Use of Extended Euclidean Algorithm in Solving a System of Linear Diophantine Equations with Bounded Variables

  • Parthasarathy Ramachandran
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4076)

Abstract

We develop an algorithm to generate the set of all solutions to a system of linear Diophantine equations with lower and upper bounds on the variables. The algorithm is based on the Euclid’s algorithm for computing the GCD of rational numbers. We make use of the ability to parametrise the set of all solutions to a linear Diophantine equation in two variables with a single parameter. The bounds on the variables are translated to bounds on the parameter. This is used progressively by reducing a n variable problem into a two variable problem. Computational experiments indicate that for a given number of variables the running times decreases with the increase in the number of equations in the system.

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References

  1. 1.
    Aardal, K., Hurkens, C.A.J., Lenstra, A.K.: Solving a system of Diophantine equation with lower and upper bounds on the variables. Mathematics of Operations Research 25, 427–442 (2000)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Beck, M., Zacks, S.: Refined upper bounds for the linear Diophantine problem of Frobenius. Advances in Applied Mathematics 32, 454–467 (2004)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bond, J.: Calculating the general solution of a linear Diophantine equation. American Mathematical Monthly 74, 955–957 (1967)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Erdös, P., Graham, R.L.: On a linear Diophantine problem of Frobenius. Acta Arithmetica 21, 399–408 (1972)MATHMathSciNetGoogle Scholar
  5. 5.
    Filgueiras, M., Tomás, A.P.: A fast method for finding the basis of non–negative solutions to a linear Diophantine equation. Journal of Symbolic Computation 19, 507–526 (1995)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Greenberg, H.: Solution to a linear Diophantine equation for nonnegative integers. Journal of Algorithms 9, 343–353 (1988)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kertzner, S.: The linear Diophantine equation. American Mathematical Monthly 88, 200–203 (1981)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Morito, S., Salkin, H.M.: Using the Blankinship algorithm to find the general solution of a linear Diophantine equation. Acta Informatica 13, 379–382 (1980)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Rödseth, Ö.J.: On a linear Diophantine problem of Frobenius. Journal für die reine und angewandte Mathematik 301, 431–440 (1978)Google Scholar
  10. 10.
    Schrijver, A.: Theory of Linear and Integer Programming. John Wiley & Sons, Chichester (1986)MATHGoogle Scholar
  11. 11.
    Selmer, E.S., Beyer, Ö.: On a linear Diophantine problem of Frobenius in three variables. Journal für die reine und angewandte Mathematik 301, 161–170 (1978)MATHMathSciNetGoogle Scholar
  12. 12.
    Vitek, Y.: Bounds for a linear Diophantine problem of Frobenius. Journal of the London Mathematical Society 10, 79–85 (1975)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Parthasarathy Ramachandran
    • 1
  1. 1.Indian Institute of ScienceBangaloreIndia

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