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E-raamat: Using R for Numerical Analysis in Science and Engineering

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Using R for Numerical Analysis in Science and EngineeringSuurem pilt
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Instead of presenting the standard theoretical treatments that underlie the various numerical methods used by scientists and engineers, Using R for Numerical Analysis in Science and Engineering shows how to use R and its add-on packages to obtain numerical solutions to the complex mathematical problems commonly faced by scientists and engineers. This practical guide to the capabilities of R demonstrates Monte Carlo, stochastic, deterministic, and other numerical methods through an abundance of worked examples and code, covering the solution of systems of linear algebraic equations and nonlinear equations as well as ordinary differential equations and partial differential equations. It not only shows how to use Rs powerful graphic tools to construct the types of plots most useful in scientific and engineering work, but also:











Explains how to statistically analyze and fit data to linear and nonlinear models Explores numerical differentiation, integration, and optimization Describes how to find eigenvalues and eigenfunctions Discusses interpolation and curve fitting Considers the analysis of time series

Using R for Numerical Analysis in Science and Engineering provides a solid introduction to the most useful numerical methods for scientific and engineering data analysis using R.

Arvustused

" the book is well organized, clearly written, and has a large amount of useful R code. It does a good job of answering the question of how to use R to perform numerical analyses of interest to scientists and engineers and, as such, can be recommended to the intended audience." Journal of the Royal Statistical Society, Series A, 2015

"I would recommend it to those seeking to improve their programming efficiency. the extensive coverage of optimization, ordinary differential equations, and partial differential equations combined with its exemplary demonstration of R coding through effective examples make this book a valuable resource for a wide audience. a good reference for scientific and engineering researchers." The American Statistician, February 2015

"... the book is well organized, clearly written, and has a large amount of useful R code. It does a good job answering the question of how to use R to perform numerical analyses of interest to scientists and engineers, and as such, can be recommended to the intended audience." Andrey Kostenko, Teaching Statistics

List of Figures
xiii
Preface xix
1 Introduction
1(6)
1.1 Obtaining and installing R
1(1)
1.2 Learning R
1(1)
1.3 Learning numerical methods
1(1)
1.4 Finding help
2(1)
1.5 Augmenting R with packages
3(2)
1.6 Learning more about R
5(2)
1.6.1 Books
5(1)
1.6.2 Online resources
5(2)
2 Calculating
7(30)
2.1 Basic operators and functions
7(1)
2.2 Complex numbers
8(1)
2.3 Numerical display, round-off error, and rounding
9(2)
2.4 Assigning variables
11(1)
2.4.1 Listing and removing variables
12(1)
2.5 Relational operators
12(1)
2.6 Vectors
13(8)
2.6.1 Vector elements and indexes
13(1)
2.6.2 Operations with vectors
14(1)
2.6.3 Generating sequences
15(1)
2.6.3.1 Regular sequences
15(1)
2.6.3.2 Repeating values
16(1)
2.6.3.3 Sequences of random numbers
16(1)
2.6.4 Logical vectors
17(1)
2.6.5 Speed of forming large vectors
18(1)
2.6.6 Vector dot product and crossproduct
19(2)
2.7 Matrices
21(13)
2.7.1 Forming matrices
21(3)
2.7.2 Operations on matrices
24(1)
2.7.2.1 Arithmetic operations on matrices
24(1)
2.7.2.2 Matrix multiplication
25(1)
2.7.2.3 Transpose and determinant
26(1)
2.7.2.4 Matrix crossproduct
26(1)
2.7.2.5 Matrix exponential
27(1)
2.7.2.6 Matrix inverse and solve
27(2)
2.7.2.7 Eigenvalues and eigenvectors
29(2)
2.7.2.8 Singular value decomposition
31(2)
2.7.3 The Matrix package
33(1)
2.7.4 Additional matrix functions and packages
34(1)
2.8 Time and date calculations
34(3)
3 Graphing
37(28)
3.1 Scatter plots
37(2)
3.2 Function plots
39(1)
3.3 Other common plots
40(4)
3.3.1 Bar charts
40(2)
3.3.2 Histograms
42(1)
3.3.3 Box-and-whisker plots
43(1)
3.4 Customizing plots
44(4)
3.4.1 Points and lines
44(1)
3.4.2 Axes, ticks, and par ()
44(2)
3.4.3 Overlaying plots with graphic elements
46(2)
3.5 Error bars
48(1)
3.6 Superimposing vectors in a plot
49(1)
3.7 Modifying axes
50(4)
3.7.1 Logarithmic axes
51(1)
3.7.2 Supplementary axes
51(1)
3.7.3 Incomplete axis boxes
52(1)
3.7.4 Broken axes
52(2)
3.8 Adding text and math expressions
54(2)
3.8.1 Making math annotations with expression()
55(1)
3.9 Placing several plots in a figure
56(2)
3.10 Two- and three-dimensional plots
58(2)
3.11 The plotrix package
60(3)
3.11.1 radial.plot and polar.plot
60(1)
3.11.2 Triangle plot
61(1)
3.11.3 Error bars in plotrix
62(1)
3.12 Animation
63(1)
3.13 Additional plotting packages
64(1)
4 Programming and functions
65(26)
4.1 Conditional execution: if and if else
65(1)
4.2 Loops
66(3)
4.2.1 for loop
66(2)
4.2.2 Looping with while and repeat
68(1)
4.3 User-defined functions
69(3)
4.4 Debugging
72(1)
4.5 Built-in mathematical functions
73(2)
4.5.1 Bessel functions
73(1)
4.5.2 Beta and gamma functions
74(1)
4.5.3 Binomial coefficients
75(1)
4.6 Special functions of mathematical physics
75(3)
4.6.1 The gsl package
75(1)
4.6.2 Special functions in other packages
75(3)
4.7 Polynomial functions in packages
78(8)
4.7.1 PolynomF package
79(4)
4.7.2 orthopolynom package
83(3)
4.8 Case studies
86(5)
4.8.1 Two-dimensional random walk
86(1)
4.8.2 Eigenvalues of a polymer chain
87(4)
5 Solving systems of algebraic equations
91(34)
5.1 Finding the zeros of a polynomial
91(1)
5.2 Finding the zeros of a function
92(4)
5.2.1 Bisection method
92(1)
5.2.2 Newton's method
93(1)
5.2.3 uniroot and uniroot.all
94(2)
5.3 Systems of linear equations: matrix solve
96(1)
5.4 Matrix inverse
97(1)
5.5 Singular matrix
97(1)
5.6 Overdetermined systems and generalized inverse
98(1)
5.7 Sparse matrices
99(5)
5.7.1 Tridiagonal matrix
99(2)
5.7.2 Banded matrix
101(1)
5.7.3 Block matrix
102(2)
5.8 Matrix decomposition
104(5)
5.8.1 QR decomposition
105(1)
5.8.2 Singular value decomposition
106(1)
5.8.3 Eigendecomposition
107(1)
5.8.4 LU decomposition
107(1)
5.8.5 Cholesky decomposition
108(1)
5.8.6 Schur decomposition
109(1)
5.8.7 backsolve and forwardsolve
109(1)
5.9 Systems of nonlinear equations
109(8)
5.9.1 multiroot in the rootSolve package
109(2)
5.9.2 nleqslv
111(1)
5.9.3 BBsolve() in the BB package
112(5)
5.10 Case studies
117(8)
5.10.1 Spectroscopic analysis of a mixture
117(3)
5.10.2 van der Waals equation
120(2)
5.10.3 Chemical equilibrium
122(3)
6 Numerical differentiation and integration
125(34)
6.1 Numerical differentiation
125(8)
6.1.1 Numerical differentiation using base R
125(1)
6.1.1.1 Using the fundamental definition
125(1)
6.1.1.2 diff()
126(1)
6.1.2 Numerical differentiation using the numDeriv package
127(1)
6.1.2.1 grad()
128(1)
6.1.2.2 jacobian()
128(1)
6.1.2.3 hessian
129(1)
6.1.3 Numerical differentiation using the pracma package
129(1)
6.1.3.1 fderiv()
129(1)
6.1.3.2 numderiv() and numdiff()
130(1)
6.1.3.3 grad() and gradient()
131(1)
6.1.3.4 jacobian()
131(1)
6.1.3.5 hessian
132(1)
6.1.3.6 laplacian()
133(1)
6.2 Numerical integration
133(19)
6.2.1 integrate: Basic integration in R
134(2)
6.2.2 Integrating discretized functions
136(1)
6.2.3 Gaussian quadrature
137(3)
6.2.4 More integration routines in pracma
140(2)
6.2.5 Functions with singularities
142(2)
6.2.6 Infinite integration domains
144(2)
6.2.7 Integrals in higher dimensions
146(2)
6.2.8 Monte Carlo and sparse grid integration
148(2)
6.2.9 Complex line integrals
150(2)
6.3 Symbolic manipulations in R
152(3)
6.3.1 D()
152(1)
6.3.2 deriv()
152(2)
6.3.3 Polynomial functions
154(1)
6.3.4 Interfaces to symbolic packages
155(1)
6.4 Case studies
155(4)
6.4.1 Circumference of an ellipse
155(1)
6.4.2 Integration of a Lorentzian derivative spectrum
156(1)
6.4.3 Volume of an ellipsoid
157(2)
7 Optimization
159(40)
7.1 One-dimensional optimization
159(3)
7.2 Multi-dimensional optimization with optim()
162(7)
7.2.1 optim() with "Nelder-Mead" default
163(2)
7.2.2 optim() with "BFGS" method
165(2)
7.2.3 optim() with "CG" method
167(1)
7.2.4 optim() with "L-BFGS-B" method to find a local minimum
167(2)
7.3 Other optimization packages
169(4)
7.3.1 nlm()
169(2)
7.3.2 ucminf package
171(1)
7.3.3 BB package
171(1)
7.3.4 optimx() wrapper
172(1)
7.3.5 Derivative-free optimization algorithms
172(1)
7.4 Optimization with constraints
173(4)
7.4.1 constrOptim to optimize functions with linear constraints
173(2)
7.4.2 External packages alabama and Rsolnp
175(2)
7.5 Global optimization with many local minima
177(6)
7.5.1 Simulated annealing
178(3)
7.5.2 Genetic algorithms
181(1)
7.5.2.1 DEoptim
181(2)
7.5.2.2 rgenoud
183(1)
7.5.2.3 Ga
183(1)
7.6 Linear and quadratic programming
183(6)
7.6.1 Linear programming
183(3)
7.6.2 Quadratic programming
186(3)
7.7 Mixed-integer linear programming
189(5)
7.7.1 Mixed-integer problems
189(1)
7.7.2 Integer programming problems
190(1)
7.7.2.1 Knapsack problems
191(1)
7.7.2.2 Transportation problems
191(1)
7.7.2.3 Assignment problems
192(1)
7.7.2.4 Subsetsum problems
193(1)
7.8 Case study
194(5)
7.8.1 Monte Carlo simulation of the 2D Ising model
194(5)
8 Ordinary differential equations
199(50)
8.1 Euler method
200(5)
8.1.1 Projectile motion
201(2)
8.1.2 Orbital motion
203(2)
8.2 Improved Euler method
205(3)
8.3 deSolve package
208(6)
8.3.1 lsoda() and lsode()
210(1)
8.3.2 "adams" and related methods
211(2)
8.3.3 Stiff systems
213(1)
8.4 Matrix exponential solution for sets of linear ODEs
214(1)
8.5 Events and roots
215(5)
8.6 Difference equations
220(1)
8.7 Delay differential equations
221(3)
8.8 Differential algebraic equations
224(3)
8.9 rootSolve for steady state solutions of systems of ODEs
227(3)
8.10 bvpSolve package for boundary value ODE problems
230(3)
8.10.1 bvpshoot()
230(1)
8.10.2 bvptwp()
231(1)
8.10.3 bvpcol()
232(1)
8.11 Stochastic differential equations: GillespieSSA package
233(7)
8.12 Case studies
240(9)
8.12.1 Launch of the space shuttle
240(1)
8.12.2 Electrostatic potential of DNA solutions
241(3)
8.12.3 Bifurcation analysis of Lotka--Volterra model
244(5)
9 Partial differential equations
249(24)
9.1 Diffusion equation
249(2)
9.2 Wave equation
251(3)
9.2.1 FTCS method
252(1)
9.2.2 Lax method
253(1)
9.3 Laplace's equation
254(2)
9.4 Solving PDEs with the ReacTran package
256(3)
9.4.1 setup.grid. 1D
257(1)
9.4.2 setup.prop. 1D
258(1)
9.4.3 tran. 1D
258(1)
9.4.4 Calling ode. 1D or steady. 1D
259(1)
9.5 Examples with the ReacTran package
259(5)
9.5.1 1-D diffusion-advection equation
259(1)
9.5.2 1-D wave equation
260(2)
9.5.3 Laplace equation
262(1)
9.5.4 Poisson equation for a dipole
263(1)
9.6 Case studies
264(9)
9.6.1 Diffusion in a viscosity gradient
264(3)
9.6.2 Evolution of a Gaussian wave packet
267(2)
9.6.3 Burgers equation
269(4)
10 Analyzing data
273(20)
10.1 Getting data into R
273(1)
10.2 Data frames
274(1)
10.3 Summary statistics for a single dataset
275(2)
10.4 Statistical comparison of two samples
277(2)
10.5 Chi-squared test for goodness of fit
279(1)
10.6 Correlation
280(1)
10.7 Principal component analysis
281(2)
10.8 Cluster analysis
283(3)
10.8.1 Using hclust for agglomerative hierarchical clustering
283(1)
10.8.2 Using diana for divisive hierarchical clustering
284(1)
10.8.3 Using kmeans for partitioning clustering
285(1)
10.8.4 Using pam for partitioning around medoids
286(1)
10.9 Case studies
286(7)
10.9.1 Chi square analysis of radioactive decay
286(3)
10.9.2 Principal component analysis of quasars
289(4)
11 Fitting models to data
293(36)
11.1 Fitting data with linear models
293(3)
11.1.1 Polynomial fitting with lm
294(2)
11.2 Fitting data with nonlinear models
296(8)
11.3 Inverse modeling of ODEs with the FME package
304(5)
11.4 Improving the convergence of series: Pade and Shanks
309(2)
11.5 Interpolation
311(5)
11.5.1 Linear interpolation
312(1)
11.5.2 Polynomial interpolation
313(1)
11.5.3 Spline interpolation
313(1)
11.5.3.1 Integration and differentiation with splines
314(1)
11.5.4 Rational interpolation
315(1)
11.6 Time series, spectrum analysis, and signal processing
316(9)
11.6.1 Fast Fourier transform: fft() function
316(1)
11.6.2 Inverse Fourier transform
317(1)
11.6.3 Power spectrum: spectrum() function
318(3)
11.6.4 findpeaks() function
321(1)
11.6.5 Signal package
322(1)
11.6.5.1 Butterworth filter
322(2)
11.6.5.2 Savitzky--Golay filter
324(1)
11.6.5.3 fft filter
324(1)
11.7 Case studies
325(4)
11.7.1 Fitting a rational function to data
325(2)
11.7.2 Rise of atmospheric carbon dioxide
327(2)
Bibliography 329(2)
Index 331
Victor A. Bloomfield is currently emeritus professor at University of Minnesota, Minneapolis, USA. His research has encompassed more than four decades and a variety of topics, including enzyme kinetics, dynamic laser light scattering, bacteriophage assembly, DNA condensation, scanning tunneling microscopy, and single molecule stretching experiments on DNA. His theoretical work on biopolymer hydrodynamics and polyelectrolyte behavior has resulted in over 200 peer-reviewed journal publications. Using R for Numerical Analysis in Science and Engineering is an extension and broadening of his 2009 book, Computer Simulation and Data Analysis in Molecular Biology and Biophysics: An Introduction Using R, for general usage in science and engineering.
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