1. What is LU Decomposition?

LU decomposition factors a matrix as the product of a lower triangular matrix (L) and an upper triangular matrix (U). This method is used to solve systems of linear equations, invert matrices, and compute determinants.

2. How Does the Calculator Work?

The calculator performs the following steps:

Where:

• \( A \) — Coefficient matrix (n×n)
• \( L \) — Lower triangular matrix
• \( U \) — Upper triangular matrix
• \( b \) — Right-hand side vector
• \( x \) — Solution vector

Explanation: The system is solved in two steps using triangular matrices which are easier to solve than the original system.

3. Importance of LU Decomposition

Details: LU decomposition is computationally efficient for solving multiple systems with the same coefficient matrix but different right-hand sides. It's more stable than Gaussian elimination and forms the basis for many numerical algorithms.

4. Using the Calculator

Tips: Enter the square matrix A with rows separated by semicolons and elements separated by commas. Enter vector b with comma-separated values. The matrix must be square and non-singular.

5. Frequently Asked Questions (FAQ)

Q1: When does LU decomposition fail?
A: LU decomposition fails when the matrix is singular or when pivoting is required (though partial pivoting can be implemented).

Q2: How is this different from Gaussian elimination?
A: LU decomposition preserves the original matrix and factors it into L and U, allowing efficient solving of multiple systems with the same A.

Q3: What's the computational complexity?
A: Decomposition is O(n³), while solving triangular systems is O(n²) per system.

Q4: Can this handle non-square matrices?
A: No, the matrix must be square for standard LU decomposition.

Q5: What about numerical stability?
A: The calculator uses partial pivoting for stability, but ill-conditioned matrices may still cause problems.