Algorithms approximate, integral

There are many algorithms for integrating the equations of motion using finite difference methods, several of which are commonly used in molecular dynamics calculations. All algorithms assume that the positions and dynamic properties (velocities, accelerations, etc.) can be approximated as Taylor series expansions ... 

Kendall, R. A., Friichtl, H. A., 1997, The Impact of the Resolution of the Identity Approximate Integral Method on Modern Ab Initio Algorithm Development , Theor. Chem. Acc., 97, 158. 

The two parameters are evaluated by means of an exact interpolation using two points inside the IntervaL The two selected points are obtained with the three-point Radau algorithm. The approximating integral is calculated analytically. This procedure is used only if the function in the three points (two internal and one extreme of the interval) is increasing or decreasing. The three-point Radau formula is adopted otherwise. 

Kendall RA, Fruchtl HA (1997) The impact of the resolution of the identity approximate integral method on modem ab initio algorithm development. Theor Chem Acc 97 158-163... 

Resolution of the Identity Approximate Integral Method on Modern Ab Initio Algorithm Development. 

In Table 1 the CPU time required by the two methods (LFV and SISM) for 1000 MD integration steps computed on an HP 735 workstation are compared for the same model system, a box of 50 water molecules, respectively. The computation cost per integration step is approximately the same for both methods so that th< syieed up of the SISM over the LFV algorithm is deter-... 

Tj = integral time constant and = derivative time constant. Upon the advent of digital control devices, this basic control algorithm was implemented as a digital approximation ... 

The third source type option in SCREEN is for area sources, which is selected by entering A or a for source type. The area source algorithm in SCREEN is based on a numerical integration approach, and allows for the area source to be approximated by a rectangular area. The inputs requested for area sources are as follows ... 

In the numerical solution the matrix structure is evaluated from Eqs. (44)-(46). Then Eqs. (47)-(49) with corresponding closure approximations are solved. Details of the solution have been presented in Refs. 32 and 33. Briefly, the numerical algorithm uses an expansion of the two-particle functions into a Fourier-Bessel series. The three-fold integrations are then reduced to sums of one-dimensional integrations. In the case of hard-sphere potentials, the BGY equation contains the delta function due to the derivative of the pair interactions. Therefore, the integrals in Eqs. (48) and (49) are onefold and contain the contact values of the functions... 

For the next higher level of integration algorithm, f(x) over segments of [a,b] can be approximated by a cubic and if this k order result is then Cote s rule can be given as... 

The MINLP-problems were implemented in GAMS [7, 8] and solved by the outer approximation/equality relaxation/augmented penalty-method [9] as implemented in DICOPT. The algorithm generates a series of NLP and MILP subproblems, which were solved by the generalized reduced gradient method [10] as implemented in CONOPT and the integrality relaxation based branch and cut method as... 

Equations (A), (B) and (C) are used in the algorithm to obtain the information required. Step (3) is used to calculate kA from equation (B), and step (4) is not required. Results are summarized in Table 12.2, for the arbitrary step-size in fA indicated G = tl[kA 1 -/A)j, and G represents the average of two consecutive values of G. The last column lists the time required to achieve the corresponding conversion in the second column. These times were obtained as approximations for the value of the integral in equation (A) by means of the trapezoidal rule ... 

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