book

Using R for Numerical Analysis in Science and Engineering

by Victor A. Bloomfield

September 2018

Intermediate to advanced

359 pages

8h 12m

English

Chapman and Hall/CRC

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1.1 Obtaining and installing R1.2 Learning R1.3 Learning numerical methods1.4 Finding help1.5 Augmenting R with packages1.6 Learning more about R1.6.1 Books1.6.2 Online resources
2.1 Basic operators and functions2.2 Complex numbers2.3 Numerical display, round-off error, and rounding2.4 Assigning variables2.4.1 Listing and removing variables2.5 Relational operators2.6 Vectors2.6.1 Vector elements and indexes2.6.2 Operations with vectors2.6.3 Generating sequences2.6.3.1 Regular sequences2.6.3.2 Repeating values2.6.3.3 Sequences of random numbers2.6.4 Logical vectors2.6.5 Speed of forming large vectors2.6.6 Vector dot product and crossproduct2.7 Matrices2.7.1 Forming matrices2.7.2 Operations on matrices2.7.2.1 Arithmetic operations on matrices2.7.2.2 Matrix multiplication2.7.2.3 Transpose and determinant2.7.2.4 Matrix crossproduct2.7.2.5 Matrix exponential2.7.2.6 Matrix inverse and solve2.7.2.7 Eigenvalues and eigenvectors2.7.2.8 Singular value decomposition2.7.3 The Matrix package2.7.4 Additional matrix functions and packages2.8 Time and date calculations
3.1 Scatter plots3.2 Function plots3.3 Other common plots3.3.1 Bar charts3.3.2 Histograms3.3.3 Box-and-whisker plots3.4 Customizing plots3.4.1 Points and lines3.4.2 Axes, ticks, and par()3.4.3 Overlaying plots with graphic elements3.5 Error bars3.6 Superimposing vectors in a plot3.7 Modifying axes3.7.1 Logarithmic axes3.7.2 Supplementary axes3.7.3 Incomplete axis boxes3.7.4 Broken axes3.8 Adding text and math expressions3.8.1 Making math annotations with expression()3.9 Placing several plots in a figure3.10 Two- and three-dimensional plots3.11 The plotrix package3.11.1 radial.plot and polar.plot3.11.2 Triangle plot3.11.3 Error bars in plotrix3.12 Animation3.13 Additional plotting packages
4.1 Conditional execution: if and ifelse4.2 Loops4.2.1 for loop4.2.2 Looping with while and repeat4.3 User-defined functions4.4 Debugging4.5 Built-in mathematical functions4.5.1 Bessel functions4.5.2 Beta and gamma functions4.5.3 Binomial coefficients4.6 Special functions of mathematical physics4.6.1 The gsl package4.6.2 Special functions in other packages4.7 Polynomial functions in packages4.7.1 PolynomF package4.7.2 orthopolynom package4.8 Case studies4.8.1 Two-dimensional random walk4.8.2 Eigenvalues of a polymer chain

5.1 Finding the zeros of a polynomial5.2 Finding the zeros of a function5.2.1 Bisection method5.2.2 Newton’s method5.2.3 uniroot and uniroot.all5.3 Systems of linear equations: matrix solve5.4 Matrix inverse5.5 Singular matrix5.6 Overdetermined systems and generalized inverse5.7 Sparse matrices5.7.1 Tridiagonal matrix5.7.2 Banded matrix5.7.3 Block matrix5.8 Matrix decomposition5.8.1 QR decomposition5.8.2 Singular value decomposition5.8.3 Eigendecomposition5.8.4 LU decomposition5.8.5 Cholesky decomposition5.8.6 Schur decomposition5.8.7 backsolve and forwardsolve5.9 Systems of nonlinear equations5.9.1 multiroot in the rootSolve package5.9.2 nleqslv5.9.3 BBsolve() in the BB package5.10 Case studies5.10.1 Spectroscopic analysis of a mixture5.10.2 van der Waals equation5.10.3 Chemical equilibrium
6.1 Numerical differentiation6.1.1 Numerical differentiation using base R6.1.1.1 Using the fundamental definition6.1.1.2 diff()6.1.2 Numerical differentiation using the numDeriv package6.1.2.1 grad()6.1.2.2 jacobian()6.1.2.3 hessian6.1.3 Numerical differentiation using the pracma package6.1.3.1 fderiv()6.1.3.2 numderiv() and numdiff()6.1.3.3 grad() and gradient()6.1.3.4 jacobian()6.1.3.5 hessian6.1.3.6 laplacian()6.2 Numerical integration6.2.1 integrate: Basic integration in R6.2.2 Integrating discretized functions6.2.3 Gaussian quadrature6.2.4 More integration routines in pracma6.2.5 Functions with singularities6.2.6 Infinite integration domains6.2.7 Integrals in higher dimensions6.2.8 Monte Carlo and sparse grid integration6.2.9 Complex line integrals6.3 Symbolic manipulations in R6.3.1 D()6.3.2 deriv()6.3.3 Polynomial functions6.3.4 Interfaces to symbolic packages6.4 Case studies6.4.1 Circumference of an ellipse6.4.2 Integration of a Lorentzian derivative spectrum6.4.3 Volume of an ellipsoid
7.1 One-dimensional optimization7.2 Multi-dimensional optimization with optim()7.2.1 optim() with “Nelder–Mead” default7.2.2 optim() with “BFGS” method7.2.3 optim() with “CG” method7.2.4 optim() with “L-BFGS-B” method to find a local minimum7.3 Other optimization packages7.3.1 nlm()7.3.2 ucminf package7.3.3 BB package7.3.4 optimx() wrapper7.3.5 Derivative-free optimization algorithms7.4 Optimization with constraints7.4.1 constrOptim to optimize functions with linear constraints7.4.2 External packages alabama and Rsolnp7.5 Global optimization with many local minima7.5.1 Simulated annealing7.5.2 Genetic algorithms7.5.2.1 DEoptim7.5.2.2 rgenoud7.5.2.3 GA7.6 Linear and quadratic programming7.6.1 Linear programming7.6.2 Quadratic programming7.7 Mixed-integer linear programming7.7.1 Mixed-integer problems7.7.2 Integer programming problems7.7.2.1 Knapsack problems7.7.2.2 Transportation problems7.7.2.3 Assignment problems7.7.2.4 Subsetsum problems7.8 Case study7.8.1 Monte Carlo simulation of the 2D Ising model
8.1 Euler method8.1.1 Projectile motion8.1.2 Orbital motion8.2 Improved Euler method8.3 deSolve package8.3.1 lsoda() and lsode()8.3.2 “adams” and related methods8.3.3 Stiff systems8.4 Matrix exponential solution for sets of linear ODEs8.5 Events and roots8.6 Difference equations8.7 Delay differential equations8.8 Differential algebraic equations8.9 rootSolve for steady state solutions of systems of ODEs8.10 bvpSolve package for boundary value ODE problems8.10.1 bvpshoot()8.10.2 bvptwp()8.10.3 bvpcol()8.11 Stochastic differential equations: GillespieSSA package8.12 Case studies8.12.1 Launch of the space shuttle8.12.2 Electrostatic potential of DNA solutions8.12.3 Bifurcation analysis of Lotka–Volterra model
9.1 Diffusion equation9.2 Wave equation9.2.1 FTCS method9.2.2 Lax method9.3 Laplace’s equation9.4 Solving PDEs with the ReacTran package9.4.1 setup.grid.1D9.4.2 setup.prop.1D9.4.3 tran.1D9.4.4 Calling ode.1D or steady.1D9.5 Examples with the ReacTran package9.5.1 1-D diffusion-advection equation9.5.2 1-D wave equation9.5.3 Laplace equation9.5.4 Poisson equation for a dipole9.6 Case studies9.6.1 Diffusion in a viscosity gradient9.6.2 Evolution of a Gaussian wave packet9.6.3 Burgers equation
10.1 Getting data into R10.2 Data frames10.3 Summary statistics for a single dataset10.4 Statistical comparison of two samples10.5 Chi-squared test for goodness of fit10.6 Correlation10.7 Principal component analysis10.8 Cluster analysis10.8.1 Using hclust for agglomerative hierarchical clustering10.8.2 Using diana for divisive hierarchical clustering10.8.3 Using kmeans for partitioning clustering10.8.4 Using pam for partitioning around medoids10.9 Case studies10.9.1 Chi square analysis of radioactive decay10.9.2 Principal component analysis of quasars
11.1 Fitting data with linear models11.1.1 Polynomial fitting with lm11.2 Fitting data with nonlinear models11.3 Inverse modeling of ODEs with the FME package11.4 Improving the convergence of series: Padé and Shanks11.5 Interpolation11.5.1 Linear interpolation11.5.2 Polynomial interpolation11.5.3 Spline interpolation11.5.3.1 Integration and differentiation with splines11.5.4 Rational interpolation11.6 Time series, spectrum analysis, and signal processing11.6.1 Fast Fourier transform: fft() function11.6.2 Inverse Fourier transform11.6.3 Power spectrum: spectrum() function11.6.4 findpeaks() function11.6.5 Signal package11.6.5.1 Butterworth filter11.6.5.2 Savitzky–Golay filter11.6.5.3 fft filter11.7 Case studies11.7.1 Fitting a rational function to data11.7.2 Rise of atmospheric carbon dioxide

Content preview from Using R for Numerical Analysis in Science and Engineering

Chapter 5Solving systems of algebraic equations

Solving equations is central to numerical analysis, and R has numerous tools for doing so. We begin by extending our consideration of polynomials from the previous chapter.

5.1 Finding the zeros of a polynomial

The base installation of R has the function polyroot to find the zeros of a real or complex polynomial, specified by the vector of its coefficients in ascending order. This function uses the algorithm of Jenkins and Traub (Jenkins and Traub (1972) TOMS Algorithm 419. Comm. ACM, 15, 9799). For example,

> polyroot(c(4,5,6))
[1] -0.4166667+0.7021791i -0.4166667-0.7021791i

Using the package PolynomF , the code is a bit more cumbersome

 > x = polynom() > (q = solve(4 + 5*x + 6*x^2)) # Solve ...

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Publisher Resources

ISBN: 9781439884485