Iter representations

The filter version of the method is a truly continuous process, as opposed to the batch process that the iterative representation implies. Furthermore, no approximation is necessary in deriving the filter version. The output of the filter is precisely the same as that of the basic iterative constrained software method. Bearing in mind that only a modest lag results, one may think of it as a real-time implementation. In an alternative approach, a laboratory computer might apply a purely software version of this filter to spectra continuously as data are acquired. The filter may also be packaged in firmware as part of a microprocessor-based instrument. Other applications also suggest themselves. 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

The subsequent representations are probably reliable within the range of data used (always less broad than 200° to 600°K), but they are only approximations outside that range. The functions are, however, always monotonic in temperature, to provide appropriate corrections when iterative programs choose temperature excursions outside the range of data. 

In many cases, the methods used to solve identification problems are based on an iterative minimization of some performance criterion measuring the dissimilarity between the experimental and the synthetic data (generated by the current estimate of the direct model). In our case, direct quantitative comparison of two Bscan images at the pixels level is a very difficult task and involves the solution of a very difficult optimization problem, which can be also ill-behaved. Moreover, it would lead to a tremendous amount of computational burden. Segmented Bscan images may be used as concentrated representations of the useful... 

Krylov Approximation of the Matrix Exponential The iterative approximation of the matrix exponential based on Krylov subspaces (via the Lanczos method) has been studied in different contexts [12, 19, 7]. After the iterative construction of the Krylov basis ui,..., Vn j the matrix exponential is approximated by using the representation A oi H(g) in this basis ... 

These methods may be called analytical, by contrast with another class of iterations that might be called arithmetic, since they exploit the fact that the number representation is finite and digital. The familiar Homer s method is an example. The first step is to establish that a root lies between a certain pair of consecutive integers. Next, if the representation is decimal, f(x) is evaluated at consecutive tenths to determine the pair of consecutive tenths between which the root lies. This is repeated for the hundredths, thousandths, etc., to as many places as may be desired and justified. 

Invariance principle, 664 Invariance properties of quantum electrodynamics, 664 Inventory problem, 252,281,286 Inverse collisions, 11 direct and, 12 Inverse operator, 688 Investment problem, 286 Irreducible representations of crystallographic point groups, 726 Isoperimetric problems, 305 Iteration for the inverse, 60... 

As the starting geometries for iterative calculation, we take all the possible structures in which bond lengths are distorted so that the set of displacement vectors may form a basis of an irreducible representation of the full symmetry group of a molecule. For example, with pentalene (I), there are 3, 2, 2 and 2 distinct bond distortions belonging respectively to a, b2 and representations of point group D21,. 

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

Equation (4) is a three-term recursion for propagating a wave packet, and, assuming one starts out with some 4>(0) and (r) consistent with Eq. (1), then the iterations of Eq. (4) will generate the correct wave packet. The difficulty, of course, is that the action of the cosine operator in Eq. (4) is of the same difficulty as evaluating the action of the exponential operator in Eq. (1), requiring many evaluations of H on the current wave packet. Gray [8], for example, employed a short iterative Lanczos method [9] to evaluate the cosine operator. However, there is a numerical simplification if the representation of H is real. In this case, if we decompose the wave packet into real and imaginary parts. 

The theory behind body-fixed representations and the associated angular momentum function expansions of the wavefunction (or wave packet) in terms of bases parameterized by the relevant constants of the motion and approximate constants of the motion is highly technical. Some pertinent results will simply be stated. The two good constants of the motion are total angular momentum, J, and parity, p = +1 or 1. An approximate constant of the motion is K, the body-fixed projection of total angular momentum on the body-fixed axis. For simplicity, we will restrict attention to the helicity-decoupled or centrifugal sudden (CS) approximation in which K can be assumed to be a constant of the motion. In terms of aU its components, and the iteration number k, the real wave packet is taken to be [21]... 

In the absence of detailed structural information about GPCRs, much of the efforts to interpret experimental results in a structural context has focused on creating molecular representations of these proteins that can incorporate directly and consistently the many types of function-related information (for a recent review, see ref. [5]). In turn, such molecular models serve as hypotheses-generators for experimental probing of functional inferences, and are continuously refined by the data obtained from such experiments. Listed below are some of the main advantages of such an iterative approach, as illustrated in this chapter ... 

Writing a specification for an operation is very different from writing an implementation. The spec is simply a Boolean expression a relation between the inputs, initial state, final state, and outputs. An implementation would choose a particular algorithmic sequence of steps, select a data representation or specific internal access functions, and work through iterations, branches, and many intermediate states before achieving the final state. Consider the specifications of these operations in contrast to their possible implementations ... 

An analysis of the stmcture of the electron correlation terms in which the reference was the antisymmetrized products of FCI -RDM elements was reported in [12], The advantage of using correlated lower order matrices for building a high order reference matrix is that in an iterative process the reference is renewed in a natural way at each iteration. However, if the purpose is to analyse the structure of the electron correlation terms in an absolute manner that is, with respect to a fixed reference with no correlation, then the Hartree Fock p-RDM"s are the apropriate references. An important argument supporting this choice is that these p-RDM s are well behaved A-representable matrices and, moreover, (as has been discussed in [15]) the set of 1-, 2-, and 3-Hartree Fock-RDM constitute a solution of the 1 -CSE. 

Hirao has also recently considered the transformation of CASSCF wavefrmctions to valence bond form [24, 25]. An orthogonal VB orbital basis was first considered, in which case the CASSCF Cl vector may be found by re-solving the Cl problem. Later he considered also the transformation to a classical VB representation. The transformation of the CASSCF space was achieved by calculating all overlap terms, (oCASscFj cASVB gjjjj golving the subsequent linear problem, using a Davidson-like iterative scheme. 

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