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Journal Articles SIAM Journal on Matrix Analysis and Applications Year : 2013

EVALUATING MATRIX FUNCTIONS BY RESUMMATIONS ON GRAPHS: THE METHOD OF PATH-SUMS

P.-L Giscard
S. Thwaite
  • Function : Author
D. Jaksch
  • Function : Author

Abstract

We introduce the method of path-sums which is a tool for analytically evaluating a primary function of a finite square discrete matrix, based on the closed-form resummation of infinite families of terms in the corresponding Taylor series. Provided the required inverse transforms are available, our approach yields the exact result in a finite number of steps. We achieve this by combining a mapping between matrix powers and walks on a weighted directed graph with a universal graph-theoretic result on the structure of such walks. We present path-sum expressions for a matrix raised to a complex power, the matrix exponential, matrix inverse, and matrix logarithm. We present examples of the application of the path-sum method.
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Dates and versions

hal-03597518 , version 1 (04-03-2022)

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P.-L Giscard, S. Thwaite, D. Jaksch. EVALUATING MATRIX FUNCTIONS BY RESUMMATIONS ON GRAPHS: THE METHOD OF PATH-SUMS. SIAM Journal on Matrix Analysis and Applications, 2013, 34 (2), pp.445-469. ⟨10.1137/120862880⟩. ⟨hal-03597518⟩
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