• Corpus ID: 17681032

Matrix factoring by fraction-free reduction

@article{Middeke2016MatrixFB,
  title={Matrix factoring by fraction-free reduction},
  author={Johannes Middeke and David J. Jeffrey},
  journal={ArXiv},
  year={2016},
  volume={abs/1603.03565}
}
We consider exact matrix decomposition by Gauss-Bareiss reduction. We investigate two aspects of the process: common row and column factors and the influence of pivoting strategies. We identify two types of common factors: systematic and statistical. Systematic factors depend on the process, while statistical factors depend on the specific data. We show that existing fraction-free QR (Gram-Schmidt) algorithms create a common factor in the last column of Q. We relate the existence of row factors… 

Figures and Tables from this paper

References

SHOWING 1-10 OF 15 REFERENCES
Fraction-free matrix factors: new forms for LU and QR factors
TLDR
The new output format for fraction free LU factoring and for QR factoring is given, which contains smaller entries than previously suggested forms, and it avoids the gcd computations required by some other partially fraction free computations.
LU factoring of non-invertible matrices
TLDR
Two new extensions to full-rank, fraction-free factoring of a matrix are proposed here: the first combines LU factoring with full-Rank factoring, and the second extension combines full- rank factored with fraction- free methods.
Fraction-free algorithms for linear and polynomial equations
TLDR
Algorithms are presented for fraction-free LU "factorization" of a matrix and for fractions-free algorithms for both forward and back substitution for solving systems of polynomial equations.
Computing the Rank Profile Matrix
TLDR
It is shown here that, from some PLUQ decompositions, it is possible to recover the row and column echelon forms of a matrix and of any of its leading sub-matrices thanks to an elementary post-processing algorithm.
Generic Gram-Schmidt orthogonalization by exact division
TLDR
This paper develops and shows how to express generic algorithms in C+so that all three possibilities are available using a single source code, and takes advantage of the genericness to test and time the algorithm using different arithmetics, including three huge-integer arithmetic packages.
Sylvester’s identity and multistep integer-preserving Gaussian elimination
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,
A fraction free Matrix Berlekamp/Massey algorithm
Fraction Free Gaussian Elimination for Sparse Matrices
A variant of the fraction free form of Gaussian elimination is presented. This algorithm reduces the amount of arithmetic involved when the matrix has many zero entries. The advantage can be great
Computing the invariant structure of integer matrices: fast algorithms into practice
TLDR
A new heuristic algorithm for computing the determinant of a nonsingular n x n integer matrix is presented and the algorithm is extended to compute the Hermite form of the input matrix.
Algorithms for computer algebra
TLDR
This book discusses Polynomials GCD Computation, the construction of bases for Polynomial Ideals, and the Risch Integration Algorithm, which automates the process of solving Systems of Equation.
...
...