Gaussian elimination method

Gaussian elimination is a numerical method used in computer science and mathematics to solve systems of linear equations. It is particularly useful when dealing with large systems of equations, such as those encountered in computer graphics, scientific simulations, and optimization problems.

The Gaussian elimination method follows a systematic procedure to transform a system of linear equations into row-echelon form or reduced row-echelon form, which makes it easier to find the solution.

Here's a step-by-step overview of the Gaussian elimination method:

1. Set up the augmented matrix: Write down the coefficients of the variables and the constant terms of the equations in a matrix form. Each row represents an equation, and the last column represents the constant terms.

2. Row operations: Perform various row operations to transform the matrix. The three primary row operations are:

   • Swapping two rows
   • Multiplying a row by a non-zero scalar
   • Adding a multiple of one row to another row

   The goal is to introduce zeros below the main diagonal (leading to row-echelon form) and reduce the coefficients to 1s (leading to reduced row-echelon form).

3. Forward elimination: Start with the first column and eliminate the coefficients below the main diagonal. To eliminate a non-zero entry in row i below the main diagonal, multiply row i by an appropriate factor and subtract it from row j (where j > i) to make the coefficient zero.

4. Back substitution: Start from the last row and substitute the solved variables back into the preceding equations to find the values of the remaining variables. This process continues until all variables are determined.

5. Interpret the solution: Once you have the reduced row-echelon form, you can interpret the solution. If the system is consistent (i.e., it has at least one solution), the variables corresponding to the pivot columns will have specific values, and the remaining variables are typically assigned free parameters.

Gaussian elimination can be implemented algorithmically using programming languages like Python, C++, or MATLAB. It is important to note that while Gaussian elimination is a powerful method, it can be computationally expensive for large systems due to its time complexity of O(n^3), where n is the number of equations. In such cases, more efficient methods like LU decomposition or iterative methods like Gauss-Seidel or Jacobi may be used.