Eigenvalue-eigenvector method

The method is also described rather clearly by Smith [95], whose description will be followed here. 

The method starts with a system of odes, represented as in (9.53), 

Instead of now discretising the left-hand side of the equation in some way (explicit BI, CN, etc.) and stepping forward in time by small time intervals, the equation is solved analytically the solution at time T is 

This has an exponential of a matrix. It is defined in terms of the expansion of the exponential function, see, for example. Smith [95, pp. 134-135]. Now, the usual eigenvalue-eigenvector equation can be written in compact form. 

Smith also shows that it follows from this that 

Yet another, quite different, approach to solving a system of odes, such as one obtains as an intermediate step when using, for example, MOL or OC, is the eigenvalue-eigenvector method. Its use for electrochemical simulations was described in two papers in 1989 and 1990 [255,332]. The method has some drawbacks, and does not appear to have seen much use since these two papers. It does have one unique feature there is no discretisation of time. A solution is generated by the algorithm, at any chosen time. So, although the method may at times be fairly inefficient, if one wants a current or concentrations at only one or a few time points, this could be faster than a time march with the usually small time intervals. 

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The PLS algorithm is relatively fast because it only involves simple matrix multiplications. Eigenvalue/eigenvector analysis or matrix inversions are not needed. The determination of how many factors to take is a major decision. Just as for the other methods the right number of components can be determined by assessing the predictive ability of models of increasing dimensionality. This is more fully discussed in Section 36.5 on validation. 

In order to find extrema of E( ui ), subject to the normalization condition, standard moves known as the Lagrange multipliers method are applied, which readily lead us to the well-known form of the generalized matrix eigenvalue/eigenvector problem ... 

Rank annihilation methods employ eigenvalue-eigenvector analyses for direct determination of analyte concentration with or without intrinsic profile determination. With the exception of rank annihilation factor analysis, these methods obtain a direct, noniterative solution by solving various reconstructions of the generalized eigenvalue-eigenvector problem. 

Alternating least squares (ALS) methods are both slower, due to their numeric intensity, and more flexible than eigenvalue-eigenvector problem-based methods for solving Equation 12.1a and Equation 12.1b. The basic PARAFAC model of Equation... 

In practice this method converges for all eigenvalues of some nontrivial matrices and at least the lowest eigenvalue of all matrices tried and is particularly suitable for Cl calculations where one normally has a very good approximation to the lowest eigenvalue/eigenvector pair namely the energy of the reference function and the vector... 

Following the determination of the hydrolytic rate constants of an epimeric pair of tricyclic nitriles (78) by a non-linear, least-squares fitting method, a novel eigenvalue-eigenvector analysis of the sensitivity coefficients permitted maximization of the kinetic dataJ ... 

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