Taylor-expansion algorithm

The most common integration algorithm used in the study of biomolecules is due to Verlet [11]. The Verlet integrator is based on two Taylor expansions, a forward expansion (t + At) and a backward expansion (t — At),... 

The Verlet algorithm is not self-starting. A lower order Taylor expansion [e.g., Eq. (13)] is often used to initiate the propagation. 

The goal of all minimization algorithms is to find a local minimum of a given function. They differ in how closely they try to mimic the way a drop of water or a small ball would roll down the slope, following the surface curvature, until it ends up at the bottom. Consider a Taylor expansion around a minimum point Xq of the general one-dimensional function F(X), which can be written as... 

As done previously, in The Newton-Raphson Algorithm (p.48), we neglect all but the first two terms in the expansion. This leaves us with an approximation that is not very accurate but, since it is a linear equation, is easy to deal with. Algorithms that include additional higher terms in the Taylor expansion, often result in fewer iterations but require longer computation times due to the calculation of higher order derivatives. 

By applying the Taylor expansion as we did in Eq. (9.8), it is possible to derive an extension of the Verlet algorithm that allows these equations to be integrated numerically. This approach to controlling the temperature is known as the Nose-Hoover thermostat... 

The Verlet scheme propagates the position vector with no reference to the particle velocities. Thus, it is particularly advantageous when the position coordinates of phase space are of more interest than the momentum coordinates, e.g., when one is interested in some property that is independent of momentum. However, often one wants to control the simulation temperature. This can be accomplished by scaling the particle velocities so that the temperature, as defined by Eq. (3.18), remains constant (or changes in some defined manner), as described in more detail in Section 3.6.3. To propagate the position and velocity vectors in a coupled fashion, a modification of Verlet s approach called the leapfrog algorithm has been proposed. In this case, Taylor expansions of the position vector truncated at second order... 

Note that in die leapfrog method, position depends on the velocities as computed one-half time step out of phase, dins, scaling of the velocities can be accomplished to control temperature. Note also that no force-deld calculations actually take place for the fractional time steps. Eorces (and thus accelerations) in Eq. (3.24) are computed at integral time steps, halftime-step-forward velocities are computed therefrom, and these are then used in Eq. (3.23) to update the particle positions. The drawbacks of the leapfrog algorithm include ignoring third-order terms in the Taylor expansions and the half-time-step displacements of the position and velocity vectors - both of these features can contribute to decreased stability in numerical integration of the trajectoiy. 

The two most widely implemented numerical integration techniques within MD are the Verlet algorithm and the use of instantaneous normal mode coordinates. The Verlet algorithm begins by writing the Taylor expansion for a coordinate at time t+ Af and f- Af ... 

Investigation 3 gave a more detailed treatment of the data considered by Box et al. (1973), using an early version of Subroutine GREG. Local Taylor expansions of f(0) were used to compute constrained New ton steps toward the minimum and to obtain interval estimates of 6. Local rank testing of v was included in the selection of working responses. A collocation algorithm was used to compute the parametric sensitivities dFi xu)/d6j. directly. 

Euler s methods can be derived from a more general Taylor s algorithm approach to numerical integration. Assuming a first-order differential equation with an initial value such as [dy/dx] = / = function of x, and y = f(x,y) with y(xo) = yo. if the f(x,y) can be differentiated with respect to x and y, then the value of y at X = (xo + h) can be found from the Taylor series expansion about the point x = xq with the help ofEq. (16) ... 

As an alternative one may use the so-called Verlet algorithm. A Taylor expansion gives... 

An alternative representation is the stochastic series expansion (SSE) [41], a generalization of Handscomb s algorithm [42] for the Heisenberg model. It starts from a Taylor expansion of the partition function in orders of / ... 

A variety of algorithms have been used for integrating the equations of motion in molecular dynamics simulations of macromolecules. Most widely employed are the algorithms due to Gear91 and Verlet.90 The algorithm introduced by Verlet in his initial studies of the dynamics of Lennard-Jones fluids is derived from the two Taylor expansions,... 

The basic idea of the algorithm stems from the observation that the requirement knum = k is too strict to fulfill for all frequencies and angles of propagation. To circumvent this problem, an approximate version of (5.39), based on Taylor expansion, is utilized and then knum = k is applied in the modified equation [28], The difference between the two sides of the expression, so extracted, is defined as the error function 2D ... 

The other main class of algorithms are the predictor-corrector algorithms (Gear, 1971). New positions, velocities, accelerations and higher time derivatives of r at (n + l)At are predicted using Taylor expansions and the current values at nAt. But these are not correct, and will eventually fail, because the forces have not been updated. So the accelerations at (n + l)Af are now calculated using the predicted positions, and hence the forces at (n + l)Af, and... 

The formal equivalence between the adopted integration algorithm and the Taylor expansion Eq. [32] is used to derive the order of the error in that is carried by the in the analytical method. This automatically yields... 

Eqs. [10] and [11], were used as input to the integration algorithm whose highest time derivative of the coordinates is of order + 2, which is equivalent to a derivative of the forces of order This integration algorithm is formally equivalent to the Taylor expansion Eq. [32]. The last term in the Taylor expansion Eq. [33] contains the [X< """>(fQ)j and the highest power Assume... 

Algorithms in this family are simple, accurate, and, as we will see below, time reversible. They are the most widely used methods for integrating the classical equations of motion. The initial form of the Verlet equations [3] is obtained by utilizing a Taylor expansion at times t — dt and t + dt... 

The problem of defining the positions and veloeities at the same time can be overcome by casting the Verlet algorithm in a different way. This is the velocity-Verlet algorithm [3,6], according to which positions are obtained through the usual Taylor expansion... 

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