This Python program solves a system of linear equations using LU decomposition. It takes the coefficients of the equations and their corresponding answers as input from the user, constructs the matrices L (lower triangular) and U (upper triangular), and then calculates the solutions for the variables (x1, x2,..., xn).
The program implements LU decomposition, a method for factoring a matrix A into the product of a lower triangular matrix L and an upper triangular matrix U. This decomposition simplifies the process of solving the linear equation system Ax = b. The program first obtains the matrix A (coefficients) and the vector b (answers) from user input. It then calculates the L and U matrices. After the decomposition, it solves the system in two steps:
- Solve Ly = b for y: This step involves forward substitution.
- Solve Ux = y for x: This step involves backward substitution.
The resulting vector x contains the solutions to the original system of linear equations.
This program is written in Python and requires a Python interpreter. It should work with Python 3.x. To run the program:
- Save: Save the code as a
.pyfile (e.g.,lu_decomposition.py). - Run: Open a terminal or command prompt, navigate to the directory where you saved the file, and run the program using the command:
python lu_decomposition.py
When you run the program, it will prompt you to enter the following information:
- Number of equations: The number of linear equations in your system.
- Answers: The constant value (answer) for each equation.
- Coefficients: The coefficients of each variable (x1, x2,..., xn) in each equation.
The program will then calculate the solutions and print the values of x1, x2,..., xn.