MATLAB Code Implementation of Cholesky Decomposition

For general linear equation systems, traditional Gaussian elimination methods may result in high computational complexity. Cholesky decomposition offers a simplified computational approach by factorizing symmetric positive-definite matrices into the product of a lower triangular matrix and its transpose. This decomposition accelerates linear system solving by converting matrix multiplication operations into transpose and multiplication operations of triangular matrices. In MATLAB implementation, the built-in chol() function performs Cholesky factorization, returning either an upper or lower triangular factor depending on the specified output format. The algorithm typically involves computing matrix elements using recursive formulas that ensure numerical stability while maintaining the symmetry property. For large-scale linear systems, Cholesky decomposition proves particularly valuable as it reduces computational complexity from O(n³) to approximately O(n³/3) while preserving memory efficiency through triangular storage. The implementation follows these key steps: first verifying matrix symmetry and positive-definiteness, then iteratively calculating diagonal elements using sqrt(a_ii - sum(L_ik²)) and off-diagonal elements via (a_ij - sum(L_ik*L_jk))/L_jj. This approach ensures that the original matrix A satisfies A = L*L' where L is the lower triangular Cholesky factor.