We present techniques which may be used to perform computations of very high accuracy using only straightforward floating point arithmetic operations of limited precision, and we prove the validity of these techniques under very general hypotheses satisfied by most implementations of floating point arithmetic. To illustrate the application of these techniques, we present an algorithm which computes the intersection of a line and a line segment. The algorithm is guaranteed to correctly decide whether an intersection exists and, if so, to produce the coordinates of the intersection point accurate to full precision. Moreover, the algorithm is usually quite efficient; only in a few cases does guaranteed accuracy necessitate an expensive computation. 1. Introduction "How accurate is a computed result if each intermediate quantity is computed using floating point arithmetic of a given precision?" The casual reader of Wilkinson's famous treatise [21] and similar roundoff error analyses might...

We present a deterministic strongly polynomial algorithm that computes the permanent of a nonnegative n x n matrix to within a multiplicative factor of e^n. To this end

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this paper we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding. The ffl-pseudozero set Z ffl (p) is the set of zeros of all polynomials p obtained by coefficientwise perturbations of p of size

Floating point arithmetics generally possess many regularity properties in addition to those that are typically used in roundoff error analyses; these properties can be exploited to produce computations that are more accurate and cost effective than many programmers might think possible. Furthermore, many of these properties are quite simple to state and to comprehend, but few programmers seem to be aware of them (or at least willing to rely on them). This dissertation presents some of these properties and explores their consequences for computability, accuracy, cost, and portability. For example, we consider several algorithms for summing a sequence of numbers and show that under very general hypotheses, we can compute a sum to full working precision at only somewhat greater cost than a simple accumulation, which can often produce a sum with no significant figures at all. This example, as well as others we present, can be generalized further by substituting still more complex algorith...

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this paper we introduce certain numbers, called collinearity indices, which are useful in detecting near collinearities in regression problems. The coefficients enter adversely into formulas concerning significance testing and the effects of errors in the regression variables. Thus they provide simple regression diagnostics, suitable for incorporation in regression packages. Keywords and phrases: collinearity, ill-conditioning, linear regression, errors in the variables, regression diagnostics. 1 Introduction

. For k + 1 power series a 0 (z); : : : ; a k (z), we present a new iterative, look-ahead algorithm for numerically computing Pad'e-Hermite systems and simultaneous Pad'e systems along a diagonal of the associated Pad'e tables. The algorithm computes the systems at all those points along the diagonal at which the associated striped Sylvester and mosaic Sylvester matrices are wellconditioned. It is shown that a good estimate for the condition numbers of these Sylvester matrices at a point is easily determined from the Pad'e-Hermite system and simultaneous Pad'e system computed at that point. The operation and the stability of the algorithm is controlled by a single parameter ø which serves as a threshold in deciding if the Sylvester matrices at a point are sufficiently wellconditioned. We show that the algorithm is weakly stable, and provide bounds for the error in the computed solutions as a function of ø . Experimental results are given which show that the bounds reflect the actual b...

This paper describes a test of a computer's floating-point arithmetic unit. The test has two goals. The first goal deals with the needs of users of computers, and the second goal deals with manufacturers of computers. The first and major goal is to determine if the machine supports a particular mathematical model of computer arithmetic. This model was developed as an aid in the design, analysis, implementation and testing of portable, high-quality numerical software. If a computer supports the arithmetic model, then software written using the model will perform correctly and to specified accuracy on that machine. The second goal of the test is to check that the basic operations perform as the manufacturer intended. For example, if division ( x x // yy ) is implemented as a composite operation ( x x × (1//yy) ), then the test should detect that fact. Also, the accuracy lost in such a division due to the extra arithmetic operations can tell the manufacturer whether it has been implemente...

In Floudas and Visweswaran (1990), a new global optimization algorithm (GOP) was proposed for solving constrained nonconvex problems involving quadratic and polynomial functions in the objective function and/or constraints. In this paper, the application of this algorithm to the special case of polynomial functions of one variable is discussed. The special nature of polynomial functions enables considerable simplification of the GOP algorithm. The primal problem is shown to reduce to a simple function evaluation, while the relaxed dual problem is equivalent to the simultaneous solution of two linear equations in two variables. In addition, the one-to-one correspondence between the x and y variables in the problem enables the iterative improvement of the bounds used in the relaxed dual problem. The simplified approach is illustrated through a simple example that shows the significant improvement in the underestimating function obtained from the application of the modified algorithm. The...