Linear Equation Solver

This Python program solves a system of linear equations using the Gaussian elimination method. It provides various functions to perform different operations on the matrix representing the system of equations.

Installation

To run the Linear Equation Solver program, make sure you have Python installed on your system. No additional libraries or dependencies are required.

1. Clone the repository:

   git clone https://github.com/PouriaAzadehR/LA-Linear-Equation-Solver.git

2. Navigate to the project directory:

   cd LA-Linear-Equation-Solver

Usage

1. Open the main.py file in a text editor.

2. Modify the getting_input function to provide the input for the system of linear equations. Enter the number of equations (num_of_equations) and the coefficients of each equation in the matrix_of_equation list.

3. Save the file.

4. Run the program:

   python main.py

5. The program will display the solution for the variables in the system of linear equations.

Functions

The Linear Equation Solver program consists of the following functions:

getting_input

This function takes input from the user for the system of linear equations. It prompts the user to enter the number of equations and the coefficients of each equation. The input is stored in the matrix variable as a NumPy matrix.

smallest_pivot_position

This function finds the smallest pivot position (row and column) for the Gaussian elimination. It iterates through the rows of the matrix starting from a given row and returns the position of the smallest pivot.

greatest_pivot_position

This function finds the greatest pivot position (row and column) for the backward phase of Gaussian elimination. It iterates through the rows of the matrix starting from a given row and returns the position of the greatest pivot.

swap_rows

This function swaps two rows in the matrix. It takes the original row and the row to be swapped as input and performs the swap operation using a temporary variable.

rows_operations

This function performs row operations on the matrix. It takes two rows and an operand as input and performs the operation row2 = row2 + row1 * operand.

row_reduce_1

This function reduces a row by dividing all its elements by a given operand. It performs the operation row = row / operand to normalize the row.

pivot_finder

This function finds all the pivot positions in the matrix. It iterates through the rows and checks for the first non-zero element in each row. It returns a list of all pivot positions.

equation_solver

This function solves the system of linear equations using the pivot positions. It calculates the solution for each variable by subtracting the product of the coefficients and the corresponding values from the constant term. It also handles the variables that are not included in the pivot positions.

find_coefficient

This function calculates the coefficient used for row operations during the Gaussian elimination. It checks if the pivot element is zero and returns None if it is. Otherwise, it calculates and returns the coefficient.

is_consistent

This function checks if the matrix is consistent, i.e., if the system of linear equations has a solution. It checks if the last row of the matrix has a non-zero element in the last column.

forward_phase

This function performs the forward phase of Gaussian elimination. It iterates through the rows of the matrix, finds the smallest pivot position, and performs row operations to eliminate the coefficients below the pivot.

backward_phase

This function performs the backward phase of Gaussian elimination. It iterates through the rows of the matrix in

reverse order, finds the greatest pivot position, and performs row operations to eliminate the coefficients above the pivot.

reduced_echelon_form

This function converts the matrix to reduced row echelon form. It swaps rows to bring the smallest pivot elements to the upper-left corner and performs row reduction to make all other elements in the same column zero.

solve

This function solves the system of linear equations. It calls the necessary functions in the required order to perform Gaussian elimination and obtain the solution for the variables. It also displays whether the matrix is consistent or not.

Example

Let's consider the following system of linear equations:

2x + 3y - z = 1
x - y + 2z = 3
3x + 2y - 4z = 4

To solve this system, modify the getting_input function as follows:

def getting_input(self):
    self.num_of_equations = 3
    matrix_of_equation = [
        [2, 3, -1, 1],
        [1, -1, 2, 3],
        [3, 2, -4, 4]
    ]
    self.matrix = np.matrix(matrix_of_equation, dtype=np.float64)

Save the file and run the program. The solution will be displayed as:

x[1]: 1.0
x[2]: 2.0
x[3]: -1.0

This indicates that x = 1.0, y = 2.0, and z = -1.0 are the solutions to the system of linear equations.

Contributing

Contributions are welcome! If you have any suggestions or improvements, please create a pull request or open an issue.

Contact

For any questions or inquiries, please contact [email protected].