Global error estimation for index-1 and -2 DAEs

In this paper, we consider the extension of three classical ODE estimation techniques (Richardson extrapolation, Zadunaisky's technique and solving for the correction) to DAEs. Their convergence analysis is carried out for semi-explicit index-1 DAEs solved by a wide set of Runge-Kutta methods. Experimentation of the estimation techniques with RADAU5 is also presented: their behaviour for index-1 and -2 problems, and for variable step size integration is investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Sharp error bounds for complex floating-point inversion

We study the accuracy of the classic algorithm for inverting a complex number given by its real and imaginary parts as floating-point numbers. Our analyses are done in binary floating-point arithmetic, with an unbounded exponent range and in precision p; we also assume that the basic arithmetic operations (+, ?, ×, /) are rounded to nearest, so that the roundoff unit is u = 2^?p. We bound the largest relative error in the computed inverse either in the componentwise or in the normwise sense. We prove the componentwise relative error bound 3u for the complex inversion algorithm (assuming p ≥ 4), and we show that this bound is asymptotically optimal (as p → ∞) when p is even, and sharp when using one of the basic IEEE 754 binary formats with an odd precision (p = 53, 113). This componentwise bound obviously leads to the same bound 3u for the normwise relative error. However, we prove that the smaller bound 2.707131u holds (assuming p ≥ 24) for the normwise relative error, and we illustrate the sharpness of this bound for the basic IEEE 754 binary formats (p = 24,53,113) using numerical examples.