Accurate and Ecient Algorithms for Floating Point Computation

1 Abstract Our goal is to find accurate and efficient algorithms, when they exist, for evaluating rational expressions containing floating point numbers, and for computing matrix factorizations (like LU, the singular value decomposition (SVD) and eigenvalue decompositions) of matrices with rational expressions as entries. More precisely, accuracy means the relative error in the output must be less than one (no matter how tiny the output is), and efficiency means that the algorithm runs in polynomial time. Our goal is challenging because our accuracy demand is much stricter than usual. The classes of floating point expressions or matrices that we can accurately and efficiently evaluate or factor depend strongly on our model of arithmetic: 1. In the " Traditional Model " (TM), the floating point result of an operation like a + b is f l(a + b) = (a + b)(1 + δ), where |δ| must be tiny. 2. In the " Long Exponent Model " (LEM) each floating point number x = f · 2 e is represented by the pair of integers (f, e), and there is no bound on the sizes of the exponents e in the input data. The LEM permits accurate and efficient computation of strictly larger classes of expressions or matrices than the TM. 3. In the " Short Exponent Model " (SEM) each floating point number x = f ·2 e is also represented by (f, e), but the input exponent sizes are bounded in terms The information presented here does not necessarily reflect the position of the Government and no official endorsement should be inferred. i i of the sizes of the input fractions f. We believe the SEM permits accurate and efficient computation of strictly more expressions or matrices than the LEM. These classes will be described by factorizability properties of the rational expressions, or of the minors of the rational matrices. For each such class, we identify new algorithms that attain our goals of accuracy and efficiency. These algorithms are often exponentially faster than prior algorithms, which would simply use a conventional algorithm with sufficiently high precision. For example, we can factorize Cauchy matrices, Vandermonde matrices, many kinds of totally positive matrices, and suitably discretized differential and integral operators in all three models much more accurately and efficiently than before. But we provably cannot add three numbers accurately in the TM, even though it is easy in the other models. …