Abstract. We have formal verified a number of algorithms for evaluating transcendental functions in double-extended precision floating point arithmetic in the Intel ® IA-64 architecture. These algorithms are used in the Itanium TM processor to provide compatibility with IA-32 (x86) hardware transcendentals, and similar ones are used in mathematical software libraries. In this paper we describe in some depth the formal verification of the sin and cos functions, including the initial range reduction step. This illustrates the different facets of verification in this field, covering both pure mathematics and the detailed analysis of floating point rounding. 1

. Many modern algorithms for the transcendental functions rely on a large table of precomputed values together with a low-order polynomial to interpolate between them. In verifying such an algorithm, one is faced with the problem of bounding the error in this polynomial approximation. The most straightforward methods are based on numerical approximations, and are not prima facie reducible to a formal HOL proof. We discuss a technique for proving such results formally in HOL, via the formalization of a number of results in polynomial theory, e.g. squarefree decomposition and Sturm's theorem, and the use of a computer algebra system to compute results that are then checked in HOL. We demonstrate our method by tackling an example from the literature. 1 Introduction Many algorithms for the transcendental functions such as exp, sin and ln in floating point arithmetic are based on table lookup. Suppose that a transcendental function f(x) is to be calculated. Values of f(a i ) are...

Abstract. In this paper we are concerned with the solution of n × n Hermitian Toeplitz systems with nonnegative generating functions f. The preconditioned conjugate gradient (PCG) method with the well–known circulant preconditioners fails in the case where f has zeros. In this paper we consider as preconditioners band–Toeplitz matrices generated by trigonometric polynomials g of fixed degree l. We use different strategies of approximation of f to devise a polynomial g which has some analytical properties of f, is easily computable and is such that the corresponding preconditioned system has a condition number bounded by a constant independent of n. For each strategy we analyze the cost per iteration and the number of iterations required for the convergence within a preassigned accuracy. We obtain different estimates of l for which the total cost of the proposed PCG methods is optimal and the related rates of convergence are superlinear. Finally, for the most economical strategy, we perform various numerical experiments which fully confirm the effectiveness of approximation theory tools in the solution of this kind of linear algebra problems. 1.

The Remez algorithm, 75 years old, is a famous method for computing minimax polynomial approximations. Most implementations of this algorithm date to an era when tractable degrees were in the dozens, whereas today, degrees of hundreds or thousands are not a problem. We present a 21st-century update of the Remez ideas in the context of the chebfun software system, which carries out numerical computing with functions rather than numbers. A crucial feature of the new method is its use of chebfun global rootfinding to locate extrema at each iterative step, based on a recursive algorithm combining ideas of Specht, Good, Boyd, and Battles. Another important feature is the use of the barycentric interpolation formula to represent the trial polynomials, which points the way to generalizations for rational approximations. We comment on available software for minimax approximation and its scientific context, arguing that its greatest importance these days is probably for fundamental studies rather than applications.