Lanczos orthogonal polynomials

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) ... 

This finding, which was mentioned in Section 4, exhibits a remarkably regular factorability of the general result for H (c0) as a direct consequence of a judicious combination of symmetry of the Hankel determinant (46) and the Lanczos orthogonal polynomials. Thus, given either the set /3 or qVi0, the Hankel determinant H (c0) or equivalently the overlap determinant detS in the Schrodinger or Krylov basis rj can be constructed at once from Eq. (175). This is very useful in practical computations. 

The differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [Lanczos, C.,/. Math. Phys. 17 123-199 (1938) and Lanczos, C., Applied Analysis, Prentice-Hall (1956)]. A major improvement was proposed by Villadsen and Stewart [Villadsen, J. V., and W. E. Stewart, Chem. Eng. Sci. 22 1483-1501 (1967)], who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. The error decreases very rapidly as N is increased since it is proportional to [1/(1 - N)]N 1. See Finlayson (2003) and Villadsen, J. V., and M. Michelsen, Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall (1978). 

The work needed to assemble the coefficient values is generally minimized when the collocation points are chosen freely. However, better convergence performance is normally achieved by the orthogonal collocation methods because a more optimal distribution of the collocation points is achieved. In the orthogonal collocation methods the discretization points are chosen as the zeros in certain orthogonal polynomials. In the historical survey of the spectral methods given by Canute et al. [28], it was assumed that Lanczos [118] was the first to reveal that a proper choice of trial functions and distribution of collocation points is crucial to the accuracy of the solution of ordinary differential equations. Villadsen and... 

The alternative recursion (57) involves the monic Lanczos states ] ) Here, the term monic" serves to indicate that for any given integer n, the highest state ) in the finite sum, which defines the vector jr ), always has an overall multiplying coefficient equal to unity similarly to a monic polynomial [2], The Lanczos states ) are orthogonal, but unnormalized, as opposed to orthonormalized Lanczos states ( linear combinations of powers of the operator U(r) acting on the initial state 0)- Therefore, due to the relation 4> ) = U (r) 0) from Eq. (36), the vectors tjrn) and [ ) are certain sums of Schrodinger states ). ... 

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). 

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 -... 

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... 

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