Abstract
In this paper we analyzed recent works on inverting Vandermonde matrix, both classical and generalized, which were unknown during the publication of Moler’s and Van Loan’s paper ‘Nineteen Dubious Ways to Compute the Exponential of a Matrix’. Upon that analysis we proposed the Vandermonde method as the fourth candidate for calculating exponent of generic matrices. On this basis we also proposed the Vandermonde based method to compute the exponential of certain class of special matrices, i.e. the companion matrices.
This work was supported by Statutory Research funds of Department of Applied Informatics, Silesian University of Technology, Gliwice, Poland (02/100/BK_21/0008).
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Notes
- 1.
In mathematical literature in some languages instead is used the term ‘Frobenius’. But in the English Literature the class of Frobenius matrices is more general, encompassing the companion matrices as its special case.
- 2.
Any function we call symmetric, if and only if after any arbitrary permutation of its independent variables we receive the same polynomial ([23] pp. 77–84).
- 3.
The problem is neatly defined in the very title of the classical in the structured matrices field monograph [35].
- 4.
The trait of generality is - surprisingly - not a standard for the algorithms available in the literature. For example the classical in the associated problem of solving the Vandermonde linear systems article [33] only sketches algorithm for a very peculiar version of the confluence, i.e. with allowed multiplicity equal to only of the first eigenvalue, with all the rest single. The same work [33] suggests that the general case cannot be easily treated, stating in the second paragraph of the page 900: “(can be treated easily)… with only the two endpoints of confluency greater than one, or that with all points of the same order of confluency.”. Significantly, all of the four Pascal-like codes in the appendix of [33] copes only with a classical Vandermonde linear systems, with single eigenvalues (pp. 901–902).
Obviously algorithms with such an artificial restrictions are worthless in the view of computing the exponential of a matrix.
- 5.
Higher dimensions `can be multiplied by the \(3 \times 3\) algorithm by recursion.
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Respondek, J. (2021). Another Dubious Way to Compute the Exponential of a Matrix. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2021. ICCSA 2021. Lecture Notes in Computer Science(), vol 12949. Springer, Cham. https://doi.org/10.1007/978-3-030-86653-2_34
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