Truncated series

Determination of confidence limits for non-linear models is much more complex. Linearization of non-linear models by Taylor expansion and application of linear theory to the truncated series is usually utilized. The approximate measure of uncertainty in parameter estimates are the confidence limits as defined above for linear models. They are not rigorously valid but they provide some idea about reliability of estimates. The joint confidence region for non-linear models is exactly given by Eqn. (B-34). Contrary to ellipsoidal contours for linear models it is generally banana-shaped. 

As noted in Section II, truncated series may give misleading results, depending upon the magnitude of the parameter, Sa say, used in the expansion and certainly the values currently used in the representation... 

Since this relative rate is numerically small, a truncated series of ln(l+x) results in the valid approximation... 

Again this equation is sensitive to uncertainties in the pressure drop. The truncated series expansion of Tolman s equation may be expected to introduce significant error for small drops. 

Finally, we present the results of the case studies for Eley-Rideal and LH reaction mechanisms illustrating the practical aspects (i.e. convergence, relation to classic approximations) of application of this new form of reaction rate equation. One of surprising observations here is the fact that hypergeometric series provides the good fit to the exact solution not only in the vicinity of thermodynamic equilibrium but also far from equilibrium. Unlike classical approximations, the approximation with truncated series has non-local features. For instance, our examples show that approximation with the truncated hypergeometric series may supersede the conventional rate-limiting step equations. For thermodynamic branch, we may think of the domain of applicability of reaction rate series as the domain, in which the reaction rate is relatively small. 

Figure 3 compares the exact solution (49) and its approximations obtained by truncating series (61) at first, second, etc. terms. It shows that the even first term (—Bq/Bj) provides reasonable approximation in the finite neighborhood of equilibrium. Addition of higher order terms increases the domain of close approximation. This is not surprising because the condition of convergence of this series is 2[Pg.74]

We assume that the series expansion (4.45) of the scattered field is uniformly convergent. Therefore, we can terminate the series after nc terms and the resulting error will be arbitrarily small for all kr if nc is sufficiently large. If, in addition, kr n, we may substitute the asymptotic expressions (4.42) and (4.44) in the truncated series the resulting transverse components of the scattered electric field are... 

The choice of the basis functions used in the truncated series (eqn. 7.112) will lead to different types of methods. The most common set of methods are the finite element and the spectral methods. For the case of finite element methods, the domain is divided into small elements, and a basis function is specified in each element to interpolate the parameters throughout the element. The functions are locally defined within each element and can handle complex geometries. 

For smaller values of this parameter and charts given in the references at the end of the chapter. Calculations using the truncated series solutions directly are discussed in Appendix C. 

When a function is defined by an infinite power series in terms of a parameter p, the traditional approach is to truncate the power series, retaining terms up to pq. However, if the power series fails to converge (i.e. outside the region of convergence of the local equation), including higher order terms does not save the truncated series from failure, and the truncated series may lead to nonphysical results in the limit of p —> oo. 

The power series may be only poorly convergent or even non-convergent, in which case the truncated series becomes a poor approximation to iji. Unlike the power series, which tries to express ij/ in terms of a single polynomial, the Pade approximation expresses i) as a ratio of two polynomials. The procedure to determine the two polynomials involves converting the power series [Eq. (342)] into another power series... 

While the relations between the inertial and planar moments are strictly linear and constant, the relation between the increments of the rotational constants Bg and the moments, say Ig, is a truncated series expression and only approximately linear, Alg = ( f/B2g)ABg. Also, the transformation coefficients f/B2g are not strictly constant (nonrandom), although usually afflicted with only a very small relative error. The approximations are, in general, good enough to satisfy the requirements for Eq. 22, and for the above statement rt = rp = rB to be true for all practical purposes. 

This then gives a direct link between the thermodynamic stress and the osmotic pressure (or the compressibility) of the suspension. As a result of this stress, the viscosity will depend directly upon the structure, and the interpartide potential, V(ry). Using this interrelationship Batchelor has been able to evaluate the ensemble averages of both the mechanical and thermodynamic stresses by renormalizing the integrals. As a result, he has developed truncated series expressions for the low shear limit viscosity, and the high shear limit viscosity, t) , corresponding to... 

The reduced partition functions of isotopic molecules determine the isotope separation factors in all equilibrium and many non-equilibrium processes. Power series expansion of the function in terms of even powers of the molecular vibrations has given explicit relationships between the separation factor and molecular structure and molecular forces. A significant extension to the Bernoulli expansion, developed previously, which has the restriction u = hv/kT < 2n, is developed through truncated series, derived from the hyper-geometric function. The finite expansion can be written in the Bernoulli form with determinable modulating coefficients for each term. They are convergent for all values of u and yield better approximations to the reduced partition function than the Bernoulli expansion. The utility of the present method is illustrated through calcidations on numerous molecular systems. 

The perturbed total energies or other properties of the system can be written as an expansion in terms of moment and polarizability components (see Section I). If different values of the field strength or charge positions are used, a system of simultaneous equations can be written from the truncated series, and these equations are solved to find the unknown polarizabilities. The system of equations must be chosen sufficiently large to ensure that the truncation error is minimized, but sometimes it is not practical to carry out the number of finite-field calculations that this might call for. 

Tgj is represented exactly and the exact electronic energy, which also includes dispersion effects correctly, is obtained. However, this comes with infinite computational costs. Hence, methods needed to be devised, which allow us to approximate the infinite expansion in Eq. (12.9) by a finite series to be as short as possible. A straightforward approach is the employment of truncated configuration interaction (CI) expansions. Note that (electronic) configuration refers to the set of molecular orbitals used to construct the corresponding Slater determinant. It is a helpful notation for the construction of the truncated series in a systematic manner and yields a classification scheme of Slater determinants with respect to their degree of excitation . Excitation does not mean physical excitation of the molecule but merely substitution of orbitals occupied in the Hartree-Eock determinant o by virtual, unoccupied orbitals. Within the LCAO representation of molecular orbitals the virtual orbitals are obtained automatically with the solution of the Roothaan equations for the occupied orbitals that enter the Hartree-Eock determinant. 

The key elements of the MWR are the expansion functions (also called the trail-, basis- or approximating functions) and the weight functions (also known as test functions). The trial functions are used as the basis functions for a truncated series expansion of the solution, which, when substituted into the differential equation, produces the residual. The test functions are used to ensure that the differential equation is satisfied as closely as possible by the truncated series expansion. This is achieved by minimizing the residual, i.e., the error in the differential equation produced by using the truncated expansion instead of the exact solution, with respect to a suitable norm. An equivalent requirement is that the residual satisfy a suitable orthogonality condition with respect to each of the test functions. 

It is often useful to expand a function into a series when direct use of the function is impractical because of complexity or when a linear approximation is sought (2, 7). In general, a particular point is chosen as a central location, and the expansion is developed as a representation of the function in the neighborhood of that point. Truncated series can be expected to give accurate descriptions near the central value and less accurate representations at more remote points. 

It can be seen that the convergence of the truncated series depends on the value of w the convergence rate is slower when frequency oj is smaller. To overcome this problem the authors of SLDM proposed a new approach for solutions to Maxwell s equations — a preconditioned modification of SLDM for low frequencies. This new method is based on the standard SLDM but with Krylov subspaces generated from the inverse powers of the Maxwell operator. In Druskin et al. (1999) this method is referred to as spectral Lanczos decomposition method using inverse powers, or SLDMINV. 

This series, in fact, diverges but it diverges so slowly that the error obtained in truncating it is always less than the last term in the truncated series. Thus, to obtain a value of the function to the same accuracy as the other series, the expansion is terminated when the last term is less than the same criterion (10 ). 

The studies reported here have been addressed to problems in theoretical physics concerning physical properties about which one has only limited analytic or computer generated data. These data may be available only for small values of the system density as, for example, in the form of a virial polynomial , or they may be limited to an analogous, truncated series in powers of some parameter gauging the strength of an interaction that causes the many-body system to behave nonideally. Alternatively, data character-... 

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