Errors / error function

One of the most efficient algorithms known for evaluating the Ewald sum is the Particle-mesh Ewald (PME) method of Darden et al. [8, 9]. The use of Ewald s trick of splitting the Coulomb sum into real space and Fourier space parts yields two distinct computational problems. The relative amount of work performed in real space vs Fourier space can be adjusted within certain limits via a free parameter in the method, but one is still left with two distinct calculations. PME performs the real-space calculation in the conventional manner, evaluating the complementary error function within a cutoff... 

Contrary to the impression that one might have from a traditional course in introductory calculus, well-behaved functions that cannot be integrated in closed form are not rare mathematical curiosities. Examples are the Gaussian or standard error function and the related function that gives the distribution of molecular or atomic speeds in spherical polar coordinates. The famous blackbody radiation cuiwe, which inspired Planck s quantum hypothesis, is not integrable in closed form over an arbitiar y inteiwal. 

Thus, we solve a two-point boundary value problem instead of a partial differential equation. When the diffiisivity is constant, the solution is the error function, a tabulated function. 

The error amplifier within the MC34025 has a totem-pole output circuit, which means that its output is not easily overriden. It will be used as a simple voltage follower and the error amplifier function will take place completely within the TT431 on the secondary side of the power supply. 

If the performance index or cost function J takes the form of a summed squared error function, then... 

In this chapter, the elution curve equation and the plate theory will be used to explain some specific features of a chromatogram, certain chromatographic operating procedures, and some specific column properties. Some of the subjects treated will be second-order effects and, therefore, the mathematics will be more complex and some of the physical systems more involved. Firstly, it will be necessary to express certain mathematical concepts, such as the elution curve equation, in an alternative form. For example, the Poisson equation for the elution curve will be put into the simpler Gaussian or Error function form. 

TABLE 7.12 Practical Modifications of the Gauss Error-Function Velocity ... 

Application of the Gauss error-function equation for velocity profile in the form proposed by Shepelev (Table 7.12) in Eq. (7.39) results in the following formula for the centerline velocity in Zone 3 of the compact jet ... 

Air velocity in each jet cross-section, described using the Reichardt Gauss error-function profile... 

The equation for the centerline temperature differential in Zone 3 of the compact jet derived" from Eq. (7.61) using the Gauss error-function temperature profile (Table 7.14) is... 

A velocity profile and a temperature difference profile have shapes that can be approximated by an error-function type curve. 

Once the question of assigning weights for the reference data has been decided, the fitting process can begin. It may be formulated in terais of an error function. 

The only remaining question is the nature of the error function. Pulay suggested the difference FDS — SDF (S is the overlap matrix), which is related to the gradient of the SCF energy with respect to the MO coefficients. This has been found to work well in practice. 

An error function depending on parameters. Only minima are of interest, and the global minimum is usually (but not always) desired. This may for example be determination of parameters in a force field, a set of atomic charges, or a set of localized Molecular Orbitals. 

Imbedded error function, 103 IND, 103 IND function, 103 Independent variables, 7 Indicator function, 102 empirical, 103 FRAC, 103 IE, 103... 

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