Recursion method equivalency

To verify the equivalence of the memory function approach to the recursion method or the Lanczos method, it is sufficient to note that the state l/ i+i) defined via Eq. (3.39d) coincides with the state... 

The expression x (J)P(j - l)x(j) in eq. (41.4) represents the variance of the predictions, y(j), at the value x(j) of the independent variable, given the uncertainty in the regression parameters P(/). This expression is equivalent to eq. (10.9) for ordinary least squares regression. The term r(j) is the variance of the experimental error in the response y(J). How to select the value of r(j) and its influence on the final result are discussed later. The expression between parentheses is a scalar. Therefore, the recursive least squares method does not require the inversion of a matrix. When inspecting eqs. (41.3) and (41.4), we can see that the variance-covariance matrix only depends on the design of the experiments given by x and on the variance of the experimental error given by r, which is in accordance with the ordinary least-squares procedure. 

Recently, Jankowiak et al. [88] proposed a new approach based on the coherent-state representation used by Doktorov et al. [47] to obtain the recursion formulas needed to compute the FC integrals. These formulas have been shown by Liang and Li [95] to be equivalent to the ones derived by Ruhoff [%] from the analytic approach of Sharp and Rosenstock [39]. The method initially limited to 0 K FC spectra has been generalized to deal with FCHT spectra [97] and finite-temperature effects [98]. 

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