Roundoff-Error-Free Algorithms for Solving Linear Systems via Cholesky and LU Factorizations

Abstract

emc@tamu.edu} L U and Cholesky factorizations are computational tools for efficiently solving linear systems that play a central role in solving linear programs and several other classes of mathematical programs. In many documented cases, however, the roundoff errors accrued during the construction and implementation of these factorizations lead to the misclassification of feasible problems as infeasible and vice versa. Hence, reducing these roundoff errors or eliminating them altogether is imperative to guarantee the correctness of the solutions provided by optimization solvers. To achieve this goal without having to use rational arithmetic, we introduce two roundofferror-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. Additionally, we present supplementary roundoff-error-free forward and backward substitution algorithms, thereby providing a complete tool set for solving systems of linear equations exactly and efficiently. An important property shared by the featured factorizations and substitution algorithms is that their individual coefficients' maximum word length-i.e., the maximum number of digits required for expression-is bounded polynomially. Unlike the rational arithmetic methods used in practice to solve linear systems exactly, however, the algorithms herein presented do not require any gcd calculations to bound the entries' word length. We also derive various other related theoretical results, including the total computational complexity of all the roundoff-error-free processes herein presented.

Solution of Dense Linear Systems via Roundoff-Error-Free Factorization Algorithms

“…Interested researchers are encouraged to use or modify our general purpose REF-LU codes (available via GitHub at https://github.com/adolfoescobedo/REF-LU) and to use them in their work by citing Escobedo and Moreno-Centeno (2015) and this article.…”

“…The REF factorization framework is founded on the integer-preserving Gaussian Elimination algorithm (IPGE), which consists, as its names suggests, of integer-preserving row-reduction operations. A standard description of this algorithm can be found in Escobedo and Moreno-Centeno (2015). To proceed with the present discussion, it is necessary to introduce only a minimal description of IPGE as well as a small number of definitions and assumptions.…”

“…The vector output at the completion of the above algorithm is the solution to the SLE LD -1 y = b. We note that Equation (6) is an equivalent but more memory-efficient version of the REF forward substitution algorithm given in Escobedo and Moreno-Centeno (2015). Specifically, this enhanced version writes over the elements of y at each iteration, whereas the original version stores an ancillary triangular array of (n 2 + n)/2 elements to output the same forward substituted vector.…”

“…As a final remark, notice from Equations (6)-(8) that it is unnecessary to store D or (Escobedo and Moreno-Centeno 2015).…”

Exact Matrix Factorization Updates for Nonlinear Programming

“…PROOF. Upon completion of the kth iteration of REF forward substitution, the individual elements of y (k) are connected to IPGE entries according to the equation (Escobedo and Moreno-Centeno 2015):…”