  Many iterative algorithms in optimization, computational geometry, computer
algebra, and other areas of computer science require repeated computation of
some algebraic expression whose input changes slightly from one iteration to
the next. Although efficient data structures have been proposed for maintaining
the solution of such algebraic expressions under low-rank updates, most of
these results are only analyzed under exact arithmetic (real-RAM model and
finite fields) which may not accurately reflect the complexity guarantees of
real computers. In this paper, we analyze the stability and bit complexity of
such data structures for expressions that involve the inversion,
multiplication, addition, and subtraction of matrices under the word-RAM model.
We show that the bit complexity only increases linearly in the number of matrix
operations in the expression. In addition, we consider the bit complexity of
maintaining the determinant of a matrix expression. We show that the required
bit complexity depends on the logarithm of the condition number of matrices
instead of the logarithm of their determinant. We also discuss rank maintenance
and its connections to determinant maintenance. Our results have wide
applications ranging from computational geometry (e.g., computing the volume of
a polytope) to optimization (e.g., solving linear programs using the simplex
algorithm).
