Aitken method

Any sequence that is linear convergent can be accelerated by a method called the Aitken method. 

For a fixed point iteration, we generate the first three terms in the sequence a , that is, a , a, and U2. Next, use the A Aitken method to generate 8q. At t stage, we can assume that the newly generated Sq is a better approximation to a than U2, and then apply the fixed point iteration to Sq to generate the next sequence of S , dj and 82- The Aitken method is now applied to the sequence d ra = 0, 1, 2 to generate y , which is a better approximation to a, and the process continues. 

If the substitution method converges, the convergence speed can be improved using the Aitken method (Buzzi-Ferraris and Manenti, 2010a). 

There is one method, however, of increasing the order of convergence, that is often extremely useful. In fact, it can be applied to an arbitrary sequence, however the sequence may have been derived it will often produce a sequence that converges more rapidly than the original, and will even, in many cases, produce a converging sequence out of one that diverges. It is due to Aitken, who called it the 82-process. Beginning with x0, let x and x2 be computed in the normal manner by Eq. (2-17), but then form x3 by... 

This oscillation may have no relation at all to the behavior of the "true" function. Therefore, we cannot recommend global interpolation except for small samples. In large samples interpolation is rarely needed. For medium size samples low order local interpolation considering 3-6 nearest neighbors of the point x of interest does the job in most cases. The most popular method is local cubic interpolation in the Aitken form programmed in the following module. 

Figure 2.24 Correlation of the oxygen permeability coefficient for a family of related polysulfones with inverse fractional free volume (calculated using the Bondi method) [33]. Reprinted with permission from C.L. Aitken, W.J. Koros and D.R. Paul, Effect of Structural Symmetry on Gas Transport Properties of Polysulfones, Macromolecules 25, 3424. Copyright 1992, American Chemical Society...

The intramolecular photodimerization and [2 + 2]-photocycloaddition in DNA involves thymine or cytosine as the chromophore. This chemistry has been intensively investigated with regards to DNA damage and repair [131]. Despite the fact that the area is of continuous interest [132], the synthetic applications are limited and are not covered here in detail. However, some preparative aspects of 4-pyrimidinone photocycloaddition chemistry will be addressed. Aitken et al. have prepared a plethora of constrained cyclobutane P-amino acids by intra- or intermolecular [2 + 2]-photocycloaddition to uracil and its derivatives [133, 134]. In a chiral adaptation of this method, the uracil-derived enone 140 was employed to prepare the diastereomeric cyclobutanes 141 in very good yield (Scheme 6.49). The compounds are easily separated and were - despite the relatively low auxiliary-induced diastereoselectivity - well suited to prepare the as-2-aminocyclobutanecarboxylic acids 142 in enantiomerically pure form. Enantioselective access to the corresponding trans-products was feasible by epimerization in a-position to the carboxyl group [135],... 

The methods of Chapter 6 are not appropriate for multiresponse investigations unless the responses have known relative precisions and independent, unbiased normal distributions of error. These restrictions come from the error model in Eq. (6.1-2). Single-response models were treated under these assumptions by Gauss (1809, 1823) and less completely by Legendre (1805), co-discoverer of the method of least squares. Aitken (1935) generalized weighted least squares to multiple responses with a specified error covariance matrix his method was extended to nonlinear parameter estimation by Bard and Lapidus (1968) and Bard (1974). However, least squares is not suitable for multiresponse problems unless information is given about the error covariance matrix we may consider such applications at another time. 

Special techniques have been used for the measurement of individual steroids. For blood progesterone Harbert et al. (H4) obtained an initial separation on an alumina column, but final purification and assay was achieved by further alumina chromatography after preparing the bis-dinitrophenylhydrazone. The method of Short (S16) involving paper chromatography and assay by ultraviolet absorption, with Allen corrections for impurity, has been used by Greig et al. (G5) and Aitken et al. (A3). Cortisol has been measured in urine and blood by fluorimetry and isotope dilution (U3, U4), and in blood by double isotope dilution derivative techniques together with cortisone (H9) and also with cortisone,... 

The third major class of analytical techniques may be called morphological methods. This identification consists of comparing the form of particles captured with the morphology of particles of known composition. It goes without saying that morphological similarity is a necessary but not always sufficient condition for compositional identity. In spite of this problem this procedure is widely employed mainly in clean atmosphere, since even Aitken size particles can be identified morphologically (A. Meszaros and Vissy, 1974 Butor, 1976). 

Lopez, A., Servant, J. and Fontan, J., 1974 Methode de mesure de I intensite des sources de noyaux d Aitken dans Tatmosphere. Atmospheric Environment 8, 733-754. 

Hot water extraction of soil boron (B) (Berger Truog 1939) has been used widely to obtain an index of soil B status. The method presents difficulties for routine analysis as it is often difficult to prepare water extracts free of colloidal material. Dilute salt solutions have largely overcome this limitation, two examples being the use of 0.01 M CaCl2-0.05 M mannitol (Cartwright etal. 1983) and hot 0.01 M CaCl2 (Aitken etal. 1987). The empirical method described is based on the use of hot water reflux for soil extraction, with alternative analytical finishes based on a colour reaction and ICPAES. 

Uranium -series dating of bone. Uranium concentrations in living bone are typically 0.1 ppm (Aitken 1990), whereas concentrations in fossil bone may exceed 1000 ppm. Theoretically, therefore. Quaternary age bones could be dated directly by the U-series method. 

By applying Aitken s method to a linearly convergent sequence obtained from fixed point (successive substitution) iteration, we can accelerate the convergence to quadratie order. This procedure is known as the Steffenson s method, which leads to Steffenson s algorithm as follows. 

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