Accurate and efficient expression evaluation and linear algebra, or why it can be easier to compute accurate eigenvalues of a Vandermonde matrix than the accurate sum of 3 numbers

We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean the computed answer has relative error less than 1, i.e. has some leading digits correct. We also address efficiency, by which we mean algorithms that run in polynomial time in the size of the input. Our results will depend strongly on the model of arithmetic: Most of our results will use what we call the <i>Traditional Model (TM)</i>, that the computed result of <i>op</i>(<i>a, b</i>), a binary operation like <i>a</i> + <i>b</i>, is given by <i>op</i>(<i>a</i>, <i>b</i>) * (1 + Δ) where all we know is that |Δ| ≤ ε ≪ 1. Here ε is a constant also known as machine epsilon.