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The exponential function that appears in the formula of the stability radius of linear time-delay differential systems is approximated by its Padé approximant. This reduces the computation of the level sets of singular values in the stability radius formula to the computation of imaginary eigenvalues of special matrix polynomials. Then a bisection method is used for computing lower and upper bounds on the stability radius. A rounding error analysis is presented. Several numerical examples are given to demonstrate the feasibility and efficiency of the bisection method. [ABSTRACT FROM AUTHOR]

Numerical errors are insidious, difficult to predict and inherent in different levels of critical systems design. Indeed, numerical algorithms generally constitute approximations of an ideal mathematical model, which itself constitutes an approximation of a physical reality which has undergone multiple measurement errors. To this are added rounding errors due to computer arithmetic implementations, often neglected even if they can significantly distort the results obtained. This applies to Runge–Kutta methods used for the numerical integration of ordinary differential equations, that are ubiquitous to model fundamental laws of physics, chemistry, biology or economy. We provide a Coq formalization of the rounding error analysis of Runge–Kutta methods applied to linear systems and implemented in floating-point arithmetic. We propose a generic methodology to build a bound on the error accumulated over the iterations, taking gradual underflow into account. We then instantiate this methodology for two classic Runge–Kutta methods, namely Euler and RK2. The formalization of the results include the definition of matrix norms, the proof of rounding error bounds of matrix operations and the formalization of the generic results and their applications on examples. In order to support the proposed approach, we provide numerical experiments on examples coming from nuclear physics applications. [ABSTRACT FROM AUTHOR]

Lubrication is a crucial tool for industrial maintenance. By reducing friction between components, it also reduces wear on equipment. Industrial lubricants have specific properties and can operate efficiently within their operating temperature range. The aim of this study is therefore to forecast the temperature variation of lubrication oil using simple exponential smoothing. In this paper, we discuss one of the most popular forecasting methods in engineering and mechanical fields. The exponential smoothing method has been successful due to the quality of the results it presents. The present contribution aims to model the temperature variation of lubricating oil in a mechanical contact by the simple exponential smoothing method. The analysis and examination of the set of results made it possible to compare the smoothing models for different values of α and to choose the model closest to the actual temperature curve, the choice of the closest model serves to calculate the optimization criteria for the forecast; this method consists in choosing the α minimizing. The forecast results obtained by the SES model with α=0.9 are judged very satisfactory, and they prove the dominance of the SES model. [ABSTRACT FROM AUTHOR]

In our previous paper (Camargo, Numer. Algor., 85:591–606, 2020), we proved that the algorithms in a certain class of divided differences schemes are backward stable and, in particular, we proved that Neville's algorithm for Lagrange interpolation is backward stable for extrapolation for monotonically ordered nodes. That proof was based on a very particular pattern of the signs of the components of the divided differences which, in the case of Neville's algorithm for monotonically ordered nodes, is not satisfied when interpolation is considered instead of extrapolation. In this note we present a different argument that shows that Neville's algorithm is backward stable on the whole real line for monotonically ordered nodes. Our reasoning is based on a pyramid algorithm for the computation of Lebesgue functions. We also explain that obtaining sharp upper bounds for the numerical error in the computation of Neville's algorithm for generic sets of nodes is difficult. [ABSTRACT FROM AUTHOR]

Abstract Most Earth‐system simulations run on conventional central processing units in 64‐bit double precision floating‐point numbers Float64, although the need for high‐precision calculations in the presence of large uncertainties has been questioned. Fugaku, currently the world's fastest supercomputer, is based on A64FX microprocessors, which also support the 16‐bit low‐precision format Float16. We investigate the Float16 performance on A64FX with ShallowWaters.jl, the first fluid circulation model that runs entirely with 16‐bit arithmetic. The model implements techniques that address precision and dynamic range issues in 16 bits. The precision‐critical time integration is augmented to include compensated summation to minimize rounding errors. Such a compensated time integration is as precise but faster than mixed precision with 16 and 32‐bit floats. As subnormals are inefficiently supported on A64FX the very limited range available in Float16 is 6 × 10−5 to 65,504. We develop the analysis‐number format Sherlogs.jl to log the arithmetic results during the simulation. The equations in ShallowWaters.jl are then systematically rescaled to fit into Float16, using 97% of the available representable numbers. Consequently, we benchmark speedups of up to 3.8x on A64FX with Float16. Adding a compensated time integration, speedups reach up to 3.6x. Although ShallowWaters.jl is simplified compared to large Earth‐system models, it shares essential algorithms and therefore shows that 16‐bit calculations are indeed a competitive way to accelerate Earth‐system simulations on available hardware.

Most Earth‐system simulations run on conventional central processing units in 64‐bit double precision floating‐point numbers Float64, although the need for high‐precision calculations in the presence of large uncertainties has been questioned. Fugaku, currently the world's fastest supercomputer, is based on A64FX microprocessors, which also support the 16‐bit low‐precision format Float16. We investigate the Float16 performance on A64FX with ShallowWaters.jl, the first fluid circulation model that runs entirely with 16‐bit arithmetic. The model implements techniques that address precision and dynamic range issues in 16 bits. The precision‐critical time integration is augmented to include compensated summation to minimize rounding errors. Such a compensated time integration is as precise but faster than mixed precision with 16 and 32‐bit floats. As subnormals are inefficiently supported on A64FX the very limited range available in Float16 is 6 × 10−5 to 65,504. We develop the analysis‐number format Sherlogs.jl to log the arithmetic results during the simulation. The equations in ShallowWaters.jl are then systematically rescaled to fit into Float16, using 97% of the available representable numbers. Consequently, we benchmark speedups of up to 3.8x on A64FX with Float16. Adding a compensated time integration, speedups reach up to 3.6x. Although ShallowWaters.jl is simplified compared to large Earth‐system models, it shares essential algorithms and therefore shows that 16‐bit calculations are indeed a competitive way to accelerate Earth‐system simulations on available hardware. Plain Language Summary: Computational performance is a major limitation to improved weather and climate forecasts. Most Earth‐system simulations run on conventional computers with every calculation being performed with 64 bits at very high precision. However, the need for high‐precision calculations in the presence of large uncertainties of the climate system has been questioned. We present results with ShallowWaters.jl, the first fluid circulation model that runs entirely with 16‐bit precision, essentially making every calculation only to four digits accurate. Furthermore, only numbers between 6 × 10−5 and 65,504 are representable and we systemically rescale all calculations to not exceed this range, making use of 97% of all representable numbers within. Simulations with ShallowWaters.jl performed on modern hardware are almost 4x faster than the conventional high‐precision calculations. Although ShallowWaters.jl is simplified compared to large Earth‐system models, it shares essential algorithms and therefore shows that 16‐bit calculations are indeed a competitive way to accelerate Earth‐system simulations on available hardware. Key Points: The first fluid circulation model entirely based on 16‐bit instead of conventional 64‐bit calculations approaches 4x speedups on hardwareSystematic rescaling squeezes all calculations into the very limited range of Float16, making use of 97% of the available numbersCompensated summation in the precision‐critical time integration minimizes rounding errors from Float16 and is faster than mixed precision [ABSTRACT FROM AUTHOR]

The Kutzmann-Butcher method is the unique implicit four-stage Runge-Kutta method of order 8. In many problems in ordinary differential equations this method realized in floating-point arithmetic gives quite good approximations to the exact solutions, but the results obtained do not contain any information on rounding errors, representation errors and the error of the method. Thus, we describe an interval version of this method, which realized in floating-point interval arithmetic gives approximations (enclosures in the form of an interval) containing all these errors. The described method can also include data uncertainties in the intervals obtained. [ABSTRACT FROM AUTHOR]

Compensated algorithms consist in computing the rounding errors of individual operations and then adding them later on to the computed result. This makes it possible to increase the accuracy of the computed result efficiently. Computing the rounding error of an individual operation is possible through the use of a so-called error-free transformation. In this article, we show that it is possible to use compensated algorithms for having tight interval inclusions. We study compensated algorithms for summation, dot product and polynomial evaluation. We prove that the use of directed rounding makes it possible to get narrow inclusions with compensated algorithms. This is due to the fact that error-free transformations are no more exact but still sufficiently accurate to improve the numerical quality of results. [ABSTRACT FROM AUTHOR]

In this note, we extend the analysis of Camargo (J. Comp. Appl. Math 365, 2020) on the numerical stability of classical Newton's divided differences to a broader a class of divided differences algorithms that includes Neville's algorithm for the Lagrange interpolation and some of its particular instances as the Richardson extrapolation and Romberg quadrature. We show that these algorithms are backward stable and we bound the overall numerical error in their computation in finite precision. Our analysis solves a subtle question in numerical Lagrange interpolation that passed unnoticed so far. On the one hand, the Richardson extrapolation and Romberg quadrature have been extensively described as divided differences schemes and their connections with Neville's algorithm were already highlighted in the literature. On the other hand, the unique algorithm for computing Lagrange interpolants for which backward stability can be ensured (so far) for extrapolation is the first barycentric formula. By showing that Neville's algorithm is also backward stable for extrapolation, our result consolidates a solid background for the usual representation of the Richardson extrapolation and Romberg quadrature as divided differences schemes. [ABSTRACT FROM AUTHOR]

A novel steganalysis method for JPEG images is introduced that is universal in the sense that it reliably detects any type of steganography as well as small payloads. It is limited to quality factors 99 and 100. The detection statistic is formed from the rounding errors in the spatial domain after decompressing the JPEG image. The attack works whenever, during compression, the discrete cosine transform is applied to integer-valued signal. Reminiscent of the well-established JPEG compatibility steganalysis, we call the new approach the “reverse JPEG compatibility attack.” While the attack is introduced and analyzed under simplifying assumptions using reasoning based on statistical signal detection, the best detection in practice is obtained with machine learning tools. Experiments on diverse datasets of both grayscale and color images, five steganographic schemes, and with a variety of JPEG compressors demonstrate the universality and applicability of this steganalysis method in practice. [ABSTRACT FROM AUTHOR]

The generalization of barycentric coordinates to arbitrary simple polygons with more than three vertices has been a subject of study for a long time. Among the different constructions proposed, mean value coordinates have emerged as a popular choice, particularly due to their suitability for the non-convex setting. Since their introduction, they have found applications in numerous fields, and several equivalent formulas for their evaluation have been presented in the literature. However, so far, there has been no study regarding their numerical stability. In this paper, we aim to investigate the numerical stability of the algorithms that compute mean value coordinates. We show that all the known methods exhibit instability in some regions of the domain. To address this problem, we introduce a new formula for computing mean value coordinates, explain how to implement it, and formally prove that our new algorithm provides a stable evaluation of mean value coordinates. We validate our results through numerical experiments. • A new algorithm for evaluating mean value coordinates. • A formal proof that this algorithm is numerically stable. • A comparison of the numerical stability of all known algorithms for mean value coordinates. [ABSTRACT FROM AUTHOR]

A number of features of today's high-performance computers make it challenging to exploit these machines fully for computational science. These include increasing core counts but stagnant clock frequencies; the high cost of data movement; use of accelerators (GPUs, FPGAs, coprocessors), making architectures increasingly heterogeneous; and multiple precisions of floating-point arithmetic, including half-precision. Moreover, as well as maximizing speed and accuracy, minimizing energy consumption is an important criterion. New generations of algorithms are needed to tackle these challenges. We discuss some approaches that we can take to develop numerical algorithms for high-performance computational science, with a view to exploiting the next generation of supercomputers. [ABSTRACT FROM AUTHOR]

This paper is concerned with normwise errors in the LS estimation for linear regression. It provides probabilistic tail bounds for the normwise error between the computed least squares estimator and the parameter vector, when the least squares problem is solved in floating point arithmetic using either the normal equations method or a backward stable method (for example, using the Householder QR factorization or the singular value decomposition). These bounds are used to provide a condition under which the computationally more efficient normal equations method can safely be used instead of a backward stable method, without any loss of accuracy in the computed estimator. [ABSTRACT FROM AUTHOR]

In the present study a new multiple-precision arithmetic environment in MATLAB is developed based on exflib, a fast multiple-precision arithmetic in the programming language C, C++ and FORTRAN. We discuss design and implementation of interface between MATLAB and exflib without any modifications to exflib. Although the proposed design incurs the overhead in execution, it is more accurate and faster than Variable Precision Arithmetic (VPA), which is the official multiple-precision arithmetic environment in MATLAB. We also exhibit an efficiency of the proposed environment for reliable computation to overcome the difficulties caused by rounding errors. [ABSTRACT FROM AUTHOR]

Summary: This paper presents a case study investigation of numerical robustness of extended Kalman filters used for estimation of stochastic chemical systems with ill‐conditioned measurements. Here, we consider both a batch reactor model and that of a continuously stirred tank reactor. Our purpose is to explore performance of extended Kalman filtering–based state estimators when the measurement model becomes increasingly ill conditioned. In this way, we determine numerically robust methods, which are suitable for accurate estimation of stochastic chemical systems in the presence of round‐off and other disturbances. We examine both conventional filters and their square‐root forms. All these algorithms are implemented by means of the conventional matrix inversion used in their measurement update steps and the Moore‐Penrose pseudoinversion as well. Furthermore, the square‐root filters under investigation are obtained in two ways, namely, by solving square‐root moment differential equations and by square rooting the filter itself. We show that only the square‐root filters grounded in the second approach (with use of stable orthogonal decompositions) are numerically robust and provide the excellent estimation accuracy within all our ill‐conditioned stochastic chemical system scenarios considered in this paper. In addition, the convectional filters and the square‐root variants based on solving moment equations are rather sensitive to round‐off and may be useful and accurate if the chemical system at hand is rather well conditioned. [ABSTRACT FROM AUTHOR]

Abstract Algorithmic robustness is important in the contact interaction analysis of polyhedral blocks using discontinuous computation methods. Several robustness issues of contact analysis associated with the identification of four contact types are discussed here, including rounding errors of floating-point operations, criteria and tolerances in contact type identification, and restrictions of input parameters. This paper also proposes revised criteria to identify contact types and general rules to specify tolerances regarding contact searches, quasi-parallel edges, overlapping angle and maximum displacement in a time step. These rules facilitate robust contact analysis of polyhedral blocks in the discontinuous deformation analysis framework. [ABSTRACT FROM AUTHOR]

The aim of this paper is to recognize the attitude of the conjugate -Gradient Algorithms for solving linear systems Ax=b under the existence of rounding errors. The effect of matrix condition number of A on the relative error of the calculated series of approximations { xk} is analyzed. An especially appealing feature of the algorithm qualified is that error rating can be obtained very easily. Some examples are presented to support the theoretical results and to demonstrate the applicability and efficiency of the methods. The paper ends with some conclusions that sum up the finding of the study. The executed program for calculation is carried using "Matlb7". [ABSTRACT FROM AUTHOR]

Several different techniques intend to improve the accuracy of results computed in floating-point precision. Here, we focus on a method to improve the accuracy of the evaluation of rational functions. We present a compensated algorithm to evaluate rational functions. This algorithm is accurate and fast. The accuracy of the computed result is similar to the one given by the classical algorithm computed in twice the working precision and then rounded to the current working precision. This algorithm runs much more faster than existing implementation producing the same output accuracy. [ABSTRACT FROM AUTHOR]

Since data for statistical analysis are always given in a discretized form, observations contain not only measurement errors but also rounding errors which are determined by the discretization step. In this paper we consider situations where the rounding errors are considerable: they are comparable to or even greater (in average) than the measurement errors. It is shown that it can be reasonable to increase the measurement errors in order to reduce the error of the final result. [ABSTRACT FROM AUTHOR]

Numerical uncertainties from noisy inputs or finite-precision roundoff errors are unavoidable on resource-constrained systems. While techniques exist to compute worst-case bounds on these errors for arithmetic operations, these approaches do not generalize to programs which take discrete decisions. In this case, the more interesting quantity is the probability of the program making the wrong decision. In this paper, we study two approaches to compute a guaranteed bound on this probability: 1) exact probabilistic inference and 2) probabilistic static analysis. By themselves, they provide accuracy and scalability, respectively, but unfortunately not at the same time. We propose an extension to the latter approach which allows us to bound the probability tightly and fully automatically while scaling to small but interesting embedded examples. [ABSTRACT FROM AUTHOR]

Software aging is a problem that was discovered two decades ago. Since then, many research studies have investigated how to manage aging problems caused by memory leakage and accumulated round-off error through resource consumption prediction or state forecasting. When applying state prediction, the performances of various aging classification algorithms are compared by the prediction error. Since forecasting error is not a precise measure and must be estimated, the forecast error variance needs to be analyzed. In this work, we carefully analyze the forecast error variance by three steps. In the first step, we propose a method to decompose the variance by considering the influence of the data sampling process and data partition procedure. In the second step, we use an enhanced Friedman test and the Nemenyi post hoc test to analyze the influence of the data sampling process on the data partitioning procedure. In the last step, a corrected t-test is proposed to compare the performance of two off-the-shelf classification algorithms. The software comparison experiment is based on a real-time web environment. We end this work by proposing a set of feasible suggestions. [ABSTRACT FROM AUTHOR]

Compensated algorithms consist in computing the rounding errors of individual operations and then adding them later on to the computed result. This makes it possible to increase the accuracy of the computed result efficiently. Computing the rounding error of an individual operation is possible through the use of a so-called error-free transformation . In this article, we show that it is possible to validate the result of compensated algorithms using stochastic arithmetic. We study compensated algorithms for summation, dot product and polynomial evaluation. We prove that the use of the random rounding mode inherent to stochastic arithmetic does not change much the accuracy of compensated methods. This is due to the fact that error-free transformations are no more exact but still sufficiently accurate to improve the numerical quality of results. [ABSTRACT FROM AUTHOR]

Feedrate scheduling is one of the most important factors of computer numerical control (CNC) machining and has become a crucial research problem. In order to improve the machining accuracy and motion smoothness, an optimized feedrate scheduling method considering round-off error compensation based on S-shaped acceleration/deceleration (ACC/DEC) algorithm is proposed in this paper. There are two main stages, namely initial feedrate scheduling and parameters calculation of round-off error compensation. In the stage of initial feedrate scheduling, a novel time rounding principle is introduced to reduce the round-off error. Meanwhile, the motion parameters of each section can be calculated based on the proposed feedrate scheduling method. The constant feedrate section used for round-off error compensation is always guaranteed to exist although the actual maximum feedrate might be smaller than the command feedrate. Then, in the stage of compensation parameters calculation, the round-off error can be obtained based on the proposed time rounding principle and the scheduled parameters should be updated. In order to maintain the continuity of the acceleration profile, the improved trapezoidal ACC/DEC algorithm is introduced to conduct the error compensation and the crucial parameters can be calculated based on its special properties. In addition, the feedrate look-ahead strategy is also tweaked to enhance the reliability of feedrate scheduling. Finally, a series of simulations and practical experiments with two non-uniform rational B-spline (NURBS) curves are conducted to verify the good performance and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

To better support multiscale modeling and simulation, we present a multiscale time representation consisting of data types, data structures, and algorithms that collectively support the recording of past events and scheduling of future events in a discrete event simulation. Our approach addresses the drawbacks of conventional time representations: limited range in the case of 32- or 64-bit fixed-point time values; problematic rounding errors in the case of floating-point numbers; and the lack of a universally acceptable precision level in the case of brute force approaches. The proposed representation provides both extensive range and fine resolution in the timing of events, yet it stores and manipulates the majority of event times as standard 64-bit numbers. When adopted for simulation purposes, the representation allows a domain expert to choose a precision level for his/her model. This time precision is honored by the simulator even when the model is integrated with other models of vastly different time scales. Making use of C++11 programming language features and the Discrete Event System Specification formalism, we implemented a simulator to test the time representation and inform a discussion on its implications for collaborative multiscale modeling efforts. [ABSTRACT FROM AUTHOR]

In this paper, an iterative algorithm for solving a generalized coupled Sylvester-conjugate matrix equations over Hermitian R -conjugate matrices given by A 1 V B 1 + C 1 W D 1 = E 1 V ¯ F 1 + G 1 and A 2 V B 2 + C 2 W D 2 = E 2 V ¯ F 2 + G 2 is presented. When these two matrix equations are consistent, the convergence theorem shows that a solution can be obtained within finite iterative steps in the absence of round-off error for any initial arbitrary Hermitian R -conjugate solution matrices V 1 , W 1 . Some lemmas and theorems are stated and proved where the iterative solutions are obtained. A numerical example is given to demonstrate the behavior of the proposed method and to support the theoretical results. [ABSTRACT FROM AUTHOR]

Rounding is a necessary step in many mathematical processes. We are taught early in our education about significant figures and how to properly round a number. So when we are given a data set and asked to find a regression line, we are inclined to offer the line with rounded coefficients to reflect our model. However, the effects are not as insignificant as they might seem at first. In this paper, we investigate some consequences of rounding the coefficients in a least squares linear regression with respect to the calculated value of R2, and consider ways to minimize the amount of error that can arise. [ABSTRACT FROM AUTHOR]

Latest developments in satellite communications encompass semitransparent transponder architectures. Such architectures have potentials: 1) to operate with enhanced frequency planning flexibility and physical layer specifications of evolving communications standards and 2) to support dynamic reconfiguration of connectivity plans. In this frame, significant on-board digital processing is involved, which calls for careful system modeling and accurate digital hardware design to achieve feasible tradeoffs between hardware efficiency and overall link-budget performance. In our recent works, we have proposed and validated a quantitative framework for performance analysis and design of digital transparent processors (DTPs): the model relies on an extended notion of noise figure that allows a radio link-budget to natively incorporate the nonideal behavior (e.g., quantization and rounding errors and linear distortions) of the various DTP segments. This paper complements the previous works by providing: 1) an explicit computation of implementation complexity of a DTP and 2) a methodology to carry out a complete design flow that moves from system requirements and brings to a detailed definition of digital hardware (HW) components in the DTP. Through numerical examples, we demonstrate that, given a requirement on the overall link budget performance in contexts of practical interest, significant differences in HW complexity are obtained for different design choices. [ABSTRACT FROM PUBLISHER]

Pipelined Krylov subspace methods typically offer improved strong scaling on par-allel HPC hardware compared to standard Krylov subspace methods for large and sparse linear systems. In pipelined methods the traditional synchronization bottleneck is mitigated by overlap- ping time-consuming global communications with useful computations. However, to achieve this communication-hiding strategy, pipelined methods introduce additional recurrence relations for a number of auxiliary variables that are required to update the approximate solution. This paper aims at studying the in uence of local rounding errors that are introduced by the additional recurrences in the pipelined Conjugate Gradient (CG) method. Specifically, we analyze the impact of local round-off effects on the attainable accuracy of the pipelined CG algorithm and compare it to the tra- ditional CG method. Furthermore, we estimate the gap between the true residual and the recursively computed residual used in the algorithm. Based on this estimate we suggest an automated residual replacement strategy to reduce the loss of attainable accuracy on the final iterative solution. The resulting pipelined CG method with residual replacement improves the maximal attainable accuracy of pipelined CG while maintaining the efficient parallel performance of the pipelined method. This conclusion is substantiated by numerical results for a variety of benchmark problems. [ABSTRACT FROM AUTHOR]

In this paper, accurate argument calculation for sine-fitting algorithms is investigated, assuming floating-point (FP) arithmetic. An easy-to-implement incremental calculation technique is suggested. In order to decrease error propagation, the algorithm is complemented with an advanced summation technique. Theoretical and numerical analyses on computational demand are performed to highlight that incremental argument calculation outperforms the method proposed in former research. Furthermore, an algorithm is implemented to mitigate the effect of imprecise representation of frequency on FP arithmetic. Monte Carlo analyses are carried out to demonstrate the accuracy of the suggested algorithms. Results show that phase information can be evaluated precisely even with single-precision FP arithmetic, applying incremental argument calculation. By this means, the cost of equipment that is needed to perform sine fitting can be reduced significantly. Finally, possible application areas are shown to demonstrate the applicability of the suggested solutions in the state-of-art measurement procedures. [ABSTRACT FROM PUBLISHER]

The algorithm by Al-Mohy and Higham (2011) [2] computes an approximation to e A b for given A and b , where A is an n -by- n matrix and b is, for example, a vector of dimension n . It uses a scaling together with a truncated Taylor series approximation to the exponential of the scaled matrix. In this paper, a method is developed for estimating the roundoff error of the computed solution. An asymptotic expansion of this error for small values of the unit roundoff is the basis of the method. The roundoff error is further expressed in terms of sums of rounding errors, which occur during the computation. A second approximation to e A b , which is computed with a lower precision than the first one, is used to evaluate these rounding errors in practice. The result is an upper bound on the normwise relative roundoff error. Further, an algorithm is proposed for computing the error bound. The cost for performing this algorithm depends on the type of problem and the accuracy, which is required for the error estimate. In case that all computations are performed with standard precisions, this cost can be expressed in terms of the number of computed matrix-vector products and is bounded from above by two times the cost for computing e A b . [ABSTRACT FROM AUTHOR]

The roundoff-error-free (REF) LU and Cholesky factorizations, combined with the REF substitution algorithms, allow rational systems of linear equations to be solved exactly and efficiently by working entirely in integer arithmetic. These REF computational tools share two key properties: their constituent divisions are exact, and their matrix entries' bit-lengths (i.e., required number of storage bits) are bounded polynomially. However, as this paper explains, updating the REF factorizations via the common delete-insert-reduce approach turns out to be inefficient in terms of operand bit-length growth and increased computational effort. In fact, we prove that this inefficiency can eliminate the computational savings expected of the factorization update process. Hence, the current work develops REF update algorithms that differ significantly from their traditional counterparts. To achieve this, it introduces the frame matrix methodology for representing and applying the REF factorizations, and then it devises new REF operations and shortcuts for transitioning between "nearby" frame matrices. The presented push-and-swap update approach utilizes these transitions to preserve the special structure of the REF factorizations and to avoid additional growth in their matrix entries' bit-lengths; this update process requires the expected O(n²) operations. The featured updates are column addition, deletion, and replacement with respect to the REF LU factorization. We also prove that analogous row updates can be performed via the column updates and discuss special considerations for updating the REF Cholesky factorization. An accompanying set of computational tests attests that for fully dense random instances, the push-and-swap column replacement update for the REF LU factorization is on average an order of magnitude faster than the corresponding delete-insert-reduce update for the exact rational arithmetic LU factorization. Altogether, the REF update algorithms represent a promising preliminary step toward enhancing the basic solution validation subroutines employed by state-of-the-art mixed-precision methodologies for exact linear and mixed-integer programming. [ABSTRACT FROM AUTHOR]

We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretization errors with a new term accounting for the propagation of numerical round-off errors, showing that the accumulated round-off error is inversely proportional to the square root of the step size. As a consequence, the numeric precision eventually sets the limit for the pointwise computability of accurate solutions of any ODE. The theoretical results are supported by numerically computed solutions and error estimates for the Lorenz system and the van der Pol oscillator. [ABSTRACT FROM AUTHOR]

In this paper, for given,, an iterative algorithm is constructed to find the solutions of generalized inverse eigenvalue problem, whereAandBshould be partially bisymmetric under a prescribed submatrix constraint. For any initial constrained matrices, a solution paircan be obtained in finite iteration steps by this iterative algorithm in the absence of roundoff errors. The least norm solution can be obtained by choosing a special kind of initial matrix pencil. In addition, the unique optimal approximation solution to a given matrix pencil in the solution set of the above problem can also be obtained by this iterative algorithm. Numerical examples are given to illustrate efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The effect of round-off errors on the solution of numerical heat transfer is illustrated by a simple example both analytically and numerically. It is found that the upper bound of the round-off error under both conditions with or without an inner heat source is proportional to the square of grid number—n2. Increase in grid number might lead to larger round-off errors. The magnitude of relative round-off error is also determined by the specific problem. Proper treatment of the computation procedure can reduce the round-off error obviously. The precision can be improved with this method without occupation of additional computational resources. [ABSTRACT FROM PUBLISHER]

This paper discusses the conjugate gradient least squares algorithm for solving the generalized coupled Sylvester matrix equations ∑ j = 1 q A i j X j B i j = F i , i = 1 , 2 , … , p . We prove that if this system is consistent then the iterative solution converges to the exact solution and if this system is inconsistent then the iterative solution converges to the least squares solution within the finite iteration steps in the absence of the roundoff errors. Also by setting the initial iterative value properly we prove that the iterative solution converges to the least squares and minimum-norm solution. [ABSTRACT FROM AUTHOR]

We describe the reversibility error method (REM) and its applications to planetary dynamics. REM is based on the time-reversibility analysis of the phase-space trajectories of conservative Hamiltonian systems. The round-off errors break the time reversibility and the displacement from the initial condition, occurring when we integrate it forward and backward for the same time interval, is related to the dynamical character of the trajectory. If the motion is chaotic, in the sense of non-zero maximal Lyapunov characteristic exponent (mLCE), thenREMincreases exponentially with time, as exp λt, while when the motion is regular (quasi-periodic), then REM increases as a power law in time, as tα, where α and λ are real coefficients. We compare the REM with a variant of mLCE, the mean exponential growth factor of nearby orbits. The test set includes the restricted three-body problem and five resonant planetary systems: HD 37124, Kepler-60, Kepler-36, Kepler-29 and Kepler-26.We found a very good agreement between the outcomes of these algorithms. Moreover, the numerical implementation of REM is astonishing simple, and is based on solid theoretical background. The REM requires only a symplectic and time-reversible (symmetric) integrator of the equations of motion. This method is also CPU efficient. It may be particularly useful for the dynamical analysis of multiple planetary systems in the Kepler sample, characterized by low-eccentricity orbits and relatively weak mutual interactions. As an interesting side result, we found a possible stable chaos occurrence in the Kepler-29 planetary system. [ABSTRACT FROM AUTHOR]

A method for experimental evaluation of the magnetic-field attenuation coefficients under a pulsed action is considered. A technique for the preliminary processing of measurement results for low useful signal-to-noise ratios is considered taking the features of the operation of digital oscilloscopes into account. The analysis of the spectral composition of measured signals and a method for their approximation in the time domain, which allows elimination of the influence of measuring-sensor noise and digital-oscilloscope rounding errors, are presented. The results of estimating the frequency dependence of the magnetic-field attenuation coefficients for a copper shield in comparison with the results of a theoretical calculation are presented. The considered method allows simultaneous determination of the coefficients of the magnetic-field attenuation by the shield in a wide frequency range. [ABSTRACT FROM AUTHOR]

Numerical software, common in scientific computing or embedded systems, inevitably uses a finite-precision approximation of the real arithmetic in which most algorithms are designed. In many applications, the roundoff errors introduced by finite-precision arithmetic are not the only source of inaccuracy, and measurement and other input errors further increase the uncertainty of the computed results. Adequate tools are needed to help users select suitable data types and evaluate the provided accuracy, especially for safety-critical applications. We present a source-to-source compiler called Rosa that takes as input a real-valued program with error specifications and synthesizes code over an appropriate floating-point or fixed-point data type. The main challenge of such a compiler is a fully automated, sound, and yet accurate-enough numerical error estimation. We introduce a unified technique for bounding roundoff errors from floating-point and fixed-point arithmetic of various precisions. The technique can handle nonlinear arithmetic, determine closed-form symbolic invariants for unbounded loops, and quantify the effects of discontinuities on numerical errors. We evaluate Rosa on a number of benchmarks from scientific computing and embedded systems and, comparing it to the state of the art in automated error estimation, show that it presents an interesting tradeoff between accuracy and performance. [ABSTRACT FROM AUTHOR]