RREF-Linear-Solver

A program implementing the Gauss-Jordan Elimination method to solve systems of linear equations. It reduces the augmented matrix to Reduced Row Echelon Form (RREF), providing a unique solution, infinite solutions, or identifying inconsistency when no solution exists.

Installation

1. Clone the repository:

   git clone https://github.com/BIBO9265/RREF-Linear-Solver.git

2. Go to the project directory:

   cd RREF-Linear-Solver

3. Build the project:

   cargo build --release

Usage

To run the program, simply provide the augmented matrix as input:

cargo run

Example 1: Unique Solution

For the system of equations:

x + y = 3
2x + 3y = 5

The augmented matrix would be represented as:

1 1 3
2 3 5

Input:

1 1 3
2 3 5

Output:

ORIGINAL MATRIX:
[
  [1.0000, 1.0000, 3.0000]
  [2.0000, 3.0000, 5.0000]
]

RREF:
[
  [1.0000, 0.0000, 4.0000]
  [0.0000, 1.0000, -1.0000]
]

RESULT: UNIQUE SOLUTION
  SOLUTION VAR_0 = 4.0000
  SOLUTION VAR_1 = -1.0000

VERIFY SOLUTION ORIGINAL EQUATION:
  EQ 0: LHS=3.0000 vs RHS=3.0000 (DIFF=0.00e0)
  EQ 1: LHS=5.0000 vs RHS=5.0000 (DIFF=0.00e0)
VERIFY FINISH. MAX ABSOLUTE DIFFERENCE: 0.00e0

This will result in a unique solution:

x = 4
y = -1

Verification:

Substitute 𝑥 = 4 and 𝑦 = − 1 into the original equations:
First equation:
4 + ( − 1 ) = 3
This is true,as 3 = 3.

Example 2: Solutions SET (INFINITELY MANY SOLUTION)

For the system of equations:

x + y = 3
2x + 2y = 6

The augmented matrix would be represented as:

1 1 3
2 2 6

Input:

1 1 3
2 2 6

Output:

ORIGINAL MATRIX:
[
  [1.0000, 1.0000, 3.0000]
  [2.0000, 2.0000, 6.0000]
]

RREF:
[
  [1.0000, 1.0000, 3.0000]
  [0.0000, 0.0000, 0.0000]
]

RESULT: SOLUTION SET (INFINITELY MANY SOLUTION)

This system has infinite solutions, as there are dependent equations, and one equation can be derived from the other.

Example 3: INCONSISTENT (NO SOLUTION)

For the system of equations:

x + y = 3
x + y = 5

The augmented matrix would be represented as:

1 1 3
1 1 5

Input:

1 1 3
1 1 5

Output:

ORIGINAL MATRIX:
[
  [1.0000, 1.0000, 3.0000]
  [1.0000, 1.0000, 5.0000]
]

RREF:
[
  [1.0000, 1.0000, 3.0000]
  [0.0000, 0.0000, 2.0000]
]

RESULT: INCONSISTENT (NO SOLUTION)

This system is inconsistent and has no solution because the two equations contradict each other.

Latest Updates

This update focuses on enhancing the solver's stability and precision, optimizing the core algorithm, and improving the clarity of the output.

Key Changes:

• Improved Precision and Numerical Stability: To handle potential minor errors in floating-point computations more reliably, a small tolerance value const EPSILON: f64 = 1e-9; (i.e., $10^{-9}$) has been introduced. This constant is used when comparing floating-point numbers, making checks for zero or equality more robust and leading to more accurate solutions. Self-learning point: Using an epsilon value is a common practice in numerical programming when dealing with floating-point comparisons, as direct equality checks (==) can be unreliable due to how these numbers are stored.

• Optimized Core Reduction Algorithm: The core algorithm for reducing matrices to Reduced Row Echelon Form (RREF), based on Gaussian-Jordan elimination, has been updated and optimized. This improves both the efficiency and accuracy of the reduction process.

• Enhanced Logic for Solution Determination: The solver now more clearly and accurately determines the nature of the linear system's solution based on the RREF result. It can precisely identify:

  • Unique Solutions: When every variable corresponds to a pivot.
  • Infinite Solutions: When free variables exist. The solver will help indicate which variables are free.
  • No Solution: When an inconsistent row (like 0 = 1) appears in the RREF.

• More Systematic and User-Friendly Output: The format for displaying the solver's findings regarding the solution(s) has been made more systematic and easier to interpret. The output now clearly indicates whether a unique solution, infinite solutions, or no solution was found, and presents the results in a more structured way for better readability.