Fraction-free matrix factors: new forms for LU and QR factors

@article{Zhou2008FractionfreeMF,
  title={Fraction-free matrix factors: new forms for LU and QR factors},
  author={Wenqin Zhou and David J. Jeffrey},
  journal={Frontiers of Computer Science in China},
  year={2008},
  volume={2},
  pages={67-80}
}
  • Wenqin Zhou, D. Jeffrey
  • Published 5 March 2008
  • Computer Science, Mathematics
  • Frontiers of Computer Science in China
Gaussian elimination and LU factoring have been greatly studied from the algorithmic point of view, but much less from the point view of the best output format. In this paper, we give new output formats for fraction free LU factoring and for QR factoring. The formats and the algorithms used to obtain them are valid for any matrix system in which the entries are taken from an integral domain, not just for integer matrix systems. After discussing the new output format of LU factoring, the… 
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16 Citations
Highly Influential Citations
3
Background Citations
6
Methods Citations
5

16 Citations

Common Factors in Fraction-Free Matrix Reduction
TLDR
LU factoring of matrices is considered in the context of exact and symbolic computation, as opposed to floating-point computation, and experimental evidence for the existence of common factors is described and analyzed.
Matrix factoring by fraction-free reduction
TLDR
It is shown that existing fraction-free QR (Gram-Schmidt) algorithms create a common factor in the last column of Q, which relates the existence of row factors in LU decomposition to factors appearing in the Smith normal form of the matrix.
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.
Roundoff-Error-Free Algorithms for Solving Linear Systems via Cholesky and LU Factorizations
TLDR
This work introduces two roundoff-error-free factorizations that require storing the same number of individual elements and performing a similar number of operations as the traditional LU and Cholesky factorizations, thereby providing a complete tool set for solving systems of linear systems.
Foundational Factorization Algorithms for the Efficient Roundoff-Error-Free Solution of Optimization Problems
TLDR
This work introduces two roundoff-error-free factorizations (REF) constructed exclusively in integer arithmetic: the REF LU and Cholesky factorizations and develops supplementary integer-preserving substitution algorithms, thereby providing a complete tool set for solving systems of linear equations exactly and efficiently.
Generalized fraction-free LU factorization for singular systems with kernel extraction
Common Factors in Fraction-Free Matrix Decompositions
TLDR
It is shown that fraction-free Gauß–Bareiss reduction leads to triangular matrices having a non-trivial number of common row factors in theLUandQRmatrix decompositions using exact computations.
Exact certification of global optimality of approximate factorizations via rationalizing sums-of-squares with floating point scalars
TLDR
This work generalizes the technique by Peyrl and Parillo to computing lower bound certificates for several well-known factorization problems in hybrid symbolic-numeric computation and certifies accurate rational lower bounds near the irrational global optima.
A fraction free Matrix Berlekamp/Massey algorithm
Berlekamp/massey algorithms for linearly generated matrix sequences
TLDR
A new early termination criterion for the Matrix Berlekamp/Massey algorithm is described and a full proof of correctness for the algorithm is given, which removes all rank and dimension constraints present in previous versions in the literature.
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References

SHOWING 1-10 OF 42 REFERENCES
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.
The Turing factorization of a rectangular matrix
TLDR
Special interest attaches to the continuity properties of the factors, and it is shown that conditions for discontinuous behaviour can be given using the factor D, which is important in computing the Moore-Penrose inverse of a matrix containing symbolic entries.
The shifted number system for fast linear algebra on integer matrices
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.
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.
A cancellation free algorithm, with factoring capabilities, for the efficient solution of large sparse sets of equations
  • J. Smit
  • Computer Science, Engineering
    SYMSAC '81
  • 1981
TLDR
The introduction of the new Factoring Recursive Minor Expansion algorithm with Memo, FDSLEM, with important properties has important properties which make the implementation of an algorithm which can generate the approximate solution of a perturbed system of equations relatively straight forward.
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,…
Factoring polynomials with rational coefficients
In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into…
Gauss Elimination: Workhorse of Linear Algebra.
TLDR
The effect of the noise in matrix entries on the effective rank of the matrix is the central aspect of Gauss elimination considered here.
Exact solution of linear equations usingP-adic expansions
SummaryA method is described for computing the exact rational solution to a regular systemAx=b of linear equations with integer coefficients. The method involves: (i) computing the inverse (modp) ofA…
...
...