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QR factorization is a key tool in mathematics, computer science, operations research, and engineering. This paper presents the roundoff-error-free (REF) QR factorization framework comprising integer-preserving versions of the standard and the thin QR factorizations and associated algorithms to compute them. Specifically, the standard REF QR factorization factors a given matrix $$A \in \Z^{m \times n}$$ as $A=QDR$, where $$Q \in \Z^{m \times m}$$ has pairwise orthogonal columns, $$D$$ is a diagonal matrix, and $$R \in \Z^{m \times n}$$ is an upper trapezoidal matrix; notably, the entries of $$Q$$ and $$R$$ are integral, while the entries of $$D$$ are reciprocals of integers. In the thin REF QR factorization, $$Q \in \Z^{m \times n}$$ also has pairwise orthogonal columns, and $$R \in \Z^{n \times n}$$ is also an upper triangular matrix. In contrast to traditional (i.e., floating-point) QR factorizations, every operation used to compute these factors is integral; thus, REF QR is guaranteed to be an exact orthogonal decomposition. Importantly, the bit-length of every entry in the REF QR factorizations (and within the algorithms to compute them) is bounded polynomially. Notable applications of our REF QR factorizations include finding exact least squares or exact basic solutions (i.e., a rational n-dimensional vector $$x$$) to any given full column rank or rank deficient linear system $A x = b$, respectively. In addition, our exact factorizations can be used as a subroutine within exact and/or high-precision quadratic programming. Altogether, REF QR provides a framework to obtain exact orthogonal factorizations of any rational matrix (as any rational/decimal matrix can be easily transformed into an integral matrix). 

Transmission switching has proven to be a highly useful post-contingency recovery technique by allowing power system operators increased levels of control through leveraging the topology of the power system. However, transmission switching remains only implemented in limited capacity because of concerns over computational complexity, uncertainty of performance in AC systems, and scalability to real-world, large-scale systems. We propose a heuristic that uses a sophisticated guided undersampling procedure combined with logistic regression to accurately identify transmission switching actions to reduce post-contingency AC power flow violations. The proposed heuristic was tested on real-world, large-scale AC power system data and consistently identified optimal or near-optimal transmission switching actions. Because the proposed heuristic is computationally inexpensive, addresses an AC system, and is validated on real-world, large-scale data, it directly addresses the aforementioned issues regarding transmission switching implementation. 

SPEX Left LU is a software package for exactly solving unsymmetric sparse linear systems. As a component of the sparse exact (SPEX) software package, SPEX Left LU can be applied to any input matrix, A , whose entries are integral, rational, or decimal, and provides a solution to the system \( Ax = b \) , which is either exact or accurate to user-specified precision. SPEX Left LU preorders the matrix A with a user-specified fill-reducing ordering and computes a left-looking LU factorization with the special property that each operation used to compute the L and U matrices is integral. Notable additional applications of this package include benchmarking the stability and accuracy of state-of-the-art linear solvers and determining whether singular-to-double-precision matrices are indeed singular. Computationally, this article evaluates the impact of several novel pivoting schemes in exact arithmetic, benchmarks the exact iterative solvers within Linbox, and benchmarks the accuracy of MATLAB sparse backslash. Most importantly, it is shown that SPEX Left LU outperforms the exact iterative solvers in run time on easy instances and in stability as the iterative solver fails on a sizeable subset of the tested (both easy and hard) instances. The SPEX Left LU package is written in ANSI C, comes with a MATLAB interface, and is distributed via GitHub, as a component of the SPEX software package, and as a component of SuiteSparse. 

To meet the growing need for extended or exact precision solvers, an efficient framework based on Integer-Preserving Gaussian Elimination (IPGE) has been recently developed, which includes dense/sparse LU/Cholesky factorizations and dense LU/Cholesky factorization updates for column and/or row replacement. This paper discusses our ongoing work developing the sparse LU/Cholesky column/row-replacement update and the sparse rank-l update/downdate. We first present some basic background for the exact factorization framework based on IPGE. Then we give our proposed algorithms along with some implementation and data-structure details. Finally, we provide some experimental results showcasing the performance of our update algorithms. Specifically, we show that updating these exact factorizations can typically be 10x to 100x faster than (re-)factorizing the matrices from scratch.