Basic Mathematics for Pharmacometrics

Outline

[Introduction]
*Machine Epsilon, Float-Point Arithmetics, Factorial Function, Square-Root Function
*Binary and Bisectional Search
*Smoothing and Spline Functions
 
[Differentiation]
Definitions of Derivative, Partial Derivative, Gradient, Jacobian, Hessian
Techniques: Chain Rule, Product Rule, Quotient Rule
Numerical Differentiation: Forward, Centered, Richardson
*Symbolic (Analytical) Differentiation with respect to eta

[Integration]
Definitions
Technique: Substitution, Integration by Parts, Partial Fraction
Integration of Error(Gaussian) Function
Numerical Integration: Trapezoidal, Simpson, Romberg, Gaussian Quadrature, Chebyshev Polynomial/Rational approximation
Differentiation under the Integral Sign
*Regularity Conditions
 
Taylor Series and Transcendental Functions
Integral Transformation: Laplace, Fourier, Convolution
Special Functions: Gamma, Incomplete Gamma, Beta, Incomplete Beta
Distribution Functions: Normal, Gamma, Chi-Square, Beta, t-, F, Binomial, Poisson
Random Number and Variate Generation

[Linear Algebra]
Basic Operations and Definitions of Matrix
Characteristic Roots of a Matrix
Positive Definite Matrix
Cholesky Decomposition
LDL' Transformation
Determinant of a Diagonal Matrix
Determinant of a Symmetric Matrix
Inverse of a Symmetric Matrix
Systems of Linear Equations
Eigen Values and Vectors of a Symmetric Matrix
*Square Root of a Symmetric Matrix
*Cramer's Rule
Singular Value Decomposition
*Generalized Inverse (Pseudo Inverse) of a Matrix

[Differential Equation]
*Coupled Ordinary Differential Equation System
*Matrix Exponential
*Runge-Kutta Method
*Handling of Stiff Differential Equation

[Root Finding and Minimization]
*Newton-Raphson Method
*Direction Vector = - Hessian x gradient
*Variable Metrics Minimization Algorithm