Abstract: This paper discusses the security problem for statistical data bases and examines in some detail a couple of techniques, namely rounding and random rounding, which have been proposed to solve the statistical data base security problem. These techniques are studied in connection with a simple model of a statistical data base which produces crosstabulations. Analysis of the effects of both techniques on data base output indicates that, with suitable choice of base, rounding may be a more secure technique than random rounding.

Abstract: It is well known that the calculation of an accurate approximate derivative f'(x) of a nontabular function f (x) on a finite-precision computer by the formula d(h) = (f (x + h) -f (x h))/2h is a delicate task. If h is too large, truncation errors cause poor answers, while if h is too small, cancellation and other "rounding" errors cause poor answers. We will show that by using simple results on the nature of the asymptotic convergence of d(h) to f', a reliable numerical method can be obtained which can yield effilciently the theoretical maximum number of accurate digits for the given machine precision.

Abstract: A stable numerical procedure is presented for updating regressions, including quantities used in the analysis of variance, when regressors or observations are added or deleted. The modified Gram-Schmidt triangular factorization is used, and the matrix with orthonormal columns is stored and updated. In this way it is possible to monitor and to minimize the effects of rounding errors through the techniques of iterative refinement and reorthogonalization. Numerical results are included.

Abstract: If the firmware calls for an operand rounding operation, apparatus in the Scientific Instruction Processor (SIP) tests the bit to the right of the low order bit of the normalized operand to determine if a rounding cycle is required. If the operand requires a normalization cycle or a mantissa overflow correction cycle, the rounding operation is performed in those cycles.

Abstract: A round off correction logic circuit is disclosed for inclusion within a floating point arithmetic binary digital multiplier implementing a modified Booth's algorithm for generating a final product of binary digits. The round off logic circuitry is connected in the multiplier for rounding its final product off to a predetermined binary digit without requiring the multiplier to generate any of the less significant binary digits to the right of the predetermined binary digit. Multiplier circuitry otherwise required to generate an unrounded final product prior to round off is eliminated without loss of accuracy in round off.

Abstract: A.n algorithm is given which finds the least square generalized inverse for classification models with arbitrary patterns. Some theorems are established to justify this algorithm and several examples are worked out. The algorithm gives the generalized inverse without any rounding off errors and can be programmed for electronic computers.

Abstract: Every linearly stable, fixed-point digital filter has a realization, using standard two's complement arithmetic with rounding, that will be free of all autonomous finite-word limit cycles. Specific structures that will achieve this condition, consisting of combinations of transversal structures and recursive structures, are derived using a state-space formalism. Freedom from limit cycles is obtained by a design procedure that reduces the norm of the matrix in the recursive feedback loop. Specific design relations for minimizing the roundoff noise in the presence of an l 2 dynamic range constraint are provided.

Abstract: Closed approximate rational arithmetic systems are described and their number theoretic foundations are surveyed. The arithmetic is shown to implicitly contain an adaptive single-to-double precision natural rounding behavior that acts to recover true simple fractional results. The probability of such recovery is investigated and shown to be quite favorable.

Abstract: The problem considered is that of the allocation and replenishment of several resources in integer quantities in such a way as to maximize the sum of the returns from activities with concave return functions. All the resources are of the same physical type and each resource has an effectiveness of 0 or 1 against each activity, depending on the geographical locations of the resources and the activities or on other constraints. Solutions with objective values arbitrarily close to the optimal value are generated by the application of resourcewise optimization to an associated problem in continuous variables, and the rounding of a continuous solution to an integer solution according to given rules. The application of other continuous methods is indicated. Some properties of optimal integer solutions are derived.

Abstract: A recent paper in SIGMICRO [1] contained a comparison of the accuracy of floating point vs. rational representations, which is very unfair to the latter. The format chosen for rational numbers utilizes 16 bits for numerators and 16 bits for denominators. This implies that the spacing between consecutive numbers in the system is in most cases of the order 2-32. Only around simple rational numbers (e.g. 1/1, 2/3) is the spacing of the order 2-16. However the rounding algorithm presented in [1] will almost certainly introduce a rounding error of the order 2-16, i.e. introduce an error which in most cases is of the order 216 larger than necessary.

Abstract: The Branch and Round algorithm uses the device (introduced as a heuristic procedure by Hillier [4] ) of locating a central point in a polyhedron and then rounding that point to the closest integer point as the fundamental search mechanism for an enumerative algorithm. No computational experience is currently available. Each unfathomed vertex separates into a comparatively large number of immediate successors. Therefore successful computation requires a low probability that many separations will be needed to achieve fathoming. Hopefully computations will show that excessive tree depth is not required or will provide useful clues for identification of (and response to) structural characteristics of polyhedra that generate “deep” trees.

Abstract: Rounding errors may change the flow of control in numerical computing processes by leading to changes in some branching decisions of the process In this paper some general topological and measure theoretical results associated with this effect of rounding errors are derived The approach is based on a model of numerical computation related to program schemes Each computing process specified by the model computes a partial functionR n →R m using rational operations and simple tests on real numbers The topological structure of input point sets inR n on which the computation follows the same execution path is studied We also investigate input points, called sensitive, on which rounding errors may change the execution path followed Conditions concerning computing processes are given which guarantee that the Lebesgue measure of sensitive points approaches zero (ie the probability of a branching error gets arbitrarily small) as the precision of the arithmetic increases Most numerical processes used in practice are easily seen to satisfy these conditions

Abstract: Data from 17 studies of stream-pebble rounding were analyzed by computing simple correlation coefficients and regression equations for distance and roundness in individual streams Of 30 cases examined, 20 showed significant downstream increases in roundness. In 13 of these 20 eases the downstream trend was described better by logarithmic or semi-logarithmic functions than by linear ones. Studies employing the Wadell-Krumbein measurement technique produce regression lines with higher Y-intercept values than do those employing Cailleux and related techniques. Regression-line slopes produced by the two methods. however, are more comparable. Slope values are also less likely than Y-intercept values to be affected by differences in operators and initial roundness values; hence, regression ines from different streams are best compared with respect to their slopes rather than their Y intercepts. Slopes tend to be steepest for limestone, most gentle for quartz and quartzite, and intermediate in steepness for other lithologies. A comparison of these slopes with those computed from Kuenen's (1956) experiments on rounding in a pebble-bedded circular flume suggests that the flume is comparable to or better than streams in rounding effectiveness.