Lanczos continued fractions

This procedure of obtaining the Green function 7Z(u) is called hereafter the method of the Lanczos continued fractions (LCF) in the symbolic notation from the rhs of Eq. (224). Then, we say that the quantity 7 LC (u) is the infinite-order LCF for the exact Green function lZ u) from Eq. (49). A truncation at the nth term in Eq. (230) leads to the approximation ... 

Employing the relationships from (277), we can rewrite the Lanczos continued fraction (230) as ... 

The second example from the general system (284) is of direct relevance to the Lanczos continued fraction (LCF). The Lanczos inhomogeneous tridiagonal system of linear equations can be identified from Eq. (284) by specifying D a and ... 

Hence, the first component of the n-dimensional column vector x = xr (1 Green function lZ(u) from Eq. (12) ... 

We recall that the Lanczos continued fraction is equivalent to the Pade-Lanczos approximant, which is defined in Eq. (207) through its polynomial quotient or equivalently via the sum of the Heaviside partial... 

The advantages of this kind of formulation stand out not only in terms of elegance and beauty (the moment method, the Lanczos method, and the recursion method are relevant but particular cases of the memory function equations), but also in the possibility of providing insight into a number of problems, such as the asymptotic behavior of continued fraction parameters and their relationship with moments, the possible inclusion of nonlinear effects, the introduction of the concept of random forces, and so on. 

The memory function formalism leads to several advantages, both from a formal point of view and from a practical point of view. It makes transparent the relationship between the recursion method, the moment method, and the Lanczos metfiod on the one hand and the projective methods of nonequiUbrium statistical mechanics on the other. Also the ad hoc use of Padd iqiproximants of type [n/n +1], often adopted in the literature without true justification, now appears natural, since the approximants of the J-frac-tion (3.48) encountered in continued fraction expansions of autocorrelation functions are just of the type [n/n +1]. The mathematical apparatus of continued fractions can be profitably used to investigate properties of Green s functions and to embody in the formalism the physical information pertinent to specific models. Last but not least, the memory function formaUsm provides a new and simple PD algorithm to relate moments to continued fraction parameters. 

We shall set N be the dimension of the finite basis subset used to represent f and v. The calculation can be performed with great efficiency using an iterative algorithm, such as the Lanczos algorithm, that transforms r into a tridiagonalized form. A continued fraction expansion is then obtained ... 

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