This paper describes an extension to the set of Basic Linear Algebra Subprograms. The extensions are targeted at matrix-vector operations which should provide for efficient and portable implementations of algorithms for high performance computers.

We describe a new method for numeric computations, which we call affine arithmetic (AA). This model is similar to standard interval arithmetic, to the extent that it automatically keeps track of rounding and truncation errors for each computed value. However, by taking into account correlations between operands and sub-formulas, AA is able to provide much tighter bounds for the computed quantities, with errors that are approximately quadratic in the uncertainty of the input variables. We also describe two applications of AA to computer graphics problems, where this feature is particularly valuable: namely, ray tracing and the construction of octrees for implicit surfaces.

This paper introduces a rigorous computer-assisted procedure for analyzing hyperbolic 3-manifolds. This procedure is used to complete the proof of several long-standing rigidity conjectures in 3-manifold theory as well as to

Abstract. For the familiar Fibonacci sequence (defined by f1 = f2 = 1, and fn = fn−1 + fn−2 for n>2), fn increases exponentially with n at a rate given by the golden ratio (1 + √ 5)/2 =1.61803398.... But for a simple modification with both additions and subtractions — the random Fibonacci sequences defined by t1 = t2 =1,andforn>2, tn = ±tn−1 ± tn−2, where each ± sign is independent and either + or − with probability 1/2 — it is not even obvious if |tn | should increase with n. Our main result is that n |tn | →1.13198824... as n →∞

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Abstract. Given a complex matrix H, we consider the decomposition H = QRP ∗ , where R is upper triangular and Q and P have orthonormal columns. Special instances of this decomposition include (a) the singular value decomposition (SVD) where R is a diagonal matrix containing the singular values on the diagonal, (b) the Schur decomposition where R is an upper triangular matrix with the eigenvalues of H on the diagonal, (c) the geometric mean decomposition (GMD) [The Geometric Mean Decomposition, Y. Jiang, W. W. Hager, and J. Li, December 7, 2003] where the diagonal of R is the geometric mean of the positive singular values. We show that any diagonal for R can be achieved that satisfies Weyl’s multiplicative majorization conditions: k� k� |ri | ≤ σi, 1 ≤ k < K, i=1 i=1 K� K� |ri | = σi, where K is the rank of H, σi is the i-th largest singular value of H, and ri is the i-th largest (in magnitude) diagonal element of R. We call the decomposition H = QRP ∗ , where the diagonal of R satisfies Weyl’s conditions, the generalized triangular decomposition (GTD). The existence of the GTD is established using a result of Horn [On the eigenvalues of a matrix with prescribed singular values, Proc. Amer. Math. Soc., 5 (1954), pp. 4–7]. In addition, we present a direct (nonrecursive) algorithm that starts with the SVD and applies a series of permutations and Givens rotations to obtain the GTD. The GMD has application to signal processing and the design of multiple-input multiple-output (MIMO) systems; the lossless filters Q and P minimize the maximum error rate of the network. The GTD is more flexible than the GMD since the diagonal elements of R need not be identical. With this additional freedom, the performance of a communication channel can be optimized, while taking into account differences in priority or differences in quality of service requirements for subchannels. Another application of the GTD is to inverse eigenvalue problems where the goal is to construct matrices with prescribed eigenvalues and singular values. Key words. Generalized triangular decomposition, geometric mean decomposition, matrix factorization, unitary factorization, singular value decomposition, Schur decomposition, MIMO systems, inverse eigenvalue problems

. A new automatic method to correct the rst-order eect of oating point rounding errors on the result of a numerical algorithm is presented. A correcting term and a condence threshold are computed using automatic dierentiation, computation of elementary rounding error and running error analysis. Algorithms for which the accuracy of the result is not aected by higher order terms are identied. The correction is applied to the nal result or to sensitive intermediate results. The properties and the eciency of the method are illustrated with a sample numerical example. AMS subject classication: 65G05. Key words: automatic error analysis, rounding error, oating point arithmetic. 1 Introduction Improving the accuracy and the stability of numerical software is one of the current challenges in scientic computing. The rounding errors introduced by oating point arithmetic contribute to the inaccuracy of the computed results and to the instability of some numerical algorithms. In t...

Abstract. The addition of two or more floating-point numbers is fundamental to numerical computations. This paper describes an efficient “distillation ” style algorithm which produces a precise sum by exploiting the natural accuracy of compensated cancellation. The algorithm is applicable to all sets of data but is particularly appropriate for ill-conditioned data, where standard methods fail due to the accumulation of rounding error and its subsequent exposure by cancellation. The method uses only standard floating-point arithmetic and does not rely on the radix used by the arithmetic model, the architecture of specific machines, or the use of accumulators.

What factors contribute to the accurate and efficient numerical solution of problems in control systems analysis and design? Although numerical methods have been used for many centuries to solve problems in science and engineering, the importance of computation grew tremendously with the advent of digital computers. It became immediately clear that many of the classical analytical and numerical methods and algorithms could not be implemented directly as computer codes, although they were well suited for hand computations. What was the reason? When doing computations by hand a person can choose the accuracy of each elementary calculation and then estimate, based on intuition and experience, its influence on the final result. In contrast, when computations are done automatically, intuitive error control is usually not possible and the effect of errors on the intermediate calculations must be estimated in a more systematic way. Due to this observation, starting

Abstract. The Hales program to prove the Kepler conjecture on sphere packings consists of five steps, which if completed, will jointly comprise a proof of the conjecture. We carry out step five of the program, a proof that the local density of a certain combinatorial arrangement, the pentahedral prism, is less than that of the face-centered cubic lattice packing. We prove various relations on the local density using computer-based interval arithmetic methods. Together, these relations imply the local density bound. 1.