The Bit Complexity of Dynamic Algebraic Formulas and their Determinants

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).