Abstract
This paper is a review of results on computational methods of linear algebra over commutative domains. Methods for the following problems are examined: solution of systems of linear equations, computation of determinants, computation of adjoint and inverse matrices, computation of the characteristic polynomial of a matrix.
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Akritas, A.: Elements of Computer Algebra with Applications. John Wiley Interscience, New York (1989)
Akritas, A.G., Akritas, E.K., Malaschonok, G.I: Various proofs of Sylvester’s (determinant) identity. Mathematics and Computations in Simulation 42, 585–593 (1996)
Akritas, A.G., Malaschonok, G.I.: Computation of the Adjoint Matrix. In: Alexandrov, V.N., van Albada, G.D., Sloot, P.M.A., Dongarra, J.J. (eds.) ICCS 2006. LNCS, vol. 3992, pp. 486–489. Springer, Heidelberg (2006)
Abdeljaoued, J.: Berkowitz Algorithm, Maple and computing the characteristic polynomial in an arbitrary commutative ring. Computer Algebra MapleTech 4(3), Birkhauser Boston (1997)
Abdeljaoued, J., Malaschonok, G.I.: Efficient Algorithms for Computing the Characteristic Polynomial in a Domain. Journal of Pure and Applied Algebra 156(2-3), 127–145 (2001)
Bareiss, E.H.: Sylvester’s Identity and Multistep Integer-Preserving Gaussian Elimination. Math. Comp. 22(103), 565–578 (1968)
Chistov, A.L.: Fast parallel calculation of the rank of matrices over a field of arbitrary characteristic. In: Budach, L. (ed.) FCT 1985. LNCS, vol. 199, pp. 147–150. Springer, Heidelberg (1985)
Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation 9, 251–280 (1990)
Dodgson, C.L.: Condensation of determinants, being a new and brief method for computing their arithmetic values. Proc. Royal Soc. Lond. A15, 150–155 (1866)
Kaltofen, E.: On Computing Determinants of Matrices Without Divisions. In: ISSAC 1992. Proc. Internat. Symp. Symbolic Algebraic Comput, pp. 342–349. ACM Press, New York (1992)
Kaltofen, E., Villard, G.: On the complexity of computing determinants. In: ASCM 2001. Proc. Fifth Asian Symposium on Computer Mathematics(extended abstract), pp. 13–27 (2001)
Malashonok, G.I.: Solution of a system of linear equations in an integral domain. USSR Journal of computational Mathematics and Mathematical Physics 23(6), 1497–1500 (1983)
Malashonok, G.I.: On the solution of a linear equation system over commutative rung. Math. Notes of the Acad. Sci. USSR 42(4), 543–548 (1987)
Malashonok, G.I.: A new solution method for linear equation systems over commutative rung. In: International Algebraic Conference, Theses on the ring theory, algebras and modules, Novosibirsk, p. 82 (August 21-26, 1989)
Malashonok, G.I.: Algorithms for the solution of systems of linear equations in commutative rings; Effective methods in algebraic geometry (Castiglioncello, 1990), Progr. Math., Birkhauser Boston, Boston, 94, 289–298 (1991)
Malashonok, G.I.: Argorithms for Computing Determinants in commutative rings; Diskretnaya Matematika 7(4), 68–76 (1995) transl. in: Discrete Math. Appl. 5(6), 557–566 (1996)
Malaschonok, G.I.: On the solution of systems of linear equations; Computational Commutative Algebra, COCOA-IV, Abstracts, Genova, 32 (May 29-June 2, 1995)
Malaschonok, G.I.: Recursive Method for the Solution of Systems of Linear Equations; Computational Mathematics. In: Proc. of the 15th IMACS World Congress, I, Berlin, August 1997, Wissenschaft & Technik Verlag, Berlin, pp. 475–480 (1997)
Malaschonok, G.I.: A Computation of the Characteristic Polynomial of an Endomorphism of a Free Module; Zap. Nauchnyh Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov (POMI), 258 (1999). Teor. Predst. Din. Sist. Komb. i Algoritm. Metody 4, 101–114 (1999)
Malaschonok, G.I.: Effective Matrix Methods in Commutative Domains; Formal Power Series and Algebraic Combinatorics, pp. 506–517. Springer, Heidelberg (2000)
Strassen, V.: Gaussian Elimination is not optimal; Numerische Mathematik 13, 354–356 (1969)
Sasaki, T., Murao, H.: Efficient Gaussian elimination method for symbolic determinants and linear systems. A.C.M. Trans. Math. Software 8(4), 277–289 (1968)
Waugh, F.V., Dwyer, P.S.: Compact computation of the inverse of a matrix. Annals of Mathemaical Statistic 16, 259–271 (1945)
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Akritas, A.G., Malaschonok, G.I. (2007). Computations in Modules over Commutative Domains. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2007. Lecture Notes in Computer Science, vol 4770. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75187-8_2
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DOI: https://doi.org/10.1007/978-3-540-75187-8_2
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