By Simons J., Sullivan D.
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Computing super-irreducible forms of systems of linear differential equations via moser-reduction: a new approach. In Proceedings of ISSAC ’07, pages 1–8, Waterloo, Canada, 2007. ACM Press.  M. A. Barkatou and E. Pfl¨ ugel. On the Moser– and super–reduction algorithms of systems of linear differential equations and their complexity. Submitted to JSC, 2007.  M. Bronstein and M. Petkovsek. An Introduction to Pseudo–Linear Algebra. Theoretical Computer Science, 157(1):3–33, 1996.  V. Dietrich.
The principle of the algorithm is given in Fig. 3, where evaluation and interpolation are performed by truncated-series products. In the case of n, n ∂ , the cor- (8) Likewise, the product of operators in K[X, X −1 ] θ reduces to some evaluation and interpolation tasks (in order to convert between operators and matrices) and to the main matrix-multiplication task (8), which is an instance of vC + dC + rC + 1, vA + dA + rC + 1, rC + 1 . The algorithm for multiplication in K[X] ∂ based on multiplication in K[X, X −1 ] θ is described in Fig.
Let now B1 and B2 be of bidegree (n, n) in K[X] ∂ . −1 Their conversion ` ´ in K[X, X ] θ can be performed using O n M(n) log n ops. and produces two operators C1 and C2 in K[X] θ , of bidegrees at most (2n, n) in (X, θ) such that B1 = X −n C1 (X, θ) and B2 = X −n C2 (X, θ). Using the commutation rule C2 (X, θ)X −n = X −n C2 (X, θ − n), we deduce the equalityPB2 B1 = X −2n C2 (X, θ − n)C1 (X, θ). Writing j C2 (X, θ) = n j=0 X cj (θ) shows that computing the coefficients of C2 (X, θ − n) amounts to n + 1 polynomial shifts in K[θ] in degree ` at most´ n.