Numerical Difficulties

As linear regression is a very fundamental operation, several methods have been developed in order to improve the numerical stability of the calculation. It is beyond the objective of this book to discuss these issues in any detail. We do feel, however, that the reader has to be aware of the potential problems and should be able to avoid them as much as possible. 

The Matlab computations invoked by the back-slash and forward-slash / operators do not perform the calculation as given in equations (4.26) and (4.31). Here a short extract from the Matlab HELP  

If F is not square and is full, then Householder reflections are used to compute an orthogonal-triangular factorization. 

Without attempting to fully understand this, the essence is important  

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The Ot) values in Eqs. (13-37) and (13-38) are effective values obtained from Eq. (13-35) or Eq. (13-36). Once these values are available, 0 can be calculated in a straightforward iteration from Eq. (13-38). Since the (ot — 0) difference can be small, 0 should be determined to four decimal places to avoid numerical difficulties. 

All stage-to-stage methods that work from both ends of the column toward the middle suffer from two other disadvantages. First, the top-down and the bottom-up calculations must me somewhere in the column. Usually the mesh is made at a feed stage, and if more than one feed stage exists, a choice of mesh point must be made for each component. When the components vary widely in volatility, the same mesh point cannot be used for all components if serious numerical difficulties are to be avoided. Second, arbitrary procedures must be set up to handle nondlstrihuted components. (A nondistributed component is one whose concentration in one of the end-product streams is smaller than the smallest number carried by the computer.) In the LM and TG equations, the concentrations for these components do not natur ly take on nonzero values at the proper point as the calculations proceed through the column. 

Because of all these numerical difficulties, neither the LM nor the TG stage-by-stage method is commonly implemented in modern computer algorithms. Nevertheless, the TG method is veiy instructive and is developed in the following example. For a single narrow-boihng feed, the TG manual method is quite efficient. 

A sum-over-states expression for the coefficient A for the expansion of the diagonal components faaaa was derived by Bishop and De Kee [20] and calculations were reported for the atoms H and He. However, the usual approach to calculate dispersion coefficients for many-electron systems by means of ab initio response methods is still to extract these coefficients from a polynomial fit to pointwise calculated frequency-dependent hyperpolarizabiiities. Despite the inefficiency and the numerical difficulties of such an approach [16,21], no ab initio implementation has yet been reported for analytic dispersion coefficients for frequency-dependent second hyperpolarizabiiities which is applicable to many-electron systems. 

The best solution to such numerical difficulties is to change methods. Integration in the reverse direction eliminates most of the difficulty. Go back to Equation (9.15). Continue to use a second-order, central difference approximation for d a/d, but now use a first-order, forward... 

The coupling of the component and energy balance equations in the modelling of non-isothermal tubular reactors can often lead to numerical difficulties, especially in solutions of steady-state behaviour. In these cases, a dynamic digital simulation approach can often be advantageous as a method of determining the steady-state variations in concentration and temperature, with respect to reactor length. The full form of the dynamic model equations are used in this approach, and these are solved up to the final steady-state condition, at which condition... 

Ongoing U.S. EPA research on radon-resistant new construction has encountered numerous difficulties in making a gastight mechanical barrier effective enough to confidently keep indoor radon levels below 4pCi/L. The types of problems encountered included... 

The obstacle to simultaneous quantum chemistry and quantum nuclear dynamics is apparent in Eqs. (2.16a)-(2.16c). At each time step, the propagation of the complex coefficients, Eq. (2.11), requires the calculation of diagonal and off-diagonal matrix elements of the Hamiltonian. These matrix elements are to be calculated for each pair of nuclear basis functions. In the case of ab initio quantum dynamics, the potential energy surfaces are known only locally, and therefore the calculation of these matrix elements (even for a single pair of basis functions) poses a numerical difficulty, and severe approximations have to be made. These approximations are discussed in detail in Section II.D. In the case of analytic PESs it is sometimes possible to evaluate these multidimensional integrals analytically. In either case (analytic or ab initio) the matrix elements of the nuclear kinetic energy... 

Van Zoonen et al. [19,20] employed an alternative approach, in an attempt to overcome the limited aqueous solubility of diaryloxalate ester-type POCL reagents. In this work, granular TCPO was mixed with controlled pore glass and packed in a flow cell, forming a solid-state TCPO reactor. When this was used in conjunction with a flow system, some of the TCPO dissolved in the carrier solution. Numerous difficulties were encountered with this approach, namely, limited reactor lifetime (approximately 8 h) and low CL emission obtained as the carrier became more aqueous (a 90% reduction of CL intensity occurred when the aqueous content of the carrier stream comprised 50% water, as compared to pure acetonitrile). The samples also required dilution with acetonitrile to increase the solubility of TCPO in the sample plug. 

It is then no longer necessary to solve a transport equation for Y and the numerical difficulties associated with treating the first reaction with a finite-rate chemistry solver are thereby avoided. 

Despite the vast amount of work on 14C dating which has already been accomplished and despite the fact that it is the best developed method available today, numerous difficulties still exist with its application. First, carbonate geochemistry which helped control 14C concentrations in the past is not simple to reconstruct. Carbonate minerals are commonly in a state of near equilibrium with groundwater, and only slight changes in water temperature or chemistry will promote either dissolution or precipitation of carbonate ions. In this way, the proportion of modern carbon in the water can be changed and some isotope... 

A numerical difficulty of the NRT description of the species in Table 3.30 is that the four specified resonance structures... 

The most difficult problem we face in deciding to use a basis of hybrids which reflects the molecular symmetry is how do we choose such a basis in view of the enormous numerical difficulties involved in optimising the non-linear parameters in molecular calculations The real question is are there any rules for this choice, can the optimisation be done (at least approximately) once and for all The chemical evidence is for us — it is the most basic concept of the theory of valence that particular electronic sub-structures tend to be largely environment-independent. How can we select our basis to reflect this chemical fact ... 

Solving the purely advective equation or even introducing an advection term into the diffusion equation is a source of numerical difficulties. The simplest advection equation of a medium moving at velocity v in one dimension can be written... 

The above equation would indicate an infinite growth rate at t=0, which is consistent with the diffusion equation (because the concentration gradient at f=0 is infinity), although in reality this would not happen. Because the growth rate quickly becomes finite, the initial infinite growth rate does not cause any numerical difficulty in solving the problem. 

We should point out that Eq. (42) indicates that the function G(s) can be obtained from the value of the friction kernel at t = 0. This is a consequence of the fact that the friction kernel is calculated in the clamping approximation. In any case, Eq. (42) allows for the calculation of G(s) without the numerical difficulties that plague the long-time tail of molecular dynamics simulations. 

Perhaps the most peculiar factor to be considered when approaching a cultural artifact, however, is its uniqueness. This actuality makes its conservation a challenge—an attractive and interesting case raising numerous difficulties—because the artifact is often the only witness of the past and cannot be sampled to perform all the adequate characterization that would be necessary to understand traditional materials and disclose ancient technologies. Furthermore the alteration compounds (corrosion products) that are considered part of the artifact because they are a testimony of its past, should not be taken away but instead should be studied and conserved in place whenever possible. 

Figures I Ob shows simulations using orthogonal collocation with possible numerical difficulties. 

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