An algorithm to solve integer linear systems exactly using numerical methods

In this paper, we present a new algorithm for the exact solutions of linear systems with integer coefficients using numerical methods. It terminates with the correct answer in well-conditioned cases or quickly aborts in ill-conditioned cases. Success of this algorithm on a linear equation requires the linear system must be sufficiently well-conditioned for the numeric linear algebra method being used to compute a solution with sufficient accuracy. Our method is to find an initial approximate solution by using a numerical method, then amplify the approximate solution by a scalar, adjust the amplified solution and corresponding residual to integers so that they can be computed without large integer arithmetic involved and can be stored exactly. Then we repeat these steps to refine the solution until sufficient accuracy is achieved, and finally reconstruct the rational solution. Our approximating, amplifying, and adjusting idea enables us to compute the solutions without high precision software floating point operations involved in the whole procedure or big integer arithmetic involved except at the final rational reconstruction step. We will expose the theoretical cost and show some experimental results. Key words: linear systems, numerical linear algebra methods, rational solvers

Foundational Factorization Algorithms for the Efficient Roundoff-Error-Free Solution of Optimization Problems

LU and Cholesky factorizations play a central role in solving linear and mixed-integer programs. In many documented cases, the round-off errors accrued during the construction and implementation of these factorizations cause the misclassification of suboptimal solutions as optimal and infeasible problems as feasible and vice versa. Such erroneous outputs bring the reliability of optimization solvers into question and, therefore, it is imperative to eliminate these round off errors altogether and to do so efficiently to ensure practicality. Firstly, this work introduces two round off-error-free factorizations (REF) constructed exclusively in integer arithmetic: the REF LU and Cholesky factorizations. Additionally, it develops supplementary integer-preserving substitution algorithms, thereby providing a complete tool set for solving systems of linear equations (SLEs) exactly and efficiently. An inherent property of the REF factorization algorithms is that their entries' bit-length--- i.e., the number of bits required for expression--- is bounded polynomially. Unlike the exact rational arithmetic methods used in practice, however, the algorithms herein presented do not require any greatest common divisor operations to guarantee this pivotal property. Secondly, this work derives various useful theoretical results and details computational tests to demonstrate that the REF factorization framework is considerably superior to the rational arithmetic LU factorization approach in computational performance and storage requirements. This is significant because the latter approach is the solution validation tool of choice of state-of-the-art exact linear programming solvers due to its ability to handle both numerically difficult and intricate problems. An additional theoretical contribution and further computational tests also demonstrate the predominance of the featured framework over Q-matrices, which comprise an alternative integer-preserving approach relying on the basis adjunct matrix. Thirdly, this work develops special algorithms for updating the REF factorizations. This is necessary because applying the traditional approach to the REF factorizations is inefficient in terms of entry growth and computational effort. In fact, these inefficiencies virtually wipe out all the computational savings commonly expected of factorization updates. Hence, the current work develops REF update algorithms that differ significantly from their traditional counterparts. The featured REF updates are column/row addition, deletion, and replacement