Lanczos vector space

The Lanczos vector space CM can be defined through its basis and the appropriate scalar product. A finite sequence of the Lanczos orthogonal polynomials of the first kind is complete, as will be shown in Section 12, and therefore, the set Q (u) with K elements represents a basis. Thus, the polynomial set Q (u) =0 will be our fixed choice for the basis in CK. Of particular importance is the set K, of the zeros uk %=1 of the Kth degree characteristic polynomial QK(u) ... 

The orthogonal characteristic polynomials or eigenpolynomials Qn(u) play one of the central roles in spectral analysis since they form a basis due to the completeness relation (163). They can be computed either via the Lanczos recursion (84) or from the power series representation (114). The latter method generates the expansion coefficients q , -r through the recursion (117). Alternatively, these coefficients can be deduced from the Lanczos recursion (97) for the rth derivative Q /r(0) since we have qni r = (l/r )Q r(0) as in Eq. (122). The polynomial set Qn(u) is the basis comprised of scalar functions in the Lanczos vector space C from Eq. (135). In Eq. (135), the definition (142) of the inner product implies that the polynomials Qn(u) and Qm(u) are orthogonal to each other (for n= m) with respect to the complex weight function dk, as per (166). The completeness (163) of the set Q (u) enables expansion of every function f(u) e C in a series in terms of the... 

In dual Lanczos transformation theory, we project onto a dynamically invariant subspace LZ or LZ called a diral Lanczos vector space. This projection is accomphshed with the projection operator... 

Given the starting vectors (ro and po), the remaining basis vectors for the dual Lanczos vector space LZ may be generated by means of the recursion relations... 

The dual Lanczos vector space generated with the starting vectors (rd and po) is finite dimensional when the Lanczos parameter vanishes for some finite value of N. For such cases, the dual Lanczos vector space is a dynamically invariant subspace of dimensionality N. This situation is realized when the operation (tm L or L pm) does not generate any new dynamical information, i.e., information not already contained in the vectors (r l x = 0, l and ft) 5 = 0,..., A - 1. ... 

One should bear in mind that a dual Lanczos vector space is a (fynamically invariant subspace by constraction, regardless of its dimensionality. 

The basis vectors for the dual Lanczos vector space may be generated with recursion relations identical inform to Eqs. (625) and (626). One only needs to replace Z with Z. As indicated earlier, the space LZ does not have to be generated when Z = Z. Even for cases not satisfying this syimnetry relation, we have not encountered a problem for which it is necessary to actually build ( z- Hence, we shall confine the remainder of our discussion to the dual Lanczos vector space LZ-... 

Lanczos vector space may be accomplished with a dual Lanczos transformation. More specifically,... 

In general, a dual Lanczos vector space depends on the choice of starting vectors (rd and po) used to build it. This is reflected in the observation that such a space does not, in general, include the whole domain of the operator L. More specifically, the operators L = Land L are not necessarily equivalent from a global point of view, i.e., in the space Q. Nonetheless, they are equivalent in the dual Lanczos vector space LZ-... 

In essence, a dual Lanczos transformation may be used to transform the retarded dynamics of a system into the closed retarded subdynamics of a dynamically invariant subspace. Of course, this dynamics is not necessarily equivalent to the retarded dynamics of the system. Nonetheless, knowledge of this subdynamics is sufficient for determining the properties of interest, provided we have properly biased the dual Lanczos vector space with the relevant dynamical information through our choice of (rol and po). 

X [X ] withX X =X X P =/. If the latter set of relations hold, the inequality must be replaced by an equality. For such cases, the dual Lanczos vector space z i fined in the whole domain of the operator L and z = ... 

Within the context of dual Lanczos transformation theory, the basic steps involved in the determination of the spectral and temporal properties of interest are as follows (i) Build a dual Lanczos vector space embedded with the appropriate dynamical information, (ii) Extract the information embedded in the dual Lanczos vector space, (iii) Utilize the extracted information to determine the spectral and temporal properties of interest. 

In order to build a dual Lanczos vector space we fo-... 

The extraction of the dynamical information embedded in the dual Lanczos vector space LZ is accomplished by forming the projected operators... 

Provided (i/f, f, Ix), and x) lie in the dual Lanczos vector space at hand, the result given by Eq. (663) applies regardless of the natnre of the underlying dynamics. Nonetheless, a dramatically simpler form may be used for the casey)f reversible systems, i.e., when the symmetry relation = -L holds. For the case of reversible systems, we can write (/ co) as follows ... 

The determination of aj (/ co) by means of Eq. (665) requires flg.oO The latter spectral function may be determined by employing Eq. (663) provided the time-reversed dynamical vectors (fo and y>o) lie in the dual Lanczos vector space at hand. As indicated earlier, this will be tme when (ro and />o) possess definite time-reversal parity. Assuming this to be the case, the normalization... 

As indicated earlier, it might not be possible and/or desirable to obtain and work with the full dual Lanczos vector space generated with the starting vectors (ro and />o). For some problems, one might And that a decent approximate treatment of the spectral and temporal properties of a system is obtained by working only with the subspace spanned by the first (5 + 1) dual Lanczos basis vectors. Such an approximate treatment is obtained by setting Pj = 0 for j > 5 + 1 and replacing and An(z) by their s + l)-dimensional approximants and... 

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