What are the different methods used for solving linear systems of equations numerically?

There are several methods used for solving linear systems of equations numerically. Some of the commonly used methods include:

1. Gaussian Elimination: This method involves transforming the system of equations into an equivalent upper triangular system by performing row operations. Once the system is in upper triangular form, back substitution is used to find the solution.

2. LU Decomposition: This method involves decomposing the coefficient matrix of the system into a lower triangular matrix (L) and an upper triangular matrix (U). The system is then solved by solving two simpler systems: Ly = b and Ux = y.

3. Iterative Methods: These methods involve iteratively improving an initial guess to the solution until a desired level of accuracy is achieved. Examples of iterative methods include Jacobi method, Gauss-Seidel method, and Successive Over-Relaxation (SOR) method.

4. Matrix Factorization Methods: These methods involve factorizing the coefficient matrix into a product of two matrices, such as Cholesky factorization, QR factorization, or Singular Value Decomposition (SVD). The factorization is then used to solve the system more efficiently.

5. Direct Methods: These methods provide an exact solution to the system of equations. Examples include Cramer's rule, which involves calculating determinants, and the method of inverse matrices, which involves finding the inverse of the coefficient matrix.

It is important to note that the choice of method depends on the specific characteristics of the linear system, such as its size, sparsity, and condition number.