Mathematics – Rings and Algebras
Scientific paper
2011-10-25
Mathematics
Rings and Algebras
25 pages, to appear in Journal of Symbolic Computation
Scientific paper
This paper is a sequel to "Computing diagonal form and Jacobson normal form of a matrix using Groebner bases", J. of Symb. Computation, 46 (5), 2011. We present a new fraction-free algorithm for the computation of a diagonal form of a matrix over a certain non-commutative Euclidean domain over a computable field with the help of Gr\"obner bases. This algorithm is formulated in a general constructive framework of non-commutative Ore localizations of $G$-algebras (OLGAs). We split the computation of a normal form of a matrix into the diagonalization and the normalization processes. Both of them can be made fraction-free. For a matrix $M$ over an OLGA we provide a diagonalization algorithm to compute $U,V$ and $D$ with fraction-free entries such that $UMV=D$ holds and $D$ is diagonal. The fraction-free approach gives us more information on the system of linear functional equations and its solutions, than the classical setup of an operator algebra with rational functions coefficients. In particular, one can handle distributional solutions together with, say, meromorphic ones. We investigate Ore localizations of common operator algebras over $K[x]$ and use them in the unimodularity analysis of transformation matrices $U,V$. In turn, this allows to lift the isomorphism of modules over an OLGA Euclidean domain to a polynomial subring of it. We discuss the relation of this lifting with the solutions of the original system of equations. Moreover, we prove some new results concerning normal forms of matrices over non-simple domains. Our implementation in the computer algebra system {\sc Singular:Plural} follows the fraction-free strategy and shows impressive performance, compared with methods which directly use fractions. Since we experience moderate swell of coefficients and obtain simple transformation matrices, the method we propose is well suited for solving nontrivial practical problems.
Levandovskyy Viktor
Schindelar Kristina
No associations
LandOfFree
Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gr{ö}bner bases does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gr{ö}bner bases, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gr{ö}bner bases will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-94409