Numerical errors

Furthermore, since analytical derivatives are subject to user input error, numerical evaluation of the derivatives can also be used in a typical computer implementation of the Gauss-Newton method. Details for a successful implementation of the method are given in Chapter 8. 

Two tests are available to ensure that the equations of motion are solved correctly in an MD simulation. The first test assesses the conservation of energy. In the absence of terms explicitly dependent on time, the Hamiltonian of the system must be a conserved quantity, and integrating Newton s equations of motion should conserve the total energy of the system E. In fact, small fluctuations of the energy will result from the finite difference approach and round-off errors. Numerical criteria often used for the energy fluctuations are... 

Error, numeric exception division by zero The limit command is applied to n = 1 ... 

Figure 5. Seasonal changes in plasma-freezing-temperature depression and Na+ and Cl concentrations of hypophysectomized (HYPEX) and sham-operated winter flounder. Values are plotted as means 1 standard error. Numerals at each point represent the number of fish sampled. The initial values for September (S) are pre-operated concentra tions. Modified from Fletcher et al...

As Kc increases, we need an iterative, trial-and-error, numerical procedure to find the roots of the characteristic equation. Such a solution is feasible through the use of a digital computer. Table 15.1 shows how the locations of the four roots change with the value of Kc. These results have been transferred in Figure 15.7, which displays the four branches of the root locus for the closed-loop reactor system. 

Referring to Fig. 171, let MP and M P be drawn at equal distances from Oy in such a way that the area bounded by these lines, the curve, and the a-axis (shaded part in the figure), is equal to half the whole area, bounded by the whole curve and the rc-axis, then it will be obvious that half the total observations will have errors numerically less than OM, and half, numerically greater... 

From W. Crookes ten determinations of the atomic weight of thallium (above) calculate the probability that the atomic weight of thallium lies between 208 632 and 208 652. Here x = + 0 01 B = 0 0023 . . t= x/B=4 4. From Table XI., P=0 997. (Note how near this number is to unity indicating certainty.) The chances are 382 to 1 that the true value of the atomic weight of thallium lies between 203 632 and 208 652. We get the same result by means of Table X. Thus h=0 4769 0 0023=207 . . hx=207 x 0 01=2 07. When hx= 2 07, P=0 997. If 1,000 observations were made under the same conditions as Crookes , we could reasonably expect 997 of them to be affeoted by errors numerically less than 0 01, and only 8 observations would be affected by errors exceeding these limits. 

A solution requires establishing the necessary and sufficient relationships between y, x, and V or L, where the rate equation can be numerically integrated, at the same time yielding the behavior of A. This involves assigning or establishing the boundary conditions. Additional considerations are as follows. The nature of the set of equations is such that trial-and-error numerical stepwise procedures are required. Furthermore, the complexity is such that even the solution for a two-component system is a formidable undertaking. 

Every care must be taken when developing an algorithm or conducting a simulation to minimize these errors. Numerical errors not only impact the accuracy of the simulation results, they also affect the numerical stability of the algorithm. Accumulation of numerical errors may result in numerical overflow (e.g., when a variable becomes larger than the maximum value a computer can handle) and a program crash. 

A new one-dimensional mierowave imaging approaeh based on suecessive reeonstruetion of dielectrie interfaees is described. The reconstruction is obtained using the complex reflection coefficient data collected over some standard waveguide band. The problem is considered in terms of the optical path length to ensure better convergence of the iterative procedure. Then, the reverse coordinate transformation to the final profile is applied. The method is valid for highly contrasted discontinuous profiles and shows low sensitivity to the practical measurement error. Some numerical examples are presented. 

E. Hairer and Ch. Lubich. The life-span of backward error analysis for numerical integrators. Numer. Math. 76 (1997) 441-462... 

The heightened appreciation of resonance problems, in particular, has been quite recent [63, 62], and contrasts the more systematic error associated with numerical stability that grows systematically with the discretization size. Ironically, resonance artifacts are worse in the modern impulse multiple-timestep methods, formulated to be symplectic and reversible the earlier extrapolative variants were abandoned due to energy drifts. 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

We have in mind trajectory calculations in which the time step At is large and therefore the computed trajectory is unlikely to be the exact solution. Let Xnum. t) be the numerical solution as opposed to the true solution Xexact t)- A plausible estimate of the errors in X um t) can be obtained by plugging it back into the differential equation. 

The last approximation is for finite At. When the equations of motions are solved exactly, the model provides the correct answer (cr = 0). When the time step is sufficiently large we argue below that equation (10) is still reasonable. The essential assumption is for the intermediate range of time steps for which the errors may maintain correlation. We do not consider instabilities of the numerical solution which are easy to detect, and in which the errors are clearly correlated even for large separation in time. Calculation of the correlation of the errors (as defined in equation (9)) can further test the assumption of no correlation of Q t)Q t )). 

Hence, we use the trajectory that was obtained by numerical means to estimate the accuracy of the solution. Of course, the smaller the time step is, the smaller is the variance, and the probability distribution of errors becomes narrower and concentrates around zero. Note also that the Jacobian of transformation from e to must be such that log[J] is independent of X at the limit of e — 0. Similarly to the discussion on the Brownian particle we consider the Ito Calculus [10-12] by a specific choice of the discrete time... 

One of the advantages of the Verlet integrator is that it is time reversible and symplectic[30, 31, 32]. Reversibility means that in the absence of numerical round off error, if the trajectory is run for many time steps, say nAt, and the velocities are then reversed, the trajectory will retrace its path and after nAt more time steps it will land back where it started. An integrator can be viewed as a mapping from one point in phase apace to another. If this mapping is applied to a measurable point set of states at on(> time, it will... 

T. R. Littell, R. D. Skeel, and M. Zhang. Error analysis of symplectic multiple time stepping. SIAM J. Numer. Anal., 34 1792-1807, 1997. 

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... 

These various techniques were recently applied to molecular simulations [11, 20]. Both of these articles used the rotation matrix formulation, together with either the explicit reduction-based integrator or the SHAKE method to preserve orthogonality directly. In numerical experiments with realistic model problems, both of these symplectic schemes were shown to exhibit vastly superior long term stability and accuracy (measured in terms of energy error) compared to quaternionic schemes. 

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