Boundary conditions matrix, orthogonality

As indicated in the last section, serious problems exist with the standard R-matrix with respect to slow convergence of the phase shifts (or S-matrix) as the number of translational basis functions is increased. This has been amply demonstrated in a number of papers (e.g.. Refs. 8-12). In all of these cases an orthogonal translational basis is used which satisfies fixed log derivative boundary conditions at the R-matrix boundary, R = A. As indicated above, the Buttle correction [9] can be added to the R-matrix to account, in an approximate fashion, for the members of the complete basis not included in the explicit R-matrix evaluation. An additional variational correction [10,11] was proposed to improve the results further. Although these procedures help a great deal, they are both expensive, at least in terms of programming effort for complex systems. 

Linear combinations of n hence form the set of vectors that are orthogonal to the stoichiometric subspace. From Chapter 6, it is known that the condition for a CSTR to lie on the AR boundary occurs when the controllability matrix E does not contain full rank. An expression may be determined for this by computing the determinant of E and setting it equal to zero. 

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