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This project implements various numerical methods for solving non-linear equations, linear equations, differential equations, and matrix inversion. Each algorithm is encapsulated in a dedicated class, with methods to handle solution approximation, convergence, and error tolerance.
File: bi_section.hpp
The Bi-Section method finds roots of non-linear equations using interval bisection.
- Initial Range Calculation: Determines an initial maximum range ((-|xmax|, |xmax|)) for root existence.
- Calculation Method: Iteratively narrows the range until the difference is within a specified error tolerance or 100 iterations are complete.
File: false_position.hpp
The False Position method refines interval boundaries using linear interpolation.
- Initial Range Calculation: Similar to the Bi-Section method, determining ((-|xmax|, |xmax|)).
- Calculation Method: Uses a linear approximation and iterates until the desired error tolerance or a maximum of 100 iterations is met.
File: secant.hpp
The Secant method uses two initial guesses and does not require the derivative of the function.
- Initial Range Calculation: Starts at an estimated maximum range.
- Calculation Method: Updates approximations iteratively until the desired tolerance is achieved.
File: newton_raphson.hpp
This method uses the function's derivative to achieve fast convergence.
- Derivative Calculation: Calculates (f'(x)) for precise results.
- Calculation Method: Updates using Newton's formula until the error tolerance or 100 iterations is reached.
File: Jacobi_iterative.hpp
A method for solving linear equations, iterating values until convergence.
- Row Swapping for Dominant Diagonal: Ensures the matrix is diagonally dominant for convergence.
- Iteration: Runs up to 100 times or until the convergence threshold is achieved.
File: gauss_seidel_iterative.hpp
An improvement over Jacobi, updating values in-place for faster convergence.
- Row Swapping for Dominant Diagonal: Ensures convergence.
- Iteration: Uses previous step values immediately for each calculation.
File: gauss_jordan_elimination.hpp
Transforms a matrix into upper triangular form for solving linear systems.
- swapp Function: Ensures non-zero diagonal entries for stability.
- gauss_elimination Function: Solves the system using elimination followed by back-substitution.
File: gauss_jordan_elimination.hpp
Extends Gaussian elimination to row-reduced echelon form.
- Row Reduction: Converts the matrix into RREF.
- Solution Check: Identifies if a system has unique, infinite, or no solutions.
File: lu_factorization.hpp
Decomposes a matrix into lower (L) and upper (U) matrices for efficient solving.
- LU Decomposition: Generates L and U matrices.
- Forward and Backward Substitution: Solves the system using L and U.
File: runge_kutta.hpp
The 4th-order Runge-Kutta method for solving differential equations.
- Equation Choices: Offers four different differential equations for demonstration.
- Step Calculation: Calculates new values of (y) at each step using four slopes.
File: matrix_inversion.hpp
Implements matrix inversion using the Gauss-Jordan elimination method.
- Matrix Augmentation: Creates an augmented matrix ([A | I]).
- Gaussian and Gauss-Jordan Elimination: Transforms the matrix to find the inverse.
Compile the application with a compatible C++ compiler. Each method outputs results to the console, allowing for easy verification and debugging.
This project is open-source and available under the MIT License.
This README.md can be saved directly in your GitHub repository for detailed documentation.