Using R for Numerical Analysis in Science and Engineering

✍ Scribed by Victor A. Bloomfield

• Publisher: CRC Press
• Year: 2014
• Tongue: English
• Leaves: 352
• Category: Library

✦ Synopsis

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 R’s 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.

✦ Table of Contents

000
001
Using R for Numerical Analysis in Science and Engineering
Contents
List of Figures
Preface
002
Chapter 1 Introduction
1.1 Obtaining and installing R
1.2 Learning R
1.3 Learning numerical methods
1.4 Finding help
1.5 Augmenting R with packages
1.6 Learning more about R
1.6.1 Books
1.6.2 Online resources
003
Chapter 2 Calculating
2.1 Basic operators and functions
2.2 Complex numbers
2.3 Numerical display, round- off error, and rounding
2.4 Assigning variables
2.4.1 Listing and removing variables
2.5 Relational operators
2.6 Vectors
2.6.1 Vector elements and indexes
2.6.2 Operations with vectors
2.6.3 Generating sequences
2.6.3.1 Regular sequences
2.6.3.2 Repeating values
2.6.3.3 Sequences of random numbers
2.6.4 Logical vectors
2.6.5 Speed of forming large vectors
2.6.6 Vector dot product and crossproduct
2.7 Matrices
2.7.1 Forming matrices
2.7.2 Operations on matrices
2.7.2.1 Arithmetic operations on matrices
2.7.2.2 Matrix multiplication
2.7.2.3 Transpose and determinant
2.7.2.4 Matrix crossproduct
2.7.2.5 Matrix exponential
2.7.2.6 Matrix inverse and solve
2.7.2.7 Eigenvalues and eigenvectors
2.7.2.8 Singular value decomposition
2.7.3 The Matrix package
2.7.4 Additional matrix functions and packages
2.8 Time and date calculations
004
Chapter 3 Graphing
3.1 Scatter plots
3.2 Function plots
3.3 Other common plots
3.3.1 Bar charts
3.3.2 Histograms
3.3.3 Box- and- whisker plots
3.4 Customizing plots
3.4.1 Points and lines
3.4.2 Axes, ticks, and par()
3.4.3 Overlaying plots with graphic elements
3.5 Error bars
3.6 Superimposing vectors in a plot
3.7 Modifying axes
3.7.1 Logarithmic axes
3.7.2 Supplementary axes
3.7.3 Incomplete axis boxes
3.7.4 Broken axes
3.8 Adding text and math expressions
3.8.1 Making math annotations with expression()
3.9 Placing several plots in a figure
3.10 Two- and three- dimensional plots
3.11 The plotrix package
3.11.1 radial.plot and polar.plot
3.11.2 Triangle plot
3.11.3 Error bars in plotrix
3.12 Animation
3.13 Additional plotting packages
005
Chapter 4 Programming and functions
4.1 Conditional execution:
4.2 Loops
4.2.1 for loop
4.2.2 Looping with while and repeat
4.3 User- defined functions
4.4 Debugging
4.5 Built- in mathematical functions
4.5.1 Bessel functions
4.5.2 Beta and gamma functions
4.5.3 Binomial coefficients
4.6 Special functions of mathematical physics
4.6.1 The gsl package
4.6.2 Special functions in other packages
4.7 Polynomial functions in packages
4.7.1 PolynomF package
4.7.2 orthopolynom package
4.8 Case studies
4.8.1 Two- dimensional random walk
4.8.2 Eigenvalues of a polymer chain
006
Chapter 5 Solving systems of algebraic equations
5.1 Finding the zeros of a polynomial
5.2 Finding the zeros of a function
5.2.1 Bisection method
5.2.2 Newton's method
5.2.3 uniroot and uniroot.all
5.3 Systems of linear equations: matrix
5.4 Matrix inverse
5.5 Singular matrix
5.6 Overdetermined systems and generalized inverse
5.7 Sparse matrices
5.7.1 Tridiagonal matrix
5.7.2 Banded matrix
5.7.3 Block matrix
5.8 Matrix decomposition
5.8.1 QR decomposition
5.8.2 Singular value decomposition
5.8.3 Eigendecomposition
5.8.4 LU decomposition
5.8.5 Cholesky decomposition
5.8.6 Schur decomposition
5.8.7 backsolve and forwardsolve
5.9 Systems of nonlinear equations
5.9.1 multiroot in the rootSolve package
5.9.2 nleqslv
5.9.3 BBsolve() in the BB package
5.10 Case studies
5.10.1 Spectroscopic analysis of a mixture
5.10.2 van der Waals equation
5.10.3 Chemical equilibrium
007
Chapter 6 Numerical differentiation and integration
6.1 Numerical differentiation
6.1.1 Numerical differentiation using base R
6.1.1.1 Using the fundamental definition
6.1.1.2 diff()
6.1.2 Numerical differentiation using the
6.1.3 Numerical differentiation using the
6.1.3.1 fderiv()
6.1.3.2 numderiv() and numdiff()
6.1.3.3 grad() and gradient()
6.1.3.4 jacobian()
6.1.3.5 hessian
6.1.3.6 laplacian()
6.2 Numerical integration
6.2.1 integrate: Basic integration in R
6.2.2 Integrating discretized functions
6.2.3 Gaussian quadrature
6.2.4 More integration routines in
6.2.5 Functions with singularities
6.2.6 Infinite integration domains
6.2.7 Integrals in higher dimensions
6.2.8 Monte Carlo and sparse grid integration
6.2.9 Complex line integrals
6.3 Symbolic manipulations in R
6.3.1 D()
6.3.2 deriv()
6.3.3 Polynomial functions
6.3.4 Interfaces to symbolic packages
6.4 Case studies
6.4.1 Circumference of an ellipse
6.4.2 Integration of a Lorentzian derivative spectrum
6.4.3 Volume of an ellipsoid
008
Chapter 7 Optimization
7.1 One- dimensional optimization
7.2 Multi- dimensional optimization with
7.2.1 optim() with “Nelder–Mead” default
7.2.2 optim() with “BFGS” method
7.2.3 optim() with “CG” method
7.2.4 optim() with “L-BFGS-B” method to find a local minimum
7.3 Other optimization packages
7.3.1 nlm()
7.3.2 ucminf package
7.3.3 BB package
7.3.4 optimx() wrapper
7.3.5 Derivative- free optimization algorithms
7.4 Optimization with constraints
7.4.1 constrOptim to optimize functions with linear constraints
7.4.2 External packages alabama and Rsolnp
7.5 Global optimization with many local minima
7.5.1 Simulated annealing
7.5.2 Genetic algorithms
7.5.2.1 DEoptim
7.5.2.2 rgenoud
7.5.2.3 GA
7.6 Linear and quadratic programming
7.6.1 Linear programming
7.6.2 Quadratic programming
7.7 Mixed- integer linear programming
7.7.1 Mixed- integer problems
7.7.2 Integer programming problems
7.7.2.1 Knapsack problems
7.7.2.2 Transportation problems
7.7.2.3 Assignment problems
7.7.2.4 Subsetsum problems
7.8 Case study
7.8.1 Monte Carlo simulation of the 2D Ising model
009
Chapter 8 Ordinary differential equations
8.1 Euler method
8.1.1 Projectile motion
8.1.2 Orbital motion
8.2 Improved Euler method
8.3 deSolve package
8.3.1 lsoda() and lsode()
8.3.2 “adams” and related methods
8.3.3 Stiff systems
8.4 Matrix exponential solution for sets of linear ODEs
8.5 Events and roots
8.6 Difference equations
8.7 Delay differential equations
8.8 Differential algebraic equations
8.9 rootSolve for steady state solutions of systems of ODEs
8.10 bvpSolve package for boundary value ODE problems
8.10.1 bvpshoot()
8.10.2 bvptwp()
8.10.3 bvpcol()
8.11 Stochastic differential equations:
8.12 Case studies
8.12.1 Launch of the space shuttle
8.12.2 Electrostatic potential of DNA solutions
8.12.3 Bifurcation analysis of Lotka– Volterra model
010
Chapter 9 Partial differential equations
9.1 Diffusion equation
9.2 Wave equation
9.2.1 FTCS method
9.2.2 Lax method
9.3 Laplace's equation
9.4 Solving PDEs with the ReacTran package
9.4.1 setup.grid.1D
9.4.2 setup.prop.1D
9.4.3 tran.1D
9.4.4 Calling ode.1D or steady.1D
9.5 Examples with the ReacTran package
9.5.1 1- D diffusion- advection equation
9.5.2 1- D wave equation
9.5.3 Laplace equation
9.5.4 Poisson equation for a dipole
9.6 Case studies
9.6.1 Diffusion in a viscosity gradient
9.6.2 Evolution of a Gaussian wave packet
9.6.3 Burgers equation
011
Chapter 10 Analyzing data
10.1 Getting data into R
10.2 Data frames
10.3 Summary statistics for a single dataset
10.4 Statistical comparison of two samples
10.5 Chi- squared test for goodness of fit
10.6 Correlation
10.7 Principal component analysis
10.8 Cluster analysis
10.8.1 Using hclust for agglomerative hierarchical clustering
10.8.2 Using diana for divisive hierarchical clustering
10.8.3 Using kmeans for partitioning clustering
10.8.4 Using pam for partitioning around medoids
10.9 Case studies
10.9.1 Chi square analysis of radioactive decay
10.9.2 Principal component analysis of quasars
012
Chapter 11 Fitting models to data
11.1 Fitting data with linear models
11.1.1 Polynomial fitting with lm
11.2 Fitting data with nonlinear models
11.3 Inverse modeling of ODEs with the
11.4 Improving the convergence of series: Pade and Shanks
11.5 Interpolation
11.5.1 Linear interpolation
11.5.2 Polynomial interpolation
11.5.3 Spline interpolation
11.5.3.1 Integration and differentiation with splines
11.5.4 Rational interpolation
11.6 Time series, spectrum analysis, and signal processing
11.6.1 Fast Fourier transform:
11.6.2 Inverse Fourier transform
11.6.3 Power spectrum: spectrum() function
11.6.4 findpeaks() function
11.6.5 Signal package
11.6.5.1 Butterworth filter
11.6.5.2 Savitzky– Golay filter
11.6.5.3 fft filter
11.7 Case studies
11.7.1 Fitting a rational function to data
11.7.2 Rise of atmospheric carbon dioxide
013
Bibliography
014
Index