Geodesic
On the Calculation of Functions of Matrices
Matematičeskie zametki, Tome 94 (2013) no. 2, pp. 175-182.

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The representation of entire functions of matrices via symmetric polynomials of $n$th order is obtained. A method of deriving analytic formulas for functions of matrices of second, third, and fourth orders is obtained. Symmetric polynomials are used to construct algorithms for the numerical calculations of entire functions of matrices, in particular, of matrix exponentials, not requiring the determination of the eigenvalues of the matrices. The efficiency of the proposed numerical methods is estimated.
Keywords: entire function of a matrix, symmetric polynomial of $n$th order, matrix exponential, numerical calculation of functions of matrices. 
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Yu. N. Belyaev. On the Calculation of Functions of Matrices. Matematičeskie zametki, Tome 94 (2013) no. 2, pp. 175-182. http://geodesic.mathdoc.fr/item/MZM_2013_94_2_a1/

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