Skip to main content

Effective Matrix Methods in Commutative Domains

  • Conference paper
Formal Power Series and Algebraic Combinatorics

Abstract

Effective matrix methods for solving standard linear algebra problems in a commutative domains are discussed. Two of them are new. There are a methods for computing adjoined matrices and solving system of linear equations in a commutative domains.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abdeljaoued, J.: Berkowitz Algorithm, Maple and computing the characteristic polynomial in an arbitrary commutative ring. Computer Algebra MapleTech 4, No. 3, Birkhauser Boston (1997)

    Google Scholar 

  2. Abdeljaoued, J., Malaschonok, G. I.: Efficient Algorithms for Computing the Characteristic Polynomial in a Domain. Journal of Pure and Applied Algebra (to appear)

    Google Scholar 

  3. Akritas, A. G., Akritas, E. K., Malaschonok, G. I.: Various proofs of Sylvester’s (determinant) identity. Mathematics and Computations in Simulation 42 (1996) 585–593

    Article  MathSciNet  MATH  Google Scholar 

  4. Akritas, A. G., Akritas, E. K., Malaschonok, G. I.: Matrix computation of subresultant polynomial remainder sequences in integral domain. Reliable Computing 1, No. 4 (1995) 375–381

    Article  MathSciNet  MATH  Google Scholar 

  5. Bareiss, E. H.: Sylvester’s Identity and Multistep Integer-Preserving Gaussian Elimination. Math. Comp. 22 (103) (1968) 565–578

    MathSciNet  MATH  Google Scholar 

  6. Bareiss, E. H.: Computational solutions of matrix problems over an integral domain. J. Inst. Maths Applics 10 (1972) 68–104

    Article  MathSciNet  MATH  Google Scholar 

  7. Chistov, A. L.: Fast parallel calculation of the rank of matrices over a field of arbitrary characteristic. Proc. FCT’85, Springer Lecture Notes in Computer Science 199 (1985) 147–150

    MathSciNet  Google Scholar 

  8. Coffins, G. E. and Encarnacion, M. J.: Efficient rational number reconstruction. Journal of Symbolic Computation. 20 (1995) 287–297

    Article  MathSciNet  Google Scholar 

  9. Coppersmith, D. and Winograd, S.: Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation 9 (1990) 251–280

    Article  MathSciNet  MATH  Google Scholar 

  10. Dixon, J.: Exact solution of linear equations using p-adic expansions. Numer. Math. 40 (1982) 137–141

    Article  MathSciNet  MATH  Google Scholar 

  11. Dodgson, C. L.: Condensation of determinants, being a new and brief method for computing their arithmetic values. Proc. Royal Soc. Lond. A.15 (1866) 150–155

    Google Scholar 

  12. Faddeev, D. K., Faddeeva, V. M.: Computational methods of linear algebra. W.H. Freeman and Co., San Francisco (1963)

    Google Scholar 

  13. Kaltofen, E.: On Computing Determinants of Matrices Without Divisions. In: Wang, P. S. (ed.): Proc. Internat. Symp. Symbolic Algebraic Comput. ISSAC’92. ACM Press (1992) 342–349

    Chapter  Google Scholar 

  14. Malaschonok, G. I.: Solution of a system of linear equations in an integral domain. USSR Journal of computational Mathematics and Mathematical Physics 23 (1983) 1497–1500

    Google Scholar 

  15. Malaschonok, G. I.: System of linear equations over a commutative ring. Academy of Sciences of Ukraine, Lvov (1986) (in Russian)

    Google Scholar 

  16. Malaschonok, G. I.: On the solution of a linear equation system over commutative rung. Math. Notes of the Acad. Sci. USSR 42, No. 4 (1987) 543–548

    MathSciNet  Google Scholar 

  17. Malaschonok, G. I.: Algorithms for the solution of systems of linear equations in commutative rings. In: Mora, T. and Traverso, C. (ed.): Effective Methods in Algebraic Geometry. Progress in Mathematics 94, Birkhauser (1991) 289–298

    Chapter  Google Scholar 

  18. Malaschonok, G. I.: Argorithms for the Computing Determinants in a commutative rings. Diskretnaya Matematika 7, No. 4 (1995) 68–76

    MathSciNet  Google Scholar 

  19. Malaschonok, G. I.: Recursive Method for the Solution of Systems of Linear Equations. In: Sydow, A. (ed.): Computational Mathematics (Proceedings of the 15th IMACS World Congress, Vol. I, Berlin, August 1997). Wissenschaft and Teclmik Verlag, Berlin (1997) 475–480

    Google Scholar 

  20. Mulders, T., Storjohann, A: Diophantine Linear System Solving. In: Proceedings of ISSAC’99: ACM International Symposium on Symbolic and Algebraic Computation, July 1999, Vancouver, Canada

    Google Scholar 

  21. Strassen, V.: Gaussian Elimination is not optimal. Numerische Mathematik 13 (1969) 354–356

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Malaschonok, G.I. (2000). Effective Matrix Methods in Commutative Domains. In: Krob, D., Mikhalev, A.A., Mikhalev, A.V. (eds) Formal Power Series and Algebraic Combinatorics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04166-6_48

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-04166-6_48

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08662-5

  • Online ISBN: 978-3-662-04166-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics