Scientific Computation using R
Preface
For whom?
Structure of the book
Software information and conventions
Acknowledgments
About the Author
1
Float Point Arithmetic
1.1
Terms
1.2
IEEE 754
1.3
Equality Test
1.4
Machine Epsilon
1.5
Round Function
1.6
Truncation Error and Roudning (Round-off) Error
1.7
References
2
Factorial and Gamma Function
2.1
Factorial Function
2.2
Gamma Function
2.3
Combination Function
2.4
References
3
Differentiation
3.1
Definition
3.2
Basic Characteristics from the definition
3.3
Chain rule
3.4
Partial derivative
3.5
Gradient vector
3.6
Hessian matrix
3.7
Jacobian matrix
3.8
derive() in R
3.9
Simple numerical differentiation - Euler method
3.10
Richardson extrapolation
4
Integration
4.1
Basic Rules
4.2
Integration by Substitution (치환적분)
4.3
Integration by Parts (부분적분)
4.4
Heaviside Cover-up Method
4.5
Integration using Partial Fraction
4.6
Differentiation under the Symbol of Integration (Leibniz Rule)
4.7
Jacobian Matrix
4.8
Trapezoidal rule
4.9
Simpson’s formula
4.10
Romberg Integration
4.11
Gaussian Quadrature
4.12
Polynomial approximation
4.13
Conclusion
4.14
References
5
Integral Transformation
5.1
Introduction
5.2
Laplace Transformation
5.3
Fourier Transformation
5.4
Convolution
5.5
Deconvolution
5.6
References
6
Special Functions
6.1
Taylor Theorem
6.2
Transcendental Number (초월수)
6.3
Elementary Transcendental Functions
6.4
SQRT function
6.5
LOG function
6.6
EXP function
6.7
Error Function
6.8
Gamma and incomplete gamma function
6.9
Beta and incomplete beta function
6.10
Intel CPU Level Float Functions (Instructions)
6.11
References
7
Distribution Functions
7.1
Basic Usage of R Distribution Functions
7.2
Normal Distribution
7.3
Log-Normal Distribution
7.4
Gamma Distribution
7.5
Chi-Square Distribution
7.6
Beta Distribution
7.7
Student t distribution
7.8
F distribution
7.9
Binomial distribution
7.10
Poisson distribution
8
Random Variate Generation
8.1
Methods of Generation
8.2
Random Uniform Distribution Deviate
8.3
Random Normal Distribution Deviate
8.4
Random Multivariate Normal Distribution Deviate
8.5
Random Number from Exponential Distribution
8.6
Random Number from Gamma Distribution
8.7
Random Number from Beta Distribution
8.8
References
9
Linear Algebra
9.1
Exercise
9.2
행렬식(determinant)의 성질
9.3
Cramer 공식
9.4
Cholesky Decomposition
9.5
LDL’ Transformation
9.6
Determinant of a Real Symmetric Matrix
9.7
Inverse of a Real Symmetric Matrix
9.8
Linear Solution of a Real Symmetric Matrix
9.9
Spectral Decomposition
9.10
References
10
Differential Equation
10.1
Terminology
10.2
Simple Solution using Integrating Factor
10.3
Analytical Solution for Linear Coupled Differential Equation
10.4
Analytical Example: Pharmacokinetics
10.4.1
Data Preparation
10.4.2
Solution using R Package ‘wnl’
10.5
Newton’s Planetary Motion
10.6
Euler Method
10.7
Runge-Kutta 4th Order with Fixed Step Size
10.8
Oral 2-Comparment Model with Michaelis-Menten Elimination
10.9
Using LSODA algorithm
10.10
References
11
Root Finding
11.1
Newton-Raphson Method
11.2
An example
12
Minimization
12.1
Terminology
12.2
Standard Test Functions for Optimization Algorithms
12.3
Rudimentary Newton Method
12.4
Revised Newton Method
12.5
Implementing BFGS: a Variable Metric Method
12.6
Test and Comparison between Routines
12.7
References
13
Basic Statistics
13.1
Classification
13.2
Basic Notations
13.3
Some Concepts
13.4
Probability Density Function (pdf)
13.5
Maximum Likelihood Estimation (MLE)
13.6
Desirable Characteristics of Estimator
13.7
Multiple Linear Regression
13.8
Run Test
13.9
References
Appendix
A
Software Tools
A.1
R and R packages
B
Software Usage
References
Kyun-Seop Bae
Scientific Computation Using R
Chapter 3
Differentiation