By Sergey Abrahamyan, Melsik Kyureghyan (auth.), Vladimir P. Gerdt, Wolfram Koepf, Ernst W. Mayr, Evgenii V. Vorozhtsov (eds.)
This e-book constitutes the refereed complaints of the thirteenth foreign Workshop on machine Algebra in clinical Computing, CASC 2011, held in Kassel, Germany, in September 2011. The 26 complete papers integrated within the publication have been conscientiously reviewed and chosen from various submissions. The articles are prepared in topical sections at the improvement of item orientated laptop algebra software program for the modeling of algebraic constructions as typed items; matrix algorithms; the research through computing device algebra; the improvement of symbolic-numerical algorithms; and the applying of symbolic computations in utilized difficulties of physics, mechanics, social technological know-how, and engineering.
Read or Download Computer Algebra in Scientific Computing: 13th International Workshop, CASC 2011, Kassel, Germany, September 5-9, 2011. Proceedings PDF
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We get from (A), (B), (C) the following. (A ) We use valx det Ar (x) + 1 ﬁrst terms of the entries of the matrix Ar (x) to compute valx det Ar (x) and valx A−1 r (x). (B ) We use no more than valx det Ar (x) + γ + 1 ﬁrst terms of the entries of the matrices A0 (x), A1 (x), . . , Ar (x) to compute γ (see (20)). (C ) We use no more than valx det Ar (x) + γ + v ﬁrst terms of the entries of the matrices A0 (x), A1 (x), . . , Ar (x) to compute the ﬁrst v terms of (21). This and Proposition 5 imply the following statement related to systems of the form (1) with invertible Ar (x).
Sp (x) “in parallel”: we generate algorithmically the sequence Higher-Order Linear Diﬀerential Systems with Truncated Coeﬃcients 21 [x0 ]s1 (x), . . , [x0 ]sp (x), [x1 ]s1 (x), . . , [x1 ]sp (x), . . until we ﬁnd i such that [xi ]sj (x) = 0 for some 1 ≤ j ≤ p. Then ν = i. (B) Let it be known in advance that among given matrices M1 (x), M2 (x), . . , Mp (x) ∈ Mat(k[[x]]), p ≥ 1, there is at least one non-zero. Then we can compute mini valx Mi (x). To do this we consider the entries of all the matrices “in parallel” (as in (A)).
We have structured the paper in four sections. The main result in the paper is provided in Section 2, together with the necessary hypotheses and some examples of application. The proof of this result is provided in Section 3. The paper ends with a brief section on open questions. 2 Statement of the Main Result In order to state the main result of the paper, we need to introduce some notation and hypotheses, ﬁrst. So, let F ∈ R[x, y, z, t] be a polynomial in the variables x, y, z, t. Moreover, we assume that the following hypotheses hold: (i) F = F (x, y, z, t) is square-free, and it explicitly depends on the variable z (otherwise the problem is reducible to that in ).