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Formal Verification of Floating Point Trigonometric Functions
 Formal Methods in ComputerAided Design: Third International Conference FMCAD 2000, volume 1954 of Lecture Notes in Computer Science
, 2000
"... Abstract. We have formal verified a number of algorithms for evaluating transcendental functions in doubleextended precision floating point arithmetic in the Intel ® IA64 architecture. These algorithms are used in the Itanium TM processor to provide compatibility with IA32 (x86) hardware transcen ..."
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Cited by 39 (8 self)
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Abstract. We have formal verified a number of algorithms for evaluating transcendental functions in doubleextended precision floating point arithmetic in the Intel ® IA64 architecture. These algorithms are used in the Itanium TM processor to provide compatibility with IA32 (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
On the accuracy of finite difference methods for elliptic problems with interfaces
 Commun. Appl. Math. Comput. Sci
"... In problems with interfaces, the unknown or its derivatives may have jump discontinuities. Finite difference methods, including the method of A. Mayo and the immersed interface method of R. LeVeque and Z. Li, maintain accuracy by adding corrections, found from the jumps, to the difference operator a ..."
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Cited by 20 (10 self)
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In problems with interfaces, the unknown or its derivatives may have jump discontinuities. Finite difference methods, including the method of A. Mayo and the immersed interface method of R. LeVeque and Z. Li, maintain accuracy by adding corrections, found from the jumps, to the difference operator at grid points near the interface and modifying the operator if necessary. It has long been observed that the solution can be computed with uniform O(h 2) accuracy even if the truncation error is O(h) at the interface, while O(h 2) in the interior. We prove this fact for a class of static interface problems of elliptic type using discrete analogues of estimates for elliptic equations. Moreover, we show that the gradient is uniformly accurate to O(h 2 log (1/h)). Various implications are discussed, including the accuracy of these methods for steady fluid flow governed by the Stokes equations. Twofluid problems can be handled by first solving an integral equation for an unknown jump. Numerical examples are presented which confirm the analytical conclusions, although the observed error in the gradient is O(h 2).
Verification of the MillerRabin Probabilistic Primality Test
, 2003
"... Using the HOL theorem prover, we apply our formalization of probability theory to specify and verify the MillerRabin probabilistic primality test. The version of the test commonly found in algorithm textbooks implicitly accepts probabilistic termination, but our own verified implementation satisfie ..."
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Cited by 19 (3 self)
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Using the HOL theorem prover, we apply our formalization of probability theory to specify and verify the MillerRabin probabilistic primality test. The version of the test commonly found in algorithm textbooks implicitly accepts probabilistic termination, but our own verified implementation satisfies the stronger property of guaranteed termination. Completing the proof of correctness requires a significant body of group theory and computational number theory to be formalized in the theorem prover. Once verified, the primality test can either be executed in the logic (using rewriting) and used to prove the compositeness of numbers, or manually extracted to Standard ML and used to find highly probable primes.
Floatingpoint verification using theorem proving
 Formal Methods for Hardware Verification, 6th International School on Formal Methods for the Design of Computer, Communication, and Software Systems, SFM 2006, volume 3965 of Lecture Notes in Computer Science
, 2006
"... Abstract. This chapter describes our work on formal verification of floatingpoint algorithms using the HOL Light theorem prover. 1 ..."
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Abstract. This chapter describes our work on formal verification of floatingpoint algorithms using the HOL Light theorem prover. 1
The Ideal Generation Problem for Fat Points
 J. Pure Appl. Alg
, 2000
"... Abstract: This paper is concerned with the problem of determining up to graded isomorphism the modules in a minimal free resolution of a fat point subscheme Z = m1p1 + · · · + mrpr ⊂ P 2 for general points p1,..., pr. We always work over an arbitrary algebraically closed field k. This paper is con ..."
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Cited by 5 (4 self)
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Abstract: This paper is concerned with the problem of determining up to graded isomorphism the modules in a minimal free resolution of a fat point subscheme Z = m1p1 + · · · + mrpr ⊂ P 2 for general points p1,..., pr. We always work over an arbitrary algebraically closed field k. This paper is concerned with determining the number νt(I(Z)) of elements in each degree t of any minimal set of homogeneous generators in the ideal I(Z) ⊂ k[P 2] defining a fat point subscheme Z = m1p1+ · · ·+mrpr ⊂ P 2, where p1,...,pr ∈ P 2 are general. Given the Hilbert function of I(Z), this is equivalent up to graded isomorphism to determining the modules
Automating elementary numbertheoretic proofs using Gröbner bases
"... Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates ..."
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Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates such as ‘divisible by’, ‘congruent ’ and ‘coprime’; one notable example in this class is the Chinese Remainder Theorem (for a specific number of moduli). The method is based on a reduction to ideal membership assertions that are then solved using Gröbner bases. As well as illustrating the usefulness of the procedure on examples, and considering some extensions, we prove a limited form of completeness for properties that hold in all rings. 1
Some facts about continued fractions that should be better known
, 1991
"... In this report I will give proofs of some simple theorems concerning continued fractions that are known to the cognoscenti, but for which proofs in the literature seem to be missing, incomplete, or hard to locate. In particular, I will give two proofs of the following “folk theorem”: if θ is an irra ..."
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Cited by 3 (2 self)
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In this report I will give proofs of some simple theorems concerning continued fractions that are known to the cognoscenti, but for which proofs in the literature seem to be missing, incomplete, or hard to locate. In particular, I will give two proofs of the following “folk theorem”: if θ is an irrational number whose continued fraction has bounded partial
Arithmetic Issues in Geometric Computations
 In Proceedings of the second Real Numbers and Computer Conference
, 1996
"... This paper first recalls by some examples the damages that the numerical inaccuracy of the floatingpoint arithmetic can cause during geometric computations, and it intends to explain why damages for geometric computations differ from those met in numerical computations. Then it surveys the various ..."
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This paper first recalls by some examples the damages that the numerical inaccuracy of the floatingpoint arithmetic can cause during geometric computations, and it intends to explain why damages for geometric computations differ from those met in numerical computations. Then it surveys the various approaches proposed to overcome inaccuracy difficulties; conservative approaches use classical geometric methods but with `exotic' arithmetics instead of the standard floatingpoint one; radical ones go farther and reject classical techniques, considering them not robust enough against inaccuracy. 1 Introduction Geometric modellers provided by commercial CADCAM softwares, and methods from the more theoretical field of Computational Geometry all perform geometric computations: for instance triangulating or meshing geometric domains for finite elements simulation, or computing intersections between geometric objects. Inaccuracy is a crucial issue for geometric computations. Not only the numer...
Cancellation in additively twisted sums on GL(n)
, 2004
"... In a previous paper with Schmid [29] we considered the regularity of automorphic distributions for GL(2, R), and its connections to other topics in number theory and analysis. In this paper we turn to the higher rank setting, establishing the nontrivial bound n≤T an e 2 π i n α = Oε(T 3/4+ε), unifor ..."
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In a previous paper with Schmid [29] we considered the regularity of automorphic distributions for GL(2, R), and its connections to other topics in number theory and analysis. In this paper we turn to the higher rank setting, establishing the nontrivial bound n≤T an e 2 π i n α = Oε(T 3/4+ε), uniformly in α ∈ R, for an the coefficients of the Lfunction of a cusp form on GL(3, Z)\GL(3, R). We also derive an equivalence (Theorem 7.1) between analogous cancellation statements for cusp forms on GL(n, R), and the sizes of certain period integrals. These in turn imply estimates for the second moment of cusp form Lfunctions. 1