Numerical precision

The sampling precision of the measured data depends on the signal amplitude. The difference between simulated and experimental data can be mainly explained by the low numerical precision of the measured data. 

The variational theorem which has been initially proved in 1907 (24), before the birthday of the Quantum Mechanics, has given rise to a method widely employed in Qnantnm calculations. The finite-field method, developed by Cohen andRoothan (25), is coimected to this method. The Stark Hamiltonian —fi.S explicitly appears in the Fock monoelectronic operator. The polarizability is derived from the second derivative of the energy with respect to the electric field. The finite-field method has been developed at the SCF and Cl levels but the difficulty of such a method is the well known loss in the numerical precision in the limit of small or strong fields. The latter case poses several interconnected problems in the calculation of polarizability at a given order, n ... 

Within the Matlab s numerical precision X is singular, i.e. the two rows (and columns) are identical, and this represents the simplest form of linear dependence. In this context, it is convenient to introduce the rank of a matrix as the number of linearly independent rows (and columns). If the rank of a square matrix is less than its dimensions then the matrix is call rank-deficient and singular. In the latter example, rank(X)=l, and less than the dimensions of X. Thus, matrix inversion is impossible due to singularity, while, in the former example, matrix X must have had full rank. Matlab provides the function rank in order to test for the rank of a matrix. For more information on this topic see Chapter 2.2, Solving Systems of Linear Equations, the Matlab manuals or any textbook on linear algebra. 

Chemistry - as a scientific and technological discipline - has some unique characteristics. In contrast to physics, where most of the underlying laws can be given in explicit and sometimes simple mathematical form, many of the laws governing chemical phenomena are either not explicitly known, or else have a mathematical form that still eludes an exact solution. Still, chemistry does provide - and rests on- quantitative data of physical or chemical properties of high numerical precision. A search for quantitative relationships is thus suggested, despite the lack of a tractable theoretical basis. 

When examined together, these values provide mathematical profiles of the drugs we studied and allow numerically precise comparisons with other agents. The use of operational definitions not only makes it simpler to characterize a drug, but allows accurate predictions of its effects at various doses. Obviously, such predictions have practical as well as academic significance. 

The third numerical detail that is important in Fig. 8.1 arises from the fact that Ag is a metal and the observation that the DOS is obtained from integrals in k space. In Section 3.1.4 we described why performing integrals in k space for metals holds some special numerical challenges. The results in Fig. 8.1 were obtained by using the Methfessel and Paxton smearing method to improve the numerical precision of integration in k space. 

The detailed analysis of this situation should include the simultaneous radial dispersion of heat and matter, and maybe axial dispersion too. In setting up the mathematical model, what simplifications are reasonable, would the results properly model the real situation, would the solution indicate unstable behavior and hot spots These questions have been considered by scores of researchers, numerous precise solutions have been claimed however, from the point of view of prediction and design the situation today still is not as we would wish. The treatment of this problem is quite difficult, and we will not consider it here. A good review of the state-of-the-art is given by Froment (1970), and Froment and Bischoff (1990). 

In short, Dalton s atomic theory allowed chemistry to become an exact science. The importance of making numerically precise measurements of chemical processes had been clear enough to Cavendish, Priestley, Lavoisier, and their contemporaries but, without an underlying theory of the elements, these numbers were simply codifications of empirical observations. They were like measurements of the depth of a river or the number of ants in a colony - they did not reveal anything about the fundamental constitution of the system. For Lavoisier, questions about the invisible particles of matter were irrelevant to chemistry s aims. 

Results. In Table 5.1 we compare a few results of classical, semi-classical and quantum moment calculations. An accurate ab initio dipole surface of He-Ar is employed (from Table 4.3 [278]), along with a refined model of the interaction potential [12]. A temperature of 295 K is assumed. The second line, Table 5.1, gives the lowest three quantum moments, computed from Eqs. 5.37, 5.38, 5.39 the numerical precision is believed to be at the 1% level. For comparison, the third line shows the same three moments, obtained from semi-classical formulae, Eqs. 5.47 along with 5.37 with the semi-classical pair distribution function inserted. We find satisfactory agreement. We note that at much lower temperatures, and also for less massive systems, the semi-classical and quantal results have often been found to differ significantly. The agreement seen in Table 5.1 is good because He-Ar at 295 K is a near-classical system. 

Riefler, R. G., and Ahlfeld, D. P. (1996). The impact of numerical precision on the solution of confined and unconfined optimal hydraulic control problems. Hazardous Waste and Hazardous Materials, 13(2), 167-176. 

The values have not been rounded to the allowed figures to avoid loss of numerical precision in subsequent calculations.)... 

Quantum mechanics has become a great tool for chemistry. For this reason, the methods of quantum mechanics used in chemistry have been grouped into a field called quantum chemistry, as in the title of this serial. The techniques of quantum chemistry were developed at a tremendous rate by the combined efforts of pure quantum theorists, application specialists, and scientific programmers, along with feedback from numerous precision experiments. These techniques have become routine in most universities worldwide, and courses on quantum chemistry at all levels are part of their curricula. 

The precision of an instrument must be considered. Many typical measurements, for example, in atomic spectroscopy, are recorded to only two significant figures. Consider a dataset in which about 95 % of the readings were recorded between 0.10 and 0.30 absorbance units, yet a statistically designed experiment tries to estimate 64 effects. The /-test provides information on the significance of each effect. However, statistical tests assume that the data are recorded to indefinite accuracy, and will not take this lack of numerical precision into account. For the obvious effects, chemo-metrics will not be necessary, but for less obvious effects, the statistical conclusions will be invalidated because of the low numerical accuracy in the raw data. 

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