Direct Elimination Methods

There are a number of methods available to solve for the solution of a given set of linear algebraic equations. One class is the direct method (i.e., requires no iteration) and the other is the iterative method, which requires iteration as the name indicates. For the second class of method, an initial guess must be provided. We will first discuss the direct methods in Section B.5 and the iterative methods will be dealt with in Section B.6. The iterative methods are preferable when the number of equations to be solved is large, the coefficient matrix is sparse and the matrix is diagonally dominant (Eqs. B.8 and B.9). 

---

Although this way of deriving a classical result seems rather awkward and time-consuming compared to more direct elimination methods, it has the advantage that it can be extended to any combination of isotopes in a decay series, o... 

The most important direct solution algorithms used in finite element computations are based on the Gaussian elimination method. 

Identification of waste characteristics that limit the effectiveness or feasibility of technologies is an important part of the screening process. Technologies clearly limited by these waste characteristics should be eliminated from consideration. Waste characteristics particularly affect the feasibility of on-site methods, direct treatment methods, and land disposal (on/off-site). 

Direct elimination of a carboxylic acid to an alkene has been accomplished by heating in the presence of palladium catalysts.Carboxylic esters in which the alkyl group has a P hydrogen can be pyrolyzed, most often in the gas phase, to give the corresponding acid and an alkene. No solvent is required. Since rearrangement and other side reactions are few, the reaction is synthetically very useful and is often carried out as an indirect method of accomplishing 17-1. The yields are excellent and the work up is easy. Many alkenes have been prepared in this manner. 

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... 

With the aid of effective Gauss method for solving linear equations with such matrices a direct method known as the elimination method has been designed and unveils its potential in solving difference equations,... 

Another way of going further is connected with re-ordering of these groups in reverse direction or inserting k = 1 and m = 1 in all of the iterations. The separate elimination method may be of assistance in minimizing the total volume of the available information in the storage of high-performance computers. 

An alternative to the measurement of the dimensions of the indentation by means of a microscope is the direct reading method, of which the Rockwell method is an example. The Rockwell hardness is based on indentation into the sample under the action of two consecutively applied loads - a minor load (initial) and a standardised major load (final). In order to eliminate zero error and possible surface effects due to roughness or scale, the initial or minor load is first applied and produce an initial indentation. The Rockwell hardness is based on the increment in the indentation depth produced by the major load over that produced by the minor load. Rockwell hardness scales are divided into a number of groups, each one of these corresponding to a specified penetrator and a specified value of the major load. The different combinations are designated by different subscripts used to express the Rockwell hardness number. Thus, when the test is performed with 150 kg load and a diamond cone indentor, the resulting hardness number is called the Rockwell C (Rc) hardness. If the applied load is 100 kg and the indentor used is a 1.58 mm diameter hardened steel ball, a Rockwell B (RB) hardness number is obtained. The facts that the dial has several scales and that different indentation tools can be filled, enable Rockwell machine to be used equally well for hard and soft materials and for small and thin specimens. Rockwell hardness number is dimensionless. The test is easy to carry out and rapidly accomplished. As a result it is used widely in industrial applications, particularly in quality situations. 

Of course, to be able to use the direct injection method of sample introduction, the analyte or the polymer system must be soluble in a solvent. Other methods of sample introduction need to be considered in order to eliminate the involatile material from the chromatographic separation. These have become extremely effective in the analysis of matrices such as polymers. 

Analogously to AAla from Ser, AAbu can be derived from Thr by using essentially the same methods.1"-104 However, because of the possibility of E- and Z-isomers with AAbu, either stereoselective synthesis or separation of isomers must be taken into consideration. Srinivasan et al. 89 reported detailed stereochemical studies on the 3-elimination of Thr derivatives. Lee et al. 122 confirmed the stereospecific formation of AAbu from O-tosyl Thr derivatives the Z-isomer from the threo type and the E-isomer from the erythro type. If DSC is used for direct elimination, only the Z-isomer 57 is formed from Z-Thr-OMe 56 in 70% yield (Scheme 19). 

A perturbation-trajectory method for determining the dynamics of gas-surface collision processes was tested on the collision and subsequent surface reactions of SiH2 on a Si(lll) surface233. The predictions of an exact classical trajectory calculation234 were confirmed the sticking probabilities were unity at all temperatures, and it was found that surface SiH2 can decompose by direct elimination of H2 or by successive dissociation of Si—H bonds. 

Protonation and Reductive Elimination Reaction of a (Diimine)platinum (II) Dimethyl Complex It has been realized for some time that direct, selective methods for catalytic conversion of hydrocarbons to products of further value are important goals for chemistry investigators. This topic is notably complex and may be reviewed for the considerable body of research conducted heretofore.348-350 Aqueous... 

This is an iterative method that calculates the next set of values by direct elimination. A small matrix [N] is added to the coefficient matrix [M] so that [M + N] is easily factored with much less arithmetic than performing elimination on [M]. An iteration parameter controls the amount of N added. The method is more economical and the convergence rate is much less sensitive to the iteration parameter than SOR. Subroutines for 2d and 3d SIP may be found in the NAG library (D03EBF and D03ECF). 

Physical and spectral properties of batrachotoxins are presented in Table I. Mass spectra have been presented and interpreted (3,13,14). The parent ion of batrachotoxin is virtually nondetectable by direct probe methods, and instead an apparent molecular ion of miz 399 is seen, probably because of pyrolytic elimination of the pyrrole carboxylate moiety. Batrachotoxin alkaloids do not chromatograph on capillary gas chromatographic columns, but a pyrolysis product has been detected at 280°C on the temperature-programmed, packed OV-1 columns used for analysis of other dendrobatid alkaloids (see Appendix). The pyrrole carboxylate moiety is responsible for major ions of C7H9N02 (m/z 139), C6H9N ... 

The overall sequence could be realized by 1,4-addition of a nucleophile to the enone and subsequent quenching at the a-position followed by B-elimination of the initial nucleophilic component. Such multistep processes will not be discussed here. However direct hydroxylation methods are scarce. Mo-riarty has reported that a,3-unsaturated ketones are oxidized by phenyliodosyldiacetate at the a-site in preference to the a -position, e.g. (138) to (139), although no yield has bWn indicated. There is no available mechanistic rationale, although the intermediacy of the a,B-epoxide is precluded. 

---