# 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} }

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…

## 16 Citations

Common Factors in Fraction-Free Matrix Reduction

- Mathematics2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
- 2013

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

- Computer ScienceArXiv
- 2016

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

- Mathematics, Computer ScienceACCA
- 2010

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

- Computer ScienceINFORMS J. Comput.
- 2015

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

- Computer Science
- 2016

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

- Mathematics, Computer Science
- 2012

Common Factors in Fraction-Free Matrix Decompositions

- MathematicsMath. Comput. Sci.
- 2021

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

- Mathematics, Computer ScienceISSAC '08
- 2008

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.

Berlekamp/massey algorithms for linearly generated matrix sequences

- Computer Science
- 2009

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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