  LU and Cholesky matrix factorization algorithms are core subroutines used to
solve systems of linear equations (SLEs) encountered while solving an
optimization problem. Standard factorization algorithms are highly efficient
but remain susceptible to the accumulation of roundoff errors, which can lead
solvers to return feasibility and optimality claims that are actually invalid.
This paper introduces a novel approach for solving sequences of closely related
SLEs encountered in nonlinear programming efficiently and without roundoff
errors. Specifically, it introduces rank-one update algorithms for the
roundoff-error-free (REF) factorization framework, a toolset built on
integer-preserving arithmetic that has led to the development and
implementation of fail-proof SLE solution subroutines for linear programming.
The formal guarantees of the proposed algorithms are established through the
derivation of theoretical insights. Their advantages are supported with
computational experiments, which demonstrate upwards of 75x-improvements over
exact factorization run-times on fully dense matrices with over one million
entries. A significant advantage of the methodology is that the length of any
coefficient calculated via the proposed algorithms is bounded polynomially in
the size of the inputs without having to resort to greatest common divisor
operations, which are required by and thereby hinder an efficient
implementation of exact rational arithmetic approaches.
