Sylvester’s identity and multistep integer-preserving Gaussian elimination

@article{Bareiss1968SylvestersIA,
  title={Sylvester’s identity and multistep integer-preserving Gaussian elimination},
  author={Erwin H. Bareiss},
  journal={Mathematics of Computation},
  year={1968},
  volume={22},
  pages={565-578}
}
  • E. Bareiss
  • Published 1 September 1968
  • Mathematics
  • Mathematics of Computation
A method is developed which permits integer-preserving elimination in systems of linear equations, AX = B, such that (a) the magnitudes of the coefficients in the transformed matrices are minimized, and (b) the computational efficiency is considerably increased in comparison with the corresponding ordinary (single-step) Gaussian elimination. The algorithms presented can also be used for the efficient evaluation of determinants and their leading minors. Explicit algorithms and flow charts are… 

Figures from this paper

Sylvester’s form of the Resultant and the Matrix-Triangularization Subresultant PRS Method
Sylvester’s form of the resultant is often encountered in the literature but is completely different from the one discussed in this paper; the form described here can be found in Sylvester’s paper of
The Algebraic Solution of Sparse Linear Systems via Minor Expansion
  • M. Griss
  • Computer Science, Mathematics
    TOMS
  • 1976
TLDR
An efficient algorithm for the Cramer's rule solution of large "sparse" linear systems with polynomial coefficients (many coefficients zero), as an alternative to sparse elimination methods is developed.
On Computing the Exact Determinant of Matrices with Polynomial Entries
The problem of computing the determinant of a matrix of polynomials is considered. Four algorithms are comparedexpansion by minors, Gausslan elimination over the integers, a method based on
Computing the Maximum Degree of Minors in Skew Polynomial Matrices
TLDR
This paper presents the first algorithms to compute the maximum degree of the Dieudonne determinant of a submatrix in a matrix $A$ whose entries are skew polynomials over a skew field $F$.
The Exact Solution of Systems of Linear Equations with Polynomial Coefficients
TLDR
An algorithm for computing exactly a general solution to a system of linear equations with coefficients that are polynomials over the integers by applying interpolation and the Chinese Remainder Theorem is presented.
Algorithms for Hermite and Smith normal matrices and linear Diophantine equations
TLDR
A modification of the Hermite algorithm gives an integer-preserving algorithm for solving linear equations with real-valued variables that is valid if the elements of the matrix are in a principal ideal domain.
Three New Methods for Computing Subresultant Polynomial Remainder Sequences (PRS’S)
Given the polynomials f, g ∈ Z[x] of degrees n, m, respectively, with n > m, three new, and easy to understand methods — along with the more efficient variants of the last two of them — are presented
Exact Algorithms for the Matrix-Triangularization Subresultant PRS Method
  • A. Akritas
  • Mathematics, Computer Science
    Computers and Mathematics
  • 1989
TLDR
This paper presents efficient, exact algorithms for the implementation of this new method for the computation of a greatest common divisor (gcd) of two polynomials, along with their polynomial remainder sequence (prs).
...
...

References

SHOWING 1-10 OF 16 REFERENCES
A method of computing exact inverses of matrices with integer coefficients
In t heory, t he problem of computing t he exac t inverse of a matrix A with integer coefficients is completely solved by solving exact ly the simultaneous equations Ax=y, in which both x and 11 are
IV. Condensation of determinants, being a new and brief method for computing their arithmetical values
  • C. Dodgson
  • Mathematics
    Proceedings of the Royal Society of London
  • 1867
If it be proposed to solve a set of n simultaneous linear equations, not being all homogeneous, involving n unknowns, or to test their compatibility when all are homogeneous, by the method of
ROOT-CUBING AND THE GENERAL ROOT-POWERING METHODS FOR FINDING THE ZEROS OF POLYNOMIALS.
A generalized mathematical-analysis technique is reported (1) that generalizes a root-squaring and rootcubing method into a general root-powering method. The introduction of partitioned polynomials
Algorithm 290: linear equations, exact solutions
TLDR
A contribution to the Algorithms Department should be in the form of an algorithm, a certific ati on, or a remark, typewritten double spaced, with particular attention to indentation and completeness of references.
A New Method for the Numerical Evaluation of Determinants
  • R. Macmillan
  • Mathematics
    The Journal of the Royal Aeronautical Society
  • 1956
In the November 1955 issue of the Journal there is a Note, “ A New Method for the Numerical Evaluation of Determinants.” I would point out that this method is not new, but is given in “ The Theory of
Note on the Method of Contractants
(1959). Note on the Method of Contractants. The American Mathematical Monthly: Vol. 66, No. 6, pp. 476-479.
Applied Mathematics Division
  • Integer-Preserving Gaussian Elimination, Program P-158 (3600F)
Condensation of determinants, being a new and brief method for computing their arithmetic values," Proc
  • Roy. Soc. Ser. A, v. 15,
  • 1866
...
...