Chebyshev

Now we replace tlie first exponential in the right-hand side of (A3.11.129) by a Chebyshev expansion as follows ... 

The Taylor series by itself is not numerically stable, since the individual temis can be very large even if the result is small, but other polynomials which are highly convergent can be found, e.g. Chebyshev [M, M and M] or Lancosz polynomials [, 68]. 

Chebyshev Approximation The well known expansion of exp(— into Chebyshev polynomials T, [23] is one of the most frequently used integration technique in numerical quantum dynamics ... 

In most cases, this Lanczos-based technique proves to be superior to the Chebyshev method introduced above. It is the method of choice for the application problems of class 2b of Sec. 2. The Chebyshev method is superior only in the case that nearly all eigenstates of the Hamiltonian are substantially occupied. 

Large stepsizes result in a strong reduction of the number of force field evaluations per unit time (see left hand side of Fig. 4). This represents the major advantage of the adaptive schemes in comparison to structure conserving methods. On the right hand side of Fig. 4 we see the number of FFTs (i.e., matrix-vector multiplication) per unit time. As expected, we observe that the Chebyshev iteration requires about double as much FFTs than the Krylov techniques. This is due to the fact that only about half of the eigenstates of the Hamiltonian are essentially occupied during the process. This effect occurs even more drastically in cases with less states occupied. 

Pig. 4. Photo dissociation of ArHCl. Left hand side the number of force field evaluations per unit time. Right hand side the number of Fast-Fourier-transforms per unit time. Dotted line adaptive Verlet with the Chebyshev approximation for the quantum propagation. Dash-dotted line with the Lanczos iteration. Solid line stepsize controlling scheme based on PICKABACK. If the FFTs are the most expensive operations, PiCKABACK-like schemes are competitive, and the Lanczos iteration is significantly cheaper than the Chebyshev approximation. 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements (see Ref. 106). Spectral methods employ Chebyshev polynomials and the Fast Fourier Transform and are quite useful for nyperbohc or parabohc problems on rec tangular domains (Ref. 125). 

The tower characteristic KaV/L can be determined by integration. Normally used is the Chebyshev method for numerically ev uating the integral, whereby... 

For example, it might be stated that there is a 5% probability of x being more than 2 a from the mean. This is true only if the distribution is known to be normal. However, if the distribution is unknown but not pathological, the Chebyshev inequality provides an estimate. 

Cliebysliev s theorem provides an interpretation of the sample standard deviation, tlie positive square root of the sample variance, as a measure of the spread (dispersion) of sample observations about tlieir mean. Chebyshev s tlieorem states tliat at least (1 - 1/k ), k > 1, of tlie sample observations lie in tlie... 

Interpolation of this type may be extremely unreliable toward the center of the region where the independent variable is widely spaced. If it is possible to select the values of x for which values of f(x) will be obtained, the maximum error can be minimized by the proper choices. In this particular case Chebyshev polynomials can be computed and interpolated [11]. 

Minimax, or Chebyshev, approximation utilizes a nonlinear criterion, and hence raises problems that are much more difficult. For a given function f(x), the problem is to find a function P(x) in a certain class such that... 

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... 

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

The explicit scheme with optimal set of Chebyshev s parameters. In... 

Thus, the well-known Chebyshev polynomial defined by... 

For the explicit scheme (14) with Chebyshev s parameters (SCP) the calculations are performed by the formula... 

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. 

For example, for doing so with the set of Chebyshev s parameters tj. in increasing order... 

Summarizing, the preceding algorithm is showing a way of obtaining the ordered set of n zeroes of Chebyshev s polynomial T (t) and a stable... 

The derivation of these estimates espressing computational stability of iterative methods with optimal sets of Chebyshev s parameters r)) is omitted in the present book. In the sequel we involve only the collection r, allowing a simpler writing of the ensuing formulcis without concern of symbols... 

It seems clear that in solving the system (72) the number of the iterations within the framework of the explicit scheme with optimal set of Chebyshev s parameters or of the simple iteration scheme is proportional to 1/x/ or l/ 7, thus causing an enormous growth as 0. 

The main goal of subsequent considerations is the comparison between ADM of the type (17) with parameters (25) and the explicit method with optimal set of Chebyshev s parameters... 

When solving the model problem concerned, the transition from the A th iteration to the (k + l)th iteration is performed either in 9 steps or in 26 steps 5 operations of addition and 4 operations of multiplication during the course of the explicit Chebyshev s method and 12 operations of addition and 14 operations of multiplication in the case of ADM in connection with the double elimination (first, along the rows and then along the columns). This provides reason enough to conclude that in the case of noncommutative operators the first method is rather economical than the second one. Both 1 1... 

It is worth noting here that the same estimate for no( ) was established before for ATM with optimal set of Chebyshev s parameters, but other formulas were used to specify r] in terms of and A - If R = —A, where A is the difference Laplace operator, and the Dirichlet problem is posed on a square grid in a unit square, then... 

Chebyshev s scheme, shows that both schemes are of the same asymptotic order as —>-0 ... 

Here n is, as usual, the number of iterations. But in practical implementations Chebyshev s scheme with known values 7j and 7, is preferable, because the extra iterations and storage are necessary for later use of the three-layer scheme concerned. What is more, the second scheme depends more significantly on the errors in specifying and fhan the first one. 

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