Chebyshev equation

However, although these equations have theoretical support, and equation (34) has fewer coefficients than have equation (25) or the tenth-order Chebyshev equation, its fit to the data for oxygen is not as good and there are systematic deviations which the polynomials do not exhibit. It therefore seems questionable whether derived thermodynamic quantities which depend upon the differential coefficients dp/dJ or d p/dr are certain to be more accurate when calculated from equation (34) than they are when calculated from equation (25) or the Chebyshev equation. 

Antilinear operator, antiunitary, 688 Antiunitary operators, 727 A-operation, 524 upon Dirac equation, 524 Approximation, 87 methods, successive minimax (Chebyshev), 96 problem of, 52 Arc, 258... 

On computational stability of iterative methods. Until recent years the iterative method with optimal set of Chebyshev s parameters was of little use in numerical solution of grid equations. This can be explained by real facts that various sequences turn out to be nonequivalent in computational procedures. 

Equation (11) represents the first iteration of the RWP idea, but it is not the most efficient. It also represents the first appearance of the damping procedure as used in Mandelshtam and Taylor s Chebyshev iteration [4]. 

Other forms of vapor pressure equations, such as Cox equation (Osborn and Douslin 1974, Chao et al. 1983), Chebyshev polynomial (Ambrose 1981), Wagner s equation (Ambrose 1986), have also been widely used. Although... 

Interestingly, the spectral transform Lanczos algorithm can be made more efficient if the filtering is not executed to the fullest extent. This can be achieved by truncating the Chebyshev expansion of the filter,76,81 or by terminating the recursive linear equation solver prematurely.82 In doing so, the number of vector-matrix multiplications can be reduced substantially. 

Chebyshev Method for Calculating State-to-State Reaction Probabilities from the Time-Independent Wavepacket Reactant-Product Decoupling Equations. 

No divergences and dependence on the contact parameters Ti 2 remain in the form for r. It shows the transmittance function (at least in the weak-coupling limit) is indeed a well-defined molecular quantity. We can rewrite equation (38), taking into account the definition of 6 (see equation (35)) and the definition of the Chebyshev polynomials of the second kind U (cos 6) — sin[(n +l)0]/sin 6 as... 

Our main concern in this section is with the actual propagation forward in time of the wavepacket. The standard ways of solving the time-dependent Schrodinger equation are the Chebyshev expansion method proposed and popularised by Kossloff [16,18,20,37 0] and the split-operator method of Feit and Fleck [19,163,164]. I will not discuss these methods here as they have been amply reviewed in the references just quoted. Comparative studies [17-19] show conclusively that the Chebyshev expansion method is the most accurate and stable but the split-operator method allows for explicit time dependence in the Hamiltonian operator and is often faster when ultimate accuracy is not required. All methods for solving the time propagation of the wavepacket require the repeated operation of the Hamiltonian operator on the wavepacket. It is this aspect of the propagation that I will discuss in this section. 

Techniques similar to the those described in this section and in Ref. 133, but used within a time-independent framework, have been developed by Kouri and coworkers [188,189] and by Mandelshtam and Taylor [62,63]. Kroes and Neuhauser [65-68] have used the methods developed in these papers to perform time-independent wavepacket calculations using only real arithmetic. The iterative equation that lies at the heart of the real wavepacket method, Eq. (4.68), is in fact simply the Chebyshev recursion relationship [187]. This was realized by Guo, who developed similar techniques based on Chebyshev iterations [50,51]. 

In concluding this discussion it is worth noting that the type of the original equation Au = f and the operator B have no influence on a universal method of numbering the parameters r1 ..., rn that can be obtained through the use of the ordered set M n of zeroes of Chebyshev s polynomial of degree n, whose description and composition were made in Section 2 of the present chapter. 

In the present case two algorithms for the best linear approximation on a discrete set were considered, one dealing with the Li norm and the other with the Lx norm (or Chebyshev approximation). The norm approximation consistently gave better results in comparison with the assumed initial molecular weight distributions. Therefore, the application of the Lx approximation was stopped, and all that follows relates to the Li approximation. In matrix notation Equation 13 may be expressed as ... 

The shifted Chebyshev polynomials of the first kind, Th (x), lead to a constant amplitude of oscillation for the differential equation... 

The bottom sections of Tables XII, XIII, XIV, and XV show results of similar calculations for the same systems, but using the best sets of Jacobi polynomials, tabulated in Tables II and IV, in place of the Chebyshev polynomials. For each best set, the calculations were carried out in the following manner using the appropriate equations. For each temperature u niax was computed from the highest frequency of the system, and the best polynomial for the range closest to u nmx was chosen—i.e, if = 4.37T the best polynomial for the range [0,47t] was selected. 

The Chebyshev (L = 0) approximations by using Equation 47 with u max corre-... 

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