LU decomposition: A = LU. L is lower triangular (1s on diagonal), U is upper triangular. Solve Ax=b in two steps: Ly=b (forward), Ux=y (backward). Cost: O(n³/3) once, then O(n²) per solve. Efficient for multiple b vectors.
How to Use
Enter a matrix:
- A: Square matrix
- b: Right-hand side
- Output: L, U, solution
Algorithm
Row reduce A to U, recording multipliers in L. lᵢⱼ = multiplier used to eliminate aᵢⱼ. L has 1s on diagonal. A = LU if no pivoting needed.
Applications
- Solving multiple systems
- Computing determinant: det(A) = ∏uᵢᵢ
- Computing inverse: solve AX=I column by column
Step-by-Step Instructions
- 1Enter matrix A.
- 2View L and U.
- 3Enter b vector.
- 4See forward sub.
- 5Get solution x.