{"knowledge_schema": {"broad_category": "Mathematics \u2192 Numerical Analysis \u2192 Finite Difference Methods", "refinement": "This problem pertains to the numerical solution of multi-dimensional heat equations using advanced techniques such as higher order finite difference methods and parallel computing.", "specific_scope": "The focus is on the transformation of a sequential algorithm into a parallel one through the approximation of the matrix exponential using fractional methods.", "goal": "Identify the crucial step that enables the transition from a sequential to a parallel algorithm in the context of approximating the matrix exponential."}, "summary": "In the context of solving multi-dimensional heat equations with higher order finite difference methods and parallel splitting, the transformation of a sequential algorithm into a parallel one hinges on the use of linear partial fraction of fractional approximation. This step is essential for effectively approximating the matrix exponential, which is a key component in the numerical solution process."}