Errors accumulated

In order to compare the efficiency of the SISM with the standard LFV method, we compared computational performance for the same level of accuracy. To study the error accumulation and numerical stability we monitored the error in total energy, AE, defined as... 

Log D predictions are more difficult as most approaches rely on the combination of estimated log P and estimated pK. Obviously, this can lead to error accumulation and errors of 2 log units or more can be found. Some algorithms, however, are designed to learn from experimental data so that the predictions improve over time. 

It has become an accepted wisdom that the use of RMs or CRMs will help to improve the accuracy and precision of an analytical process. This belief has led to a rapid growth in the use of RMs and CRMs in commercial laboratories. The authors and many analysts the world over support this view, but also recognize that in far too many cases inexperience and carelessness conspire together with the result that error accumulates and often unreliable data are produced. 

Uneven distributions of residuals. The MaxEnt calculations in presence of an overall chi-square constraint suffer from highly non-uniform distributions of residuals, first reported and discussed by Jauch and Palmer [29, 30] the error accumulates on a few strong reflexions at low-resolution. The phenomenon is only partially cured by devising an ad hoc weighting scheme [20,31, 32]. Carvalho et al. have discussed this topic, and suggested that the recourse to as many constraints as degrees of freedom would cure the problem [33]. 

Error propagation analysis is the estimation of error accumulation in a final result as a consequence of error in the individual components used to obtain the result. Given an equation explicitly expressing a result, the error propagation equation can be used to estimate the error in the result as a function of error in the other variables. 

Both the gravimetric and volumetric methods are subject to cumulative errors. The continuous flow method produces each data point independently and is not subject to error accumulation. 

It replaces Eq. (A.l) by a similar formula for XN + l. The Runga-Kutta-Gill method can be programmed to minimize storage and rounding error accumulation.47 When this is done, it requires storage equivalent to XN, XN-t, VN, VN-U and f(XN, tN). It has a very small truncation error and is self-starting. However, it requires 4 derivative evaluations per step. 

The solution to the first problem is limited by the increase in time or the computer capacity available to solve more complete or more advanced equations. The second problem is even more difficult to acknowledge. It may be due to error accumulation through the nonlinear domain. The numerical solution of a differential equation is based on the approximation of time and, in the case of PDEs, space partial derivatives, by finite-difference equivalents. 

The error accumulated during a calculation due to rounding intermediate results, rounding... 

There are many ways to combine the various finite differences that may be used for each of the terms of the mass balance equation, and there are as many ways to approximate a partial differential equation by a finite-difference scheme. The choice is limited in practice, however, for two reasons. First, we need the numerical calculation to be stable, and there is a condition to satisfy to achieve numerical stability. Second, we need to control the numerical errors that are made during the calculations. Replacing a partial difference term with any of the possible finite difference terms gives a tnmcation error. These tnmcation errors accumulate during the calculation of a numerical solution. The error contribution... 

Ability to take many data points without compromising accuracy through error accumulation. 

In many physical applications the Hamiltonian is explicitly time dependent. The common solution for propagation in these explicitly time-dependent problems is to use very small grid spacing in time, such that within each time step the Hamiltonian H(r) is almost stationary. Under these semistationary conditions a short-time propagation method in the time-energy phase space is employed. The drawback of this solution is that it is based on extrapolation therefore the errors accumulate. Moreover, time ordering errors add with the usual numerical dispersion errors (108). 

A global solution to the error accumulation problem is obtained by embedding in the time-energy phase space of the system in a larger phase space where interpolation in the energy domain becomes possible (111,114). The first step is to add a grid in a new coordinate The relation between the embedded wave function and the usual one subject to an initial state (jf, 0) is defined as... 

Ans. Yes. The error in replication is not a problem since many correct copies (99,999 out of each 100,000) of the mRNA and polypeptide will be produced and this is sufficient for the proper functioning of the cell. The one incorrect (out of 100,000) copy of the mRNA and polypeptide will not be used by the cell. Furthermore, the error is not propagated to subsequent generations of cells or to offspring. A much higher accuracy is needed for replication because errors accumulate and are transmitted to subsequent generations of cells and offspring. 

In conclusion, the linearization of the evolution equations for the analysis of chemical and biochemical networks is unpredictably limited. For the linearization to be valid it is necessary to use small perturbations, for which the experimental errors are very large. In the papers that have appeared on this subject, insufficient (or no) attention has been given to error accumulation and propagation [25]. The response approaches developed here avoid linearization and hence are to be preferred. 

Unsteady development of such a flow w also investigated numerically and analytically in Ref. [92] both for a constant flow rate and constant pressure drop. In the analytical approach, it was considered that the fiow stops after a ten-fold viscosity growth. A step model of viscosity variation was also used, but of more complex form than is assumed by Eq. (3). A comparison with the experiment has shown that the analytical model describes better longer periods of time, while the numerical approach works better in the initial stages. This indicates only an unsuccessful choice of the computational algorithm and demonstrates a typical process of error accumulation during calculations caused first of aU by a too roi fixed two-dimtai-sional network (17 x 13 in the most exact case). 

Kragten and Decnop-Weever (1978, 1979, 1980, 1982, 1983a, b, 1984, 1987) conducted a series of solubility measurements over a wide range of pH (ca. 6 15). The dependence of the precipitation on pH allowed calculation of formation constants of mononuclear and polynuclear species which were internally consistent. The accuracy of the calculated constants was estimated to yield uncertainties of ca. +0.1-0.2 log units. The accuracy of successive stability constants (as p and/or q increased) decreased as a result of error accumulation in the fitting procedure. Nevertheless, the results, in general, indicate that the hydrolysis constants for stepwise formation of higher hydroxo species are of the same order of magnitude as those for the formation of Ln(OH) +. 

The rate of accumulation of error due to the thermostat is also difficult to quantify analytically, since it requires calculation of the error in an autocorrelation function over a time interval which may not be short compared to the simulation stepsize, particularly for slow-relaxing systems. Again, an estimate is obtainable by considering the rate of error accumulation at short times (obtained by Taylor expansion of the exact and approximate autocorrelation functions). The result of these two estimates is then an analytical estimate for the efficiency that relates various schemes for various values of the parameters involved. 

The equation is still undefined for pure species condition, however, the round-off error accumulates to the same degree in the numerator and denominator. The Knudsen diffusion coefficient for the species k in Eq. 4.5 is defined as... 

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