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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]

This article considers the numerical inversion of the power series p(x)=1 + b1x + b²x² +· · · to compute the inverse series q(x) satisfying p(x)q(x)= 1. Numerical inversion is a special case of triangular back-substitution, which has been known for its beguiling numerical stability since the classic work of Wilkinson (1961, Error analysis of direct methods of matrix inversion. J. Assoc. Comput. Mach., 8, 281-330). We prove the numerical stability of inversion of power series and obtain bounds on numerical error. A range of examples show that these bounds overestimate the error by only a few digits. When p(x) is a polynomial and x = a is a root with p(a) = 0, we show that root deflation via the simple division p(x)/(x - a) can trigger instabilities relevant to polynomial root finding and computation of finite-difference weights. When p(x) is a polynomial, the accuracy of the computed inverse q(x) is connected to the pseudozeros of p(x). [ABSTRACT FROM AUTHOR]

Financial prices are often discretized--with smallest tick size of one cent, for example. Thus prices involve rounding errors. Rounding errors affect the estimation of volatility, and understanding them is critical, particularly when using high frequency data. We study the asymptotic behavior of realized volatility (RV), which is commonly used as an estimator of integrated volatility. We prove the convergence of the RV and scaled RV under varous conditions on the rounding level and the number of observations. A bias-corrected volatility estimator is proposed and an associated central limit theorem is shown. The simulation and empirical results demonstrate that the proposed method can yield substantial statistical improvement. [ABSTRACT FROM AUTHOR]

The asymptotic behaviour of a class of least squares problems when subjected to structured perturbations is considered. It is permitted that the number of rows (observations) in the design matrix can be unbounded while the number of degrees of freedom (variables) is fixed. It is shown that for certain classes of random data the solution sensitivity depends asymptotically on the condition number of the design matrix rather than on its square, which is the generic result for inconsistent systems when the norm of the residual is not small. Extension of these results to the case where the perturbations are due to rounding errors is considered. [ABSTRACT FROM PUBLISHER]

With auxiliary information that is well correlated with the primary variable of interest, ratio estimation of the finite population total may be much more efficient than alternative estimators that do not make use of the auxiliary variate. The well-known properties of ratio estimators are perturbed when the auxiliary variate is measured with error. In this contribution we examine the effect of measurement error in the auxiliary variate on the design-based statistical properties of three common ratio estimators. We examine the case of systematic measurement error as well as measurement error that varies according to a fixed distribution. Aside from presenting expressions for the bias and variance of these estimators when they are contaminated with measurement error we provide numerical results based on a specific population. Under systematic measurement error, the biasing effect is asymmetric around zero, and precision may be improved or degraded depending on the magnitude of the error. Under variable measurement error, bias of the conventional ratio-of-means estimator increased slightly with increasing error dispersion, but far less than the increased bias of the conventional mean-of-ratios estimator. In similar fashion, the variance of the mean-of-ratios estimator incurs a greater loss of precision with increasing error dispersion compared with the other estimators we examine. Overall, the ratio-of-means estimator appears to be remarkably resistant to the effects of measurement error in the auxiliary variate. [ABSTRACT FROM AUTHOR]

Discusses the stability and performance analysis of a block elimination solver for bordered linear systems. Use of perturbation approach to enhance the stability; Experiments illustrating the performance of the algorithm; Govaerts-Pryce algorithm; Roundoff error analysis.