Regularized Cholesky decomposition method for finite bit width computing

This research focuses on the Cholesky decomposition of symmetric positive definite matrices. While the Cholesky decomposition is known for its computational efficiency and numerical robustness, it may encounter decomposition failures when applied to ill-conditioned matrices with large condition numbers. To address these computational challenges, this paper proposes an improved probabilistic rounding error analysis method. This method can more accurately estimate the rounding errors and thereby guide the selection of the optimal diagonal loading value. The main contribution of this research is the determination of a diagonal loading value applicable to all positive definite matrices, ensuring the successful completion of Cholesky decomposition. In addition, taking into account the binary representation of numbers in computers, the diagonal loading value is converted to exponential form, allowing multiplication to be replaced by the floating-point bitwise operations. This approach is both practical and efficient, effectively solving the challenges posed by ill-conditioned matrices and limited computational precision.