AbRtract This paper deals with the domain decomposition-based preconditioned conjugate gradient method. The Schur complement is expressed as a function of & simple interface matrix. This function is approximated by a simple rational function to generate a simple matrix that is then used 8.8 & preconditioner for the Schur complement. Extensive experiments are performed to examine the effectivene...

Abstra t. We introdu e a new omputationally e ient algorithm for onstru ting near optimal rational approximations of large (onedimensional) data sets. In ontrast to wavelet-type approximations, these new approximations are e e tively shift invariant. We note that the omplexity of urrent algorithms for omputing near optimal rational approximations prevents their use for large data sets. In order...

The first course is devoted to the basic setup of Diophantine approximation: we start with rational approximation to a single real number. Firstly, positive results tell us that a real number x has “good” rational approximation p/q, where “good” is when one compares |x − p/q| and q. We discuss Dirichlet’s result in 1842 (see [6] Course N◦2 §2.1) and the Markoff–Lagrange spectrum ([6] Course N◦1...

This paper gives explicit rational functions for interpolating and approximating functions on the intervals [-1,1], [0, oo], and [-oo, oo]. The rational functions are linear in the functions to be approximated, and they have preassigned poles. The error of approximation of these rationals is nearly as small as the error of best rational approximation with numerator and denominator polynomials o...

We apply Padé approximation techniques to deduce lower bounds for simultaneous rational approximation to one or more algebraic numbers. In particular, we strengthen work of Osgood, Fel’dman and Rickert, proving, for example, that max {∣∣∣√2− p1/q∣∣∣ , ∣∣∣√3− p2/q∣∣∣} > q−1.79155 for q > q0 (where the latter is an effective constant). Some of the Diophantine consequences of such bounds will be d...

If a, b and n are positive integers with b ≥ a and n ≥ 3, then the equation of the title possesses at most one solution in positive integers x and y, with the possible exceptions of (a, b, n) satisfying b = a + 1, 2 ≤ a ≤ min{0.3n, 83} and 17 ≤ n ≤ 347. The proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometri...

We give a short introduction to Padé approximation (rational approximation to a function with close contact at one point) and to Hermite-Padé approximation (simultaneous rational approximation to several functions with close contact at one point) and show how orthogonality plays a crucial role. We give some insight into how logarithmic potential theory helps in describing the asymptotic behavio...

Rational expectations models make stringent assumptions on the agent’s knowledge about the true model. This paper introduces a model in which the rational agent realizes that using a given model involves approximation errors, and adjusts behavior accordingly. If the researcher accounts for this empirical rationality on part of the agent, the resulting empirical model assigns likelihood to the d...

If a, b and n are positive integers with b ≥ a and n ≥ 3, then the equation of the title possesses at most one solution in positive integers x and y, with the possible exceptions of (a, b, n) satisfying b = a + 1, 2 ≤ a ≤ min{0.3n, 83} and 17 ≤ n ≤ 347. The proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometri...