Fast Fourier transform method

Heather, R.W. and Metiu, H. (1987). An efficient procedure for calculating the evolution of the wave function by fast Fourier transform methods for systems with spatially extended wave function and localized potential, J. Chem. Phys. 86, 5009-5017. 

The Konnert-Hendrickson method is relatively expensive in terms of computing power [131], but recent developments that combine the use of fast Fourier transform methods (section iii below) have provided dramatic increases in speed. 

Figure 8 Structures of docked salicylhydrazine inhibitors in the IN catalytic core. Salicylhydrazines positioned by the MOE program are shown as stick models, those positioned by the fast Fourier transform method are shown as ball and stick models. SHI, SH2, and SH30 are shown on the top left, top right, and bottom, respectively.

We will describe integral equation approximations for the two-particle correlation functions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27. 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the h q)emetted chain (HNC) approximation for charged systems, are readily obtained by fast Fourier transform methods... 

The solutions to this approximation are obtained numerically. Fast Fourier transform methods and a reformulation of the HNC (and other integral equation approximations) in terms of the screened Coulomb potential by Allnatt [64] are especially useful in the numerical solution. Figure A2.3.12 compares the osmotic coefficient of a 1-1 RPM electrolyte at 25°C with each of the available Monte Carlo calculations of Card and Valleau [M]. 

A very simple procedure for time evolving the wavepacket is the second order differencing method. Here we illustrate how this method is used in conjunction with a fast Fourier transform method for evaluating the spatial coordinate derivatives in the Hamiltonian. 

GPS combined with gravity measurements The geoid determination and the height transfer across the Belt can also be based on gravity measurements. In the area of concern gravimetric measurement on land and sea have been performed in the past. Based on this data the Danish Geodetic Institute has carried out computations by Least-Squares Collocation (LSC) and by Fast Fourier Transform Method (FFT). A geoid, computed by the LSC... 

Figure 6 shows the Fourier transformed and phase-corrected spectrum in the frequency domain. The fast Fourier transform method yields in this case 512 points with a resolution of 1 KHz. The splitting of the lines occurs due to the nuclear quadrupole interaction. 

C. Fast Fourier Transform Methods for Poisson Equations... 

The LSQ method finds a solution that gives the best fit to the expraimental data. However, it suffers from the problem that the noise in the experimental data are carried through to the final result. To minimize the noise in the deconvoluted distribution, a maximum entropy method (MEM) can be applied [40], Figure 7.9 shows the isotope deconvolution of the experimental spectrum of a deuterated peptide with both LSQ method and MEM method. While LSQ gives a noisy distribution with some negative abundances, MEM produces a smoother and more realistic distribution of the trimodal isotope distribution. Similar performances were also obtained by a fast Fourier transform method [6] and an LI regularization method [8] with significantly improved computation speed. 

On the basis of the above method, the algorithm of the calculation of the Fourier transforms expressed by Eqs (2.85) and (2.100) is calculated by using the Fast Fourier Transform Method [68]. 

The basic technique used to propagate the wave packet in the spatial domain is the fast Fourier transform method [287, 288, 299, 300]. The time-dependent Schrodinger equation is solved numerically, employing the second-order differencing approach [299, 301]. In this approach the wave function Sit t = t St is constructed recursively from the wave functions at t and t" = t — St. The operator including the potential energy is applied in phase space and that of the kinetic energy in momentum space. Therefore, for each... 

Ewald summation presented above calls for the calculation of AP terms for each of the periodic boxes, a computationally demanding requirement for large biomolecular systems. Recently, Darden et al. proposed an N log N method, called particle mesh Ewald (PME), which incorporates a spherical cutoff R. This method uses lookup tables to calculate the direa space sum and its derivatives. The reciprocal sum is implemented by means of multidimensional piecewise interpolation methods, which permit the calculation of this sum and its first derivative at predefined grids with fast Fourier transform methods. The overhead for this calculation in comparison to Coulomb interactions ranges from 16 to 84% of computer time, depending on the reciprocal sum grid size and the order of polynomial used in calculating this sum. 

For the experimental model in Fig. 13, white Gaussian noise was used as the input signal to exeite the dynamies of the aetuator, whose displaeement as the output signal was measured using a laser displaeement sensor foeused 0.5 mm from its tip. The fast Fourier transform method was employed to obtain the transfer funetion in Eq. 27. The validity of this model was realized under a dynamie signal shown in Fig. 14 (John et al. 2008b). It must be noted the model validation is the last step in a typieal system identifieation proeedure. 

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