Chebyshev approximation

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 ... 

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. 

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... 

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

The problem [Eq. (15)] is a minimax optimization problem. For the case (as it is here) where the approximating function depends linearly on the coefficients, the optimization problem [Eq. (15)] has the form of the Chebyshev approximation problem and has a known solution (Murty, 1983). Indeed, it can be easily shown that with the introduction of the dummy variables z, z, z the minimax problem can be transformed to the following linear program (LP) ... 

This procedure is also known as uniform or Chebyshev approximation. We have the introduce the single auxiliary variable s > 0 to translate the minimization of (1.83) into the problem... 

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 ... 

Much more precise approximation forms were given by Cody, who used Chebyshev approximations. Three rational approximations for the function exp(x )erfc(x) with maximal relative errors ranging down to 10 were derived. These forms and the corresponding intervals are. 

The Chebyshev-Jackson approximation smooths the ripples that occur in a standard Chebyshev approximation for any function with sharp features. This effect is obtained by damping the high order coefficients, c,, in the sum (6.8) by the damping factors, gf e [0,1], given by (6.10). 

Parks, T.W. and McClellan, J.H. 1972a. Chebyshev approximations for non recursive digital filters with linear phase. IEEE Trans. Circuit Theory CT-19 189-194. 

Fox L (1962) Chebyshev approximation. In Fox L (Ed) Numerical Solution of Ordinary and Partial Differential Equations. Pergammon Press,... 

In ref 139 the authors presented variable-stepsize Chebyshev-type methods for the integration of second-order initial-value problems. More specifically, Panovsky and Richardson in ref. 140 presented a method based on Chebyshev approximations for the numerical solution of the problem y" = f(x,y), with constant stepsize. In ref. 141 Coleman and Booth analyzed the method developed in ref 140 and proposed the convenience to design a variable stepzesize methods of Chebyshev-type. The development of the new methods is based on the test equation ... 

The authors have given the exact solution of the above test equation at y t). Based on the above exact solution, they calculate the exact solution at y t At). After some algebraic manipulations (in order to avoid the existence of y t)) and using the finite Chebyshev approximations in order to replace the/, they produce a system of n equations with n unknowns. The determination of these unknowns gives the coefficients of the new methods. We note that finally the developed method is of second algebraic order. For comparison purposes the authors use the following methods ... 

In ref. 161 the authors develop numerical methods based on Chebyshev approximations with variable stepsize. This was an extension of the papers of J. Panovsky and D. L. Richardson and of John P. Coleman, Andrew S. Booth. ... 

In ref. 164 the authors consider a new BDF fourth-order method for solving stiff initial-value problems, based on Chebyshev approximation. The authors prove that the developed method may be presented as a Runge-Kutta method having stage order four. They examine the stability properties of the method and they presented a strategy for changing the step size based on embedded pair of the Runge-Kutta schemes. 

A. Higinio Ramos and A. Jesus Vigo-Aguiar, A fourth-order Runge-Kutta method based on BDF-type Chebyshev approximations, Journal of Computational and Applied Mathematics, 2007, 204, 124 136. 

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