{"knowledge_schema": {"broad_category": "Mathematics \u2192 Numerical Analysis \u2192 Finite Difference Methods", "refinement": "This problem pertains to the numerical solution of higher dimensional heat equations using finite difference methods and parallel computing techniques.", "specific_scope": "The focus is on the approximation of the matrix exponential function in the context of parallel algorithms for solving heat equations, particularly through fractional approximations.", "goal": "Identify the key factor that enables the conversion of a sequential algorithm into a parallel algorithm in the context of approximating the matrix exponential."}, "summary": "This question addresses the numerical analysis of higher dimensional heat equations using finite difference methods and the transition from sequential to parallel algorithms. The key factor in this transition is the use of linear partial fractions in the fractional approximation of the matrix exponential function, which facilitates parallel computation. Understanding this concept is crucial for effectively implementing parallel algorithms in numerical simulations."}