Numerical Methods C++ Programs

This repository contains C++ implementations of various numerical methods commonly used in engineering, mathematics, and scientific computing. Each program is self-contained and demonstrates a fundamental numerical technique.

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Table of Contents

1. Bisection Method
2. Fixed Point Iteration
3. Newton–Raphson Method
4. Gauss Elimination
5. Gauss–Jordan Elimination
6. Jacobi Method
7. LU Decomposition
8. Matrix Inversion
9. Power Method
10. Monte Carlo Integration
11. Runge–Kutta 4th Order
12. Relaxation (SOR) Method
13. Simpson 1/3 Rule
14. Matrix Multiplication

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1. Bisection Method

Purpose: Find a root of an equation f(x) = 0 using interval halving.
File: bisection_method.cpp

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2. Fixed Point Iteration

Purpose: Solve x = g(x) iteratively.
File: fixed_point_iteration.cpp

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3. Newton–Raphson Method

Purpose: Solve f(x) = 0 using derivative-based iteration.
File: newton-rapson.cpp

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4. Gauss Elimination

Purpose: Solve a system of linear equations Ax = b using forward elimination and back substitution.
File: gauss_elim.cpp

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5. Gauss–Jordan Elimination

Purpose: Solve Ax = b using Gauss–Jordan elimination (Reduced Row Echelon Form).
File: gauss_jordon.cpp

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6. Jacobi Method

Purpose: Iteratively solve Ax = b using Jacobi iteration.
File: jacobi_nethod.cpp

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7. LU Decomposition

Purpose: Solve Ax = b by decomposing A = L * U.
File: LU.cpp

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8. Matrix Inversion

Purpose: Find the inverse of a square matrix A using Gauss–Jordan elimination.
File: matrix_inversion.cpp

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9. Power Method

Purpose: Find the dominant eigenvalue and eigenvector of a matrix.
File: power_method.cpp

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10. Monte Carlo Integration

Purpose: Approximate definite integral ∫ f(x) dx using random sampling.
File: monte_carglo.cpp

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11. Runge–Kutta 4th Order (RK4)

Purpose: Solve first-order ODE dy/dx = f(x, y) with initial condition y(x0) = y0.
File: range_kutta.cpp

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12. Relaxation (SOR) Method

Purpose: Solve Ax = b iteratively with relaxation factor ω to accelerate convergence.
File: relaxtion_method.cpp

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13. Simpson 1/3 Rule

Purpose: Numerical integration using Simpson’s 1/3 formula.
File: simpson.cpp

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14. Matrix Multiplication

Purpose: Multiply two matrices C = A * B.
File: matrix_multiplication.cpp

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Usage

Compile any C++ file

g++ filename.cpp -o program

Run the executable

./program

Follow on-screen prompts to input the data

Notes

• Some methods (like Simpson 1/3, Bisection, Monte Carlo) use example functions in the code which can be modified.
• Jacobi and Relaxation methods require diagonally dominant matrices for guaranteed convergence.
• Always ensure matrix dimensions are compatible for multiplication or inversion.
• The Table of Contents links directly to the corresponding .cpp files in this repository.