Implement numerical algorithms in Java using NM Dev, an object-oriented and high-performance programming library for mathematics.Yoüll see how it can help you easily create a solution for your complex engineering problem by quickly putting together classes. Numerical Methods Using Java covers a wide range of topics, including chapters on linear algebra, root finding, curve fitting, differentiation and integration, solving differential equations, random numbers and simulation, a whole suite of unconstrained and constrained optimization algorithms, statistics, regression and time series analysis. The mathematical concepts behind the algorithms are clearly explained, with plenty of code examples and illustrations to help even beginners get started. What You Will Learn
Program in Java using a high-performance numerical library
Learn the mathematics for a wide range of numerical computing algorithms
Convert ideas and equations into code
Put together algorithms and classes to build your own engineering solution
Build solvers for industrial optimization problems
Do data analysis using basic and advanced statistics
Who This Book Is For
Programmers, data scientists, and analysts with prior experience with programming in any language, especially Java.
Table of Contents
About the Authors...........................................................................................................i
Preface............................................................................................................................ii
1. Why Java?..............................................................................................................6
1.1. Java in 2020.....................................................................................................6
1.2. Java vs. C++....................................................................................................6
1.3. Java vs. Python................................................................................................6
1.4. Java in the future .............................................................................................6
2. Data Structures.......................................................................................................7
2.1. Function...........................................................................................................7
2.2. Polynomial ......................................................................................................7
3. Linear Algebra .......................................................................................................8
3.1. Vector and Matrix ...........................................................................................8
3.1.1. Vector Properties .....................................................................................8
3.1.2. Element-wise Operations.........................................................................8
3.1.3. Norm ........................................................................................................9
3.1.4. Inner product and angle ...........................................................................9
3.2. Matrix............................................................................................................10
3.3. Determinant, Transpose and Inverse.............................................................10
3.4. Diagonal Matrices and Diagonal of a Matrix................................................10
3.5. Eigenvalues and Eigenvectors.......................................................................10
3.5.1. Householder Tridiagonalization and QR Factorization Methods..........10
3.5.2. Transformation to Hessenberg Form (Nonsymmetric Matrices)...........10
4. Finding Roots of Single Variable Equations .......................................................11
4.1. Bracketing Methods ......................................................................................11
4.1.1. Bisection Method ...................................................................................11
4.2. Open Methods...............................................................................................11
4.2.1. Fixed-Point Method ...............................................................................11
4.2.2. Newton's Method (Newton-Raphson Method) .....................................11
4.2.3. Secant Method .......................................................................................11
4.2.4. Brent's Method ......................................................................................11
5. Finding Roots of Systems of Equations...............................................................12
5.1. Linear Systems of Equations.........................................................................12
5.2. Gauss Elimination Method............................................................................12
5.3. LU Factorization Methods ............................................................................12
5.3.1. Cholesky Factorization ..........................................................................12
5.4. Iterative Solution of Linear Systems.............................................................12
5.5. System of Nonlinear Equations.....................................................................12
6. Curve Fitting and Interpolation............................................................................14
6.1. Least-Squares Regression .............................................................................14
6.2. Linear Regression..........................................................................................14
6.3. Polynomial Regression..................................................................................14
6.4. Polynomial Interpolation...............................................................................14
6.5. Spline Interpolation .......................................................................................14
7. Numerical Differentiation and Integration...........................................................15
7.1. Numerical Differentiation .............................................................................15
7.2. Finite-Difference Formulas...........................................................................15
7.3. Newton-Cotes Formulas................................................................................15
7.3.1. Rectangular Rule....................................................................................15
7.3.2. Trapezoidal Rule....................................................................................15
7.3.3. Simpson's Rules.....................................................................................15
7.3.4. Higher-Order Newton-Coles Formulas..................................................15
7.4. Romberg Integration .....................................................................................15
7.4.1. Gaussian Quadrature..............................................................................15
7.4.2. Improper Integrals..................................................................................15
8. Numerical Solution of Initial-Value Problems....................................................16
8.1. One-Step Methods.........................................................................................16
8.2. Euler's Method..............................................................................................16
8.3. Runge-Kutta Methods...................................................................................16
8.4. Systems of Ordinary Differential Equations.................................................16
9. Numerical Solution of Partial Differential Equations..........................................17
9.1. Elliptic Partial Differential Equations...........................................................17
9.1.1. Dirichlet Problem...................................................................................17
9.2. Parabolic Partial Differential Equations........................................................17
9.2.1. Finite-Difference Method ......................................................................17
9.2.2. Crank-Nicolson Method.........................................................................17
9.3. Hyperbolic Partial Differential Equations.....................................................17
10..................................................................................................................................18
11..................................................................................................................................19
12. Random Numbers and Simulation ....................................................................20
12.1. Uniform Distribution .................................................................................20
12.2. Normal Distribution...................................................................................20
12.3. Exponential Distribution............................................................................20
12.4. Poisson Distribution ..................................................................................20
12.5. Beta Distribution........................................................................................20
12.6. Gamma Distribution ..................................................................................20
12.7. Multi-dimension Distribution ....................................................................20
13. Unconstrainted Optimization ............................................................................21
13.1. Single Variable Optimization ....................................................................21
13.2. Multi Variable Optimization .....................................................................21
14. Constrained Optimization .................................................................................22
14.1. Linear Programming..................................................................................22
14.2. Quadratic Programming ............................................................................22
14.3. Second Order Conic Programming............................................................22
14.4. Sequential Quadratic Programming...........................................................22
14.5. Integer Programming.................................................................................22
15. Heuristic Optimization......................................................................................23
15.1. Genetic Algorithm .....................................................................................23
15.2. Simulated Annealing .................................................................................23
16. Basic Statistics..................................................................................................24
16.1. Mean, Variance and Covariance................................................................24
16.2. Moment......................................................................................................24
16.3. Rank...........................................................................................................24
17. Linear Regression .............................................................................................25
17.1. Least-Squares Regression..........................................................................25
17.2. General Linear Least Squares....................................................................25
18. Time Series Analysis ........................................................................................26
18.1. Univariate Time Series..............................................................................26
18.2. Multivariate Time Series ...........................................................................26
18.3. ARMA .......................................................................................................26
18.4. GARCH .....................................................................................................26
18.5. Cointegration .............................................................................................26
19. Bibliography .....................................................................................................27
20. Index .....................................................................................................
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