Polynomials Lanczos

The Jacobi matrix and the Lanczos polynomial matrix within... 

In Eq. (162), the result (156) is used to identify the residue dk. Note that the upper limit K — 1 in Eq. (163) could be replaced by K on account of the characteristic equation Qk(u>c) = 0. The expression (163) represents the local completeness relation or closure for the Lanczos polynomial Qn(uk) -Next, we are interested in considering the weighted products dkQn(uk)QM summed over all the eigenvalues uk k=1 for two arbitrary degrees n and m ... 

This is the orthogonality relation of the two Lanczos polynomials Q (m) and Qm(u) with the weight function, which is the residue dk [48]. We recall that the sequence Q = (Q ( z-) coincides with the set of eigenvectors of the Jacobi matrix (60). 

Matrix elements of the evolution operator in terms of the Lanczos polynomials... 

This duality enables switching from the work with the Lanczos state vectors fn) to the analysis with the Lanczos polynomials Q (m). A change from one representation to the other is readily accomplished along the lines indicated in this section, together with the basic relations from Sections 11 and 12, in particular, the definition (142) of the inner product in the Lanczos space CK, the completeness (163) and orthogonality (166) of the polynomial basis Qn,k -... 

Inspecting the explicit expressions for the first few Lanczos polynomials, we deduced the following expression for the general degree n ... 

Further, regarding ijrn), it is also possible to have the explicit Lanczos algorithm by deriving the expression that holds the whole result with no recourse to recurrence relations. For example, applying the explicit Lanczos polynomial operator Q (U) from Eq. (177 to tq) = 0) will generate the wave packet IV n) according to ijrn) = Q (U) V o) as in Eq. (91). Therefore, the final result is the following expression for the explicit Lanczos states Vr ) ... 

If the set of the couplings anrp is precomputed, it is clear from the preceding section that the LCF is technically more efficient than the PLA. This is because the PLA still needs to generate the Lanczos polynomials Pk(u),Qk(u) to arrive at Eq. (205), whereas LCF does not. The LCF is an accurate, robust, and fast processor for computation of shape spectra with an easy way of programming implementations in practice. For parametric estimations of spectra, there are two options. We can search for the poles in Eq. (230) from the inherent polynomial equation after the LCF is reduced to its polynomial quotient, which is precisely the PLA. In such a case, the efficiency of the LCF is the same as that of the PLA. However, the poles in Eq. (230) can be obtained without reducing 7< cf(m) to the polynomial quotient. Since A 0, as per Eq. (61), we can rewrite the characteristic equation (112) as Qk(u) = 0. This can be stated as the tridiagonal secular equation ... 

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