complete function implementation in R, along with examples to demonstrate the function and its use.

Computational Methods for Numerical Analysis with R is intended for those who already know R, but are interested in learning more about how the underlying algorithms work. As such, it is suitable for statisticians, economists, and engineers, and others with a computational and numerical background.

Arvustused

"The author says that the book is written for advanced undergraduate or first year graduate student as a collateral text for numerical analysis courses. The theoretical part of numerical analysis is mostly omitted, the focus is to present a working R code for many basic tasks of numerical computation including linear algebra, interpolation, numerical integration, root finding and optimisation and differential equations. The presentation is very clear and reader friendly." Matti Vuorinen (Turku), in Zentralblatt Mathematik, April 2018

List of Figures
xiii
List of Tables
xv
List of R Functions
xvii
Preface xix
1 Introduction to Numerical Analysis
1(28)
1.1 Numerical Analysis
1(7)
1.1.1 The Goals of Numerical Analysis
1(2)
1.1.2 Numerical Analysis in R
3(2)
1.1.3 Efficiency
5(3)
1.2 Data Types in R
8(8)
1.2.1 Data Types
8(3)
1.2.2 Data Structures
11(5)
1.3 Elementary Problems
16(13)
1.3.1 Summation Algorithms
16(4)
1.3.2 Evaluating Polynomials
20(4)
1.3.3 The nth Root Algorithm
24(2)
Comments
26(1)
Exercises
27(2)
2 Error Analysis
29(30)
2.1 True Values
29(6)
2.1.1 Accuracy
30(2)
2.1.2 Precision
32(3)
2.2 Internal Data Storage
35(7)
2.2.1 Binary Numbers
35(2)
2.2.2 Floating Point Numbers
37(5)
2.3 Numerical Error
42(11)
2.3.1 Round-Off Error and Machine ε
42(2)
2.3.2 Loss of Significance
44(4)
2.3.3 Overflow and Underflow
48(2)
2.3.4 Error Propagation and Stability
50(3)
2.4 Applications
53(6)
2.4.1 Simple Division Algorithms
53(2)
2.4.2 Binary Long Division
55(2)
Comments
57(1)
Exercises
58(1)
3 Linear Algebra
59(36)
3.1 Vectors and Matrices
59(8)
3.1.1 Vector and Matrix Operations
59(5)
3.1.2 Elementary Row Operations
64(3)
3.2 Gaussian Elimination
67(9)
3.2.1 Row Echelon Form
67(6)
3.2.2 Tridiagonal Matrices
73(3)
3.3 Matrix Decomposition
76(6)
3.3.1 LU Decomposition
76(4)
3.3.2 Cholesky Decomposition
80(2)
3.4 Iterative Methods
82(7)
3.4.1 Jacobi Iteration
83(3)
3.4.2 Gauss--Seidel Iteration
86(3)
3.5 Applications
89(6)
3.5.1 Least Squares
89(2)
Comments
91(1)
Exercises
92(3)
4 Interpolation and Extrapolation
95(38)
4.1 Polynomial Interpolation
95(7)
4.1.1 Linear Interpolation
95(2)
4.1.2 Higher-Order Polynomial Interpolation
97(5)
4.2 Piecewise Interpolation
102(13)
4.2.1 Piecewise Linear Interpolation
103(2)
4.2.2 Cubic Spline Interpolation
105(5)
4.2.3 Bezier Curves
110(5)
4.3 Multidimensional Interpolation
115(7)
4.3.1 Bilinear Interpolation
115(4)
4.3.2 Nearest Neighbor Interpolation
119(3)
4.4 Applications
122(11)
4.4.1 Time Series Interpolation
122(3)
4.4.2 Computer Graphics
125(5)
Comments
130(1)
Exercises
131(2)
5 Differentiation and Integration
133(42)
5.1 Numerical Differentiation
133(5)
5.1.1 Finite Differences
133(4)
5.1.2 The Second Derivative
137(1)
5.2 Newton--Cotes Integration
138(11)
5.2.1 Multipanel Interpolation Rules
139(6)
5.2.2 Newton--Cotes Errors
145(2)
5.2.3 Newton--Cotes Forms, Generally
147(2)
5.3 Gaussian Integration
149(4)
5.3.1 The Gaussian Method
149(2)
5.3.2 Implementation Details
151(2)
5.4 More Techniques
153(10)
5.4.1 Adaptive Integrators
153(3)
5.4.2 Romberg's Method
156(3)
5.4.3 Monte Carlo Methods
159(4)
5.5 Applications
163(12)
5.5.1 Revolved Volumes
163(5)
5.5.2 Gini Coefficients
168(2)
Comments
170(3)
Exercises
173(2)
6 Root Finding and Optimization
175(38)
6.1 One-Dimensional Root Finding
175(10)
6.1.1 Bisection Method
175(4)
6.1.2 Newton--Raphson Method
179(4)
6.1.3 Secant Method
183(2)
6.2 Minimization and Maximization
185(7)
6.2.1 Golden Section Search
185(3)
6.2.2 Gradient Descent
188(4)
6.3 Multidimensional Optimization
192(8)
6.3.1 Multidimensional Gradient Descent
192(3)
6.3.2 Hill Climbing
195(2)
6.3.3 Simulated Annealing
197(3)
6.4 Applications
200(13)
6.4.1 Least Squares
200(3)
6.4.2 The Traveling Salesperson
203(5)
Comments
208(2)
Exercises
210(3)
7 Differential Equations
213(36)
7.1 Initial Value Problems
213(14)
7.1.1 Euler Method
213(6)
7.1.2 Runge--Kutta Methods, Generally
219(5)
7.1.3 Linear Multistep Methods
224(3)
7.2 Systems of Ordinary Differential Equations
227(7)
7.2.1 Solution Systems and Initial Value Problems
228(3)
7.2.2 Boundary Value Problems
231(3)
7.3 Partial Differential Equations
234(8)
7.3.1 The Heat Equation
234(4)
7.3.2 The Wave Equation
238(4)
7.4 Applications
242(7)
7.4.1 Carbon Dating
242(2)
7.4.2 Lotka--Volterra Equations
244(1)
Comments
245(2)
Exercises
247(2)
Suggested Reading 249(4)
Index 253
James P Howard, II
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