Chebychev

The preceding discussion of the variance as a measure of the spread of a distribution about its mean has been largely qualitative. One way in which the variance can be used to give quantitative information about the distribution is through use of the Chebychev inequality, which gives an upper bound on the amount of area contained outside an interval centered at the mean. The formal statement of the Chebychev inequality is... 

Fig. 3-5 compares the bound the Chebychev inequality places on the probability P X(i) — m > e with the actual value of this quantity when the underlying distribution is gaussian. The agreement between... 

Chaise-current four vector, 545 Chebyshev approximation, 96 Chebychev inequality, 124 Chemoft, H., 102,151 Cholesky method, 67 Circuit, 256 matrix, 262... 

For smectic phases the defining characteristic is their layer structure with its one dimensional translational order parallel to the layer normal. At the single molecule level this order is completely defined by the singlet translational distribution function, p(z), which gives the probability of finding a molecule with its centre of mass at a distance, z, from the centre of one of the layers irrespective of its orientation [19]. Just as we have seen for the orientational order it is more convenient to characterise the translational order in terms of translational order parameters t which are the averages of the Chebychev polynomials, T (cos 2nzld)-, for example... 

Strategies for Spectral Analysis in Dissipative Systems Filter Diagonalization in the Lanczos Representation and Harmonic Inversion of the Chebychev Order Domain Autocorrelation Function. 

Chebychev Inequality For the following two probability distributions, find the lower limit of the probability of the indicated event using the Chebychev inequality (3-18) and the exact probability using the appropriate table ... 

If no assumption of the type of distribution can be made the values of k may be calculated from CHEBYCHEV s inequality [DOERFFEL, 1990]. It provides an absolute lower bound of probability that P % of the values of variable x are in the interval from /i - ka to fi + k a ... 

Following Tal-Ezer and Kosloff (1984) the time evolution-operator is expanded in terms of Chebychev polynomials ipk according to... 

The fulfil the same recursion relation as the Chebychev polynomials, namely... 

The time evolution operator exp(—/HAf/ft) acting on ( ) propagates the wave function forward in time. A number of propagation methods have been developed and we will briefly describe the following the split operator method [91,94,95], the Lanzcos method [96] and the polynomial methods such as Chebychev [93,97], Newtonian [98], Faber [99] and Hermite [100,101]. A classical comparison between the three first mentioned methods was done by Leforestier et al. [102]. 

The numerical study of the Orr-Sommerfeld equations requires to discretize the dy operators in equation (25). As in [84], the spectral tau-Chebychev approximations are often used, though pseudo-spectral [85] or finite element techniques [86] may be chosen too. 

For the numerical study of the whole spectrum (for g R fixed), [79] uses a spectral tau-Chebychev discretization in y and the Arnold method (see [88]) to solve the generalized eigenvalue problem (see [89]). This numerical method is based on the orthonormalization of the Krylov space of the iterates of the inverse of the matrix A B. This method has been used more recently in [90]. It has been proven efficient in the stiff problems arising in the study of spectral stability of viscoelastic fluids. 

Based on the Christoffel-Darboux formula (20) it can be shown that this procedure leads to a functional expansion which becomes an interpolation formula on the integration points, vK<7 ) = (<7 )- As an example consider an expansion by the Chebychev orthogonal polynomials g,(q) = T (q) with the constant weights W(qi) = 2/tt. The quadrature points q, are the zeros of the Chebychev polynomial of degree N + 1. On inserting Eq. (16) into the functional expansion Eq. (2) becomes... 

Figure 4 Direct grid interpolation function g iq) (n - 15, Ng = 40) for Chebychev interpolation points. The function g (q) equals one on grid point n = 15 and zero at all other grid points. The 40 interpolation points are shown as stars at the bottom of the plot.

If the expansion functions are derived from orthogonal polynomials, the matrix d can be obtained from the recursion relation for the orthogonal polynomials (32). If there is a fast transform for G (which is true for the Chebychev polynomial expansion), then applying Eq. (38) will scale as 0(Ng log Ng). 

Figure 23 The evolution of an unharmonic oscillator on a position-time grid (q, t). The potential is V(q) = q2/2 — 2 q 5. Comparison between the evolution generated by the Chebychev (left) and SOD (right) propagation schemes. The interpolation grid consists of 32 Chebychev points with a time step At = 0.05. The propagation by the SOD scheme required the same numerical effort. The upper panel represents the first two cycles. The middle panels show the evolution after 10 cycles, and the lower panels shows the evolution after 50 cycles.

Another widely used propagation scheme is the Chebychev polynomial expansion method introduced by Kosloff and Kosloff (8). This is a global propagator in the sense that it expands the propagator e u h)H in the interval [0, /]. The method is based on the Chebychev expansion relation for the function exp(iRX) (X E [—1, 1]) (13),... 

The Chebychev method converges exponentially with the number of expansion terms n for a given step size A and is particularly advantageous and efficient when A is large. However, unlike short-time propagators such as SOD or SP, the Chebychev method is not directly applicable to time-dependent or non-Hermitian Hamiltonians. 

Here, Tn is the nth Chebychev polynomial satisfying the recursion relation... 

An important variation of this procedure is to apply a window to the basic solution, h(E — H)Xi(0) (47). This focuses the calculation to a very specific, finite range of energies (even without the truncation of H). It has the consequence of improving the rate of convergence of the Chebychev expansion, since the windowed delta function is no longer singular. As a result, the coefficients decay as the summation index increases, even for a Hermitian Hamiltonian. 

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