Linear Equations Solver in Java

1. Gauss Elimination

A direct method for solving linear systems by:

• Transforming the matrix into an upper triangular form.
• Performing back substitution to compute the solution.

2. Gauss-Jordan Elimination

An extension of Gauss Elimination that:

• Transforms the matrix into reduced row echelon form (RREF).
• Provides the solution directly without the need for back substitution.

3. LU Decomposition

A method that decomposes a matrix into the product of:

• A lower triangular matrix (L).
• An upper triangular matrix (U).

This facilitates easier computation of linear systems. There are multiple approaches to LU decomposition:

• Doolittle Method: Produces a unit lower triangular matrix (L) and a regular upper triangular matrix (U).
• Crout Method: Produces a lower triangular matrix (L) with non-zero diagonal elements and an upper triangular matrix (U) with unit diagonal elements.
• Cholesky Decomposition: Specialized for symmetric positive-definite matrices, decomposing them into the product of a lower triangular matrix and its transpose.

4. Gauss-Seidel Method

An iterative technique for solving systems of linear equations:

• Particularly useful for large systems where direct methods become computationally expensive.
• Converges efficiently for diagonally dominant matrices.

5. Jacobi Iteration

Another iterative method designed for solving systems of linear equations:

• Works well with matrices that are diagonally dominant.
• Often used as a simple iterative approach in numerical computations.

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