Numerical Analysis

This repository contains various numerical methods and algorithms implemented in a variety of programming languages, including Julia, MATLAB, Python, and Fortran. The code is organized into different sections according to topics like interpolation, optimization, root-finding algorithms, linear algebra, and more.

Table of Contents

• Interpolation and Extrapolation
• Linear Algebra
• Monte Carlo Simulations
• Navier-Stokes Equations
• Numerical Differentiation and Integration
• Optimization
• Ordinary Differential Equations (ODEs)
• Partial Differential Equations (PDEs)
• Root-Finding Algorithms
• License

Interpolation and Extrapolation

Interpolation and extrapolation are techniques to estimate unknown values between or beyond a set of known data points. The following methods are implemented:

• Cubic Spline Interpolation (CubicSplineInterpolation.jl)
• Hermite Interpolation (Hermitelnterpolation.jl)
• Lagrange Interpolation (Lagrangelnterpolation.f90)
• Linear Spline Interpolation (LinearSplineInterpolation.jl)
• Newton’s Backward Interpolation (NewtonsBackwardInterpolation.m)
• Newton’s Divided Differences (NewtonsDividedDifferencesInterpolation.jl)
• Newton’s Forward Interpolation (NewtonsForwardInterpolation.m)
• Quadratic Spline Interpolation (QuadraticSplineInterpolation.jl)
• General Spline Interpolation in Python (SplineInterpolation.py)

Use Cases: These methods are commonly used in numerical computing for curve fitting and data smoothing.

Linear Algebra

Linear algebra algorithms are fundamental for solving systems of equations and matrix operations.

• Gauss Jordan Method (GaussJordan.m)
• Gauss-Seidel Method (GaussSeidel.m)
• Jacobi Method (Jacobi.m)
• LU Decomposition (Cholesky's Method) (LUCholeskysMethod.m)
• LU Decomposition (Crout's Method) (LUCroutsMethod.m)
• LU Decomposition (Doolittle's Method) (LUDoolittleMethod.m)
• Power Method (PowerMethod.m)

Use Cases: These methods are often used in solving linear systems, matrix decomposition, and eigenvalue problems.

Monte Carlo Simulations

Monte Carlo methods are used to perform numerical integration and simulate random processes.

• 1D Monte Carlo Simulation (MonteCarlo1d.py)
• 2D Monte Carlo Simulation (MonteCarlo2d.py)
• Monte Carlo Markov Chains (MonteCarloMarkovChains.ipynb)

Use Cases: Widely used in numerical integration, stochastic modeling, and simulations in physics and finance.

Navier-Stokes Equations

This section contains implementations for solving the Navier-Stokes equations, primarily in fluid dynamics.

• CFD: Lid Driven Cavity Problem (CFD_Lid_Driven_Cavity.py)

Use Cases: These equations describe the motion of viscous fluid substances and are used in computational fluid dynamics (CFD).

Numerical Differentiation and Integration

Numerical methods for differentiation and integration are useful for approximating the results of derivatives and integrals.

• Gaussian Quadrature (GaussianQuadrature.f90, GaussianQuadrature.m)
• Monte Carlo Integration (1D) (MonteCarlolntegration1d.f90)
• Monte Carlo Integration (2D) (MonteCarlolntegration2d.f90)
• Simpson's Rule (SimpsonsRule.f90)
• Trapezoidal Rule (TrapezoidalRule.f90)

Use Cases: These methods are crucial for numerical integration when exact methods are impractical.

Optimization

Optimization techniques help in finding maxima, minima, or optimal solutions.

• Linear Programming (Big-M Method) (BigM.m)
• Dual Simplex Method (DualSimplex.m)
• Graphical Method (GraphicalMethod.m)
• Simplex Method (Simplex.m)
• Gradient Descent for Linear Problems (GradDescentLin.ipynb)

Use Cases: These methods are widely used in operational research, machine learning, and economics to solve optimization problems.

Ordinary Differential Equations (ODEs)

Methods for numerically solving ordinary differential equations.

• Euler's Method (Fortran) (EulersMethod.f90)
• Euler's Method (Python) (EulersMethod.py)
• Runge-Kutta 2nd Order Method (RungeKutta2ndOrder.f90)
• Runge-Kutta 4th Order Method (RungeKutta4thOrder.f90)

Use Cases: These methods are widely used in physics, engineering, and biological modeling to solve time-dependent processes.

Partial Differential Equations (PDEs)

PDEs describe functions with multiple variables and their partial derivatives.

• Crank-Nicholson Method (CrankNicholsonMethod.m)
• DuFort-Frankel Method (DuFortFrankelMethod.m)
• Backward Time Central Space (BTCS) for Heat Equation (PDE-BTCS-HeatEquation.m)
• Forward Time Central Space (FTCS) for Heat Equation (PDE-FTCS-HeatEquation.m)
• Richardson Method (RichardsonMethod.m)

Use Cases: PDEs are used in modeling heat transfer, fluid dynamics, and electromagnetism.

Root-Finding Algorithms

Algorithms for finding roots of nonlinear equations.

• Bisection Method (Bisection.f90)
• Fixed Point Iteration (FixedPointlteration.m)
• Newton-Raphson Method (NewtonRaphson.f90)
• Regula Falsi Method (Regula-Falsi.m)
• Secant Method (SecantMethod.f90)

Use Cases: Used in numerical analysis to solve equations in engineering and science problems.

License

This project is licensed under the Apache 2.0 License. See the LICENSE file for details.