PLSSVM, an LS-SVM implementation, now only uses the Conjugate Gradient algorithm for solving a set of linear equations. However, for an ill-conditioned matrix, it especially gets into trouble, as the converged solution drifts away from the actual solution due to rounding errors. Therefore, this thesis implements a different solver, e.g., the Cholesky Decomposition, which will be implemented in SYCL. We will implement multiple variations of the Cholesky Decomposition algorithm, including a blocked version, and utilize many different features of SYCL. The focus will primarily be on the fastest implementations. In the end, the fastest implementation will be integrated into PLSSVM alongside a Forward and Backward Substitution implementation for solving the set of linear equations. We will conclude with a runtime comparison between the implementations, a comparison of our best Cholesky Decomposition with the Conjugate Gradient using a dataset and a small discussion about numerical errors.