Converged eigenvalues

The Cullum-Willoughby test,27,37 on the other hand, was designed to identify the so-called spurious eigenvalues, rather than the converged eigenvalues. In particular, the tridiagonal matrix T and its submatrix, obtained... 

The two vectors in Eq. [43] are used to initiate two Lanczos recursions that yield not only converged eigenvalues but also the residues R j . In particular, Wyatt and Scott have shown that these residues are simply the squared first elements of the Lanczos eigenvectors in a K-step recursion 46... 

For this algorithm to converge, eigenvalues of the Jacobian (0g/0 ) must have absolute values less than 1. Thus, the choice of the function g(jc), on the basis of physico-chemical considerations, is of particular importance. Since the convergence of a direct iterative method is only linear, good initial approximations to the solution are needed. Otherwise, Newton s algorithm must be preferred. 

Figure 11 Sampling efficiency defined by the ratio of converged eigenvalues within a fixed number of digits to the number of sampling points, as a function of the logarithm of the number of sampling points. The heavy line represents the asymptotic value of tt/4.

Figure 12 Sampling efficiency defined as the ratio of converged eigenvalues divided by NK, for a fixed number of points (Ng = 64) as a function of the grid spacing Aq. The optimal sampling spacing is marked. At Aqapl the sampling efficiency approaches tt/4 at the cusp point Aq. ...

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