Numerical Methods Using Kotlin: For Data Science, Analysis, and Engineering

Description: 1: Introduction to Numerical Methods in Kotlin.- 2: Linear Algebra.- 3: Finding Roots of Equations.-4: Finding Roots of Systems of Equations.- 5: Curve Fitting and Interpolation.- 6: Numerical Differentiation and Integration.- 7: Ordinary Differential Equations.- 8: Partial Differential Equations.- 9: Unconstrained Optimization.- 10: Constrained Optimization.- 11: Heuristics.- 12: Basic Statistics.- 13: Random Numbers and Simulation.- 14: Linear Regression.- 15: Time Series Analysis.- References. Table of ContentsAbout the Authors...........................................................................................................iPreface............................................................................................................................ii1. Why Kotlin?..............................................................................................................61.1. Kotlin in 2022.....................................................................................................61.2. Kotlin vs. C++....................................................................................................61.3. Kotlin vs. Python................................................................................................61.4. Kotlin in the future .............................................................................................62. Data Structures.......................................................................................................72.1. Function...........................................................................................................72.2. Polynomial ......................................................................................................73. Linear Algebra .......................................................................................................83.1. Vector and Matrix ...........................................................................................83.1.1. Vector Properties .....................................................................................83.1.2. Element-wise Operations.........................................................................83.1.3. Norm ........................................................................................................93.1.4. Inner product and angle ...........................................................................93.2. Matrix............................................................................................................103.3. Determinant, Transpose and Inverse.............................................................103.4. Diagonal Matrices and Diagonal of a Matrix................................................103.5. Eigenvalues and Eigenvectors.......................................................................103.5.1. Householder Tridiagonalization and QR Factorization Methods..........103.5.2. Transformation to Hessenberg Form (Nonsymmetric Matrices)...........104. Finding Roots of Single Variable Equations .......................................................114.1. Bracketing Methods ......................................................................................114.1.1. Bisection Method ...................................................................................114.2. Open Methods...............................................................................................114.2.1. Fixed-Point Method ...............................................................................114.2.2. Newton's Method (Newton-Raphson Method) .....................................114.2.3. Secant Method .......................................................................................114.2.4. Brent's Method ......................................................................................115. Finding Roots of Systems of Equations...............................................................125.1. Linear Systems of Equations.........................................................................125.2. Gauss Elimination Method............................................................................125.3. LU Factorization Methods ............................................................................125.3.1. Cholesky Factorization ..........................................................................125.4. Iterative Solution of Linear Systems.............................................................125.5. System of Nonlinear Equations.....................................................................126. Curve Fitting and Interpolation............................................................................146.1. Least-Squares Regression .............................................................................146.2. Linear Regression..........................................................................................146.3. Polynomial Regression..................................................................................146.4. Polynomial Interpolation...............................................................................146.5. Spline Interpolation .......................................................................................147. Numerical Differentiation and Integration...........................................................157.1. Numerical Differentiation .............................................................................157.2. Finite-Difference Formulas...........................................................................157.3. Newton-Cotes Formulas................................................................................157.3.1. Rectangular Rule....................................................................................157.3.2. Trapezoidal Rule....................................................................................157.3.3. Simpson's Rules.....................................................................................157.3.4. Higher-Order Newton-Coles Formulas..................................................157.4. Romberg Integration .....................................................................................157.4.1. Gaussian Quadrature..............................................................................157.4.2. Improper Integrals..................................................................................158. Numerical Solution of Initial-Value Problems....................................................168.1. One-Step Methods.........................................................................................168.2. Euler's Method..............................................................................................168.3. Runge-Kutta Methods...................................................................................168.4. Systems of Ordinary Differential Equations.................................................169. Numerical Solution of Partial Differential Equations..........................................179.1. Elliptic Partial Differential Equations...........................................................179.1.1. Dirichlet Problem...................................................................................179.2. Parabolic Partial Differential Equations........................................................179.2.1. Finite-Difference Method ......................................................................179.2.2. Crank-Nicolson Method.........................................................................179.3. Hyperbolic Partial Differential Equations.....................................................1710..................................................................................................................................1811..................................................................................................................................1912. Random Numbers and Simulation ....................................................................2012.1. Uniform Distribution .................................................................................2012.2. Normal Distribution...................................................................................2012.3. Exponential Distribution............................................................................2012.4. Poisson Distribution ..................................................................................2012.5. Beta Distribution........................................................................................2012.6. Gamma Distribution ..................................................................................2012.7. Multi-dimension Distribution ....................................................................2013. Unconstrainted Optimization ............................................................................2113.1. Single Variable Optimization ....................................................................2113.2. Multi Variable Optimization .....................................................................2114. Constrained Optimization .................................................................................2214.1. Linear Programming..................................................................................2214.2. Quadratic Programming ............................................................................2214.3. Second Order Conic Programming............................................................2214.4. Sequential Quadratic Programming...........................................................2214.5. Integer Programming.................................................................................2215. Heuristic Optimization......................................................................................2315.1. Genetic Algorithm .....................................................................................2315.2. Simulated Annealing .................................................................................2316. Basic Statistics..................................................................................................2416.1. Mean, Variance and Covariance................................................................2416.2. Moment......................................................................................................2416.3. Rank...........................................................................................................2417. Linear Regression .............................................................................................2517.1. Least-Squares Regression..........................................................................2517.2. General Linear Least Squares....................................................................2518. Time Series Analysis ........................................................................................2618.1. Univariate Time Series..............................................................................2618.2. Multivariate Time Series ...........................................................................2618.3. ARMA .......................................................................................................2618.4. GARCH .....................................................................................................2618.5. Cointegration .............................................................................................2619. Bibliography .....................................................................................................2720. Index .....................................................................................................

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