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Algorithms for summation and dot product of floating point numbers are presented which are fast in terms of measured computing time. We show that the computed results are as accurate as if computed in twice or K-fold working precision, K 3. For twice the working precision our algorithms for summation and dot product are some 40 % faster than the corresponding XBLAS routines while sharing similar error estimates. Our algorithms are widely applicable because they require only addition, subtraction and multiplication of floating point numbers in the same working precision as the given data. Higher precision is unnecessary, algorithms are straight loops without branch, and no access to mantissa or exponent is necessary.

. A perturbation analysis shows that if a numerically stable procedure is used to compute the matrix sign function, then it is competitive with conventional methods for computing invariant subspaces. Stability analysis of the Newton iteration improves an earlier result of Byers and confirms that ill-conditioned iterates may cause numerical instability. Numerical examples demonstrate the theoretical results. 1. Introduction. If A 2 R n\Thetan has no eigenvalue on the imaginary axis, then the matrix sign function sign(A) may be defined as sign(A) = 1 ßi Z fl (zI \Gamma A) \Gamma1 dz \Gamma I; (1) where fl is any simple closed curve in the complex plane enclosing all eigenvalues of A with positive real part. The sign function is used to compute eigenvalues and invariant subspaces [2, 4, 6, 13, 14] and to solve Riccati and Sylvester equations [9, 15, 16, 28]. The matrix sign function is attractive for machine computation, because it can be efficiently evaluated by relatively simp...

Contents Table of Contents v List of Tables x List of Figures xi 1. Introduction 1 1.1 Detecting Instability In Numerical Algorithms : : : : : : : : : : 1 1.2 Overview of Functional Stability Analysis : : : : : : : : : : : : : 2 1.3 Results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.4 Organization : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2. Theoretical Background 7 2.1 Problems and Conditioning : : : : : : : : : : : : : : : : : : : : 8 2.1.1 Definitions : : : : : : : : : : : : : : : : : : : : : : : : : : 8 2.1.2 Problems and Conditioning : : : : : : : : : : : : : : : : 9 2.1.3 Alternative Treatments and Descriptions : : : : : : : : : 12 2.2 Approximations and Stability : : : : : : : : : : : : : : : : : : : 12 2.2.1 Definitions : : : : : : : : : : : : : : : : : : : : : : : : :

This paper uses a forward and backward error analysis to try to identify some classes of matrices for . P which the matrix sign function is a numerically stable algorithm for extracting invariant subspaces roper scaling is essential to numerical stability as well as to rapid convergence. g a Roberts [21] and Beavers and Denman [7] introduced the matrix sign function as a means of solvin lgebraic Riccati equations and Lyapunov equations. The matrix sign function has since attracted the ] a attention of control engineers and some applied mathematicians ([1] to [21]). Balzer [3], Barraud [5 nd Byers [9] have suggested strategies for accelerating convergence. Denman and Beavers [11] - - 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002 extended matrix sign function algorithms to a list of invariant subspace related calculations. Howland e a [16] used the matrix sign function to count eigenvalues in boxes in the complex plane. Some of th lgorithms have been refined and extended by Attarzadeh [2], Bierman [8], Byers [9]. Gardiner and d d Laub [14] have extended the use of the matrix sign function to generalized Riccati equations an iscrete Riccati equations. Higham [15] has used matrix sign function techniques to calculate polar decompositions

ABSTRACT: Limiting consideration to algorithms satisfying various numerical stability requirements may change lower bounds for computational complexity and/or make lower bounds easier to prove. We will show that, under a sufficiently strong restriction upon numerical stability, any algorithm for multiplying two n x n matrices using only +,- and x requires at least n 3 multiplications. We conclude with a survey of results concerning the numerical stability of several algorithms which have been considered by complexity theorists. I.

this paper tries to do

Given a vector of floating-point numbers with exact sum s, we present an algorithm for calculating a faithful rounding of s into the set of floating-point numbers, i.e. one of the immediate floating-point neighbors of s. If the s is a floating-point number, we prove that this is the result of our algorithm. The algorithm adapts to the condition number of the sum, i.e. it is very fast for mildly conditioned sums with slowly increasing computing time proportional to the condition number. All statements are also true in the presence of underflow. Furthermore algorithms with K-fold accuracy are derived, where in that case the result is stored in a vector of K floating-point numbers. We also present an algorithm for rounding the sum s to the nearest floating-point number. Our algorithms are fast in terms of measured computing time because they neither require special operations such as access to mantissa or exponent, they contain no branch in the inner loop, nor do they require extra precision: The only operations used are standard floating-point addition, subtraction and multiplication in one working precision, for example double precision. Moreover, in contrast to other approaches, the algorithms are ideally suited for parallelization. We also sketch dot product algorithms with similar properties.

Abstract. A recent paper, published in Algorithms—ESA2004, presented examples designed to illustrate that using floating-point arithmetic in algorithms for computational geometry may cause implementations to fail. The stated purpose was to demonstrate, to students and implementors, the inadequacy of floating-point arithmetic for geometric computations. The examples presented were both useful and insightful, but certain of the accompanying remarks were misleading. One such remark is that researchers in numerical analysis may believe that simple approaches are available to overcome the problems of finite-precision arithmetic. Another is the reference, as a general statement, to the inadequacy of floating-point arithmetic for geometric computations. In this paper it will be shown how the now-classical backward error analysis can be applied in the area of computational geometry. This analysis is relevant in the context of uncertain data, which may well be the practical context for computational-geometry algorithms such as, say, those for computing convex hulls. The exposition will illustrate the fact that the backward error analysis does not pretend to overcome the problem of finite precision: it merely provides a tool to distinguish, in a fairly routine way, those algorithms that overcome the problem to whatever extent it is possible to do so. It will also be shown, by using one of the examples of failure presented in the principal reference, that often the situation in computational geometry is exactly parallel to other areas, such as the numerical solution of linear equations, or the algebraic eigenvalue problem. Indeed, the example mentioned can be viewed simply as an example of an unstable algorithm, for a problem for which computational geometry has already discovered provably stable algorithms. 1

In this paper, we present a recent algorithm given by Ogita, Rump and Oishi [39] for accurately computing the sum of n floating point numbers. They also give a computational error bound for the computed result. We apply this algorithm in computing determinant and more particularly in computing robust geometric predicates used in computational geometry. We improve existing results that use either a multiprecision libraries or extended large accumulators.