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A Divide-and-Conquer Algorithm for Functions of Triangular Matrices

We propose a divide-and-conquer algorithm for computing arbitrary functions of upper triangular matrices, which requires approximately the same number of arithmetic operations as Parlett’s algorithm. However, the new algorithm has better performance on computers with two levels of memory due to its block structure and thus, less memory-cache traffic requirements. Like Parlett’s algorithm, the new algorithm also requires that the eigenvalues (main diagonal elements) of the input matrix be distinct, and computes the matrix function nearly as accurately.