Cholesky Decomposition Method
Cholesky Decomposition Method – Solved Problems The Cholesky Decomposition Method is used to solve systems of linear equations when the coefficient matrix is symmetric and positive definite . The matrix is decomposed as A = LL T where L is a lower triangular matrix. The solution is obtained in two steps: Forward substitution: LY = B Backward substitution: L T X = Y Problem 1 Solve using Cholesky decomposition 4x + 2y = 6 2x + 3y = 7 Show Solution Matrix form A = [4 2 2 3] L = [2 0 1 √2] Forward substitution: 2y₁ = 6 y₁ = 3 3 + √2 y₂ = 7 y₂ = 2√2 Backward substitution: √2 y = 2√2 y = 2 2x + 2 = 3 x = 1/2 Solution: x = 1/2 y = 2 Problem 2 Solve 4x + 2y + 2z = 8 2x + 5y + z = 3 2x + y + 3z = 5 Show Solution Matrix A = [4 2 2 2 5 1 2 1 3] Cholesky factor L = [2 0 0 1 2 0 1 0 √2] Forward substitution gives Y = [4 -1/2 1/√2] Backwar...