Step lengths

Aq becomes asymptotically a g/ g, i.e., the steepest descent fomuila with a step length 1/a. The augmented Hessian method is closely related to eigenvector (mode) following, discussed in section B3.5.5.2. The main difference between rational fiinction and tmst radius optimizations is that, in the latter, the level shift is applied only if the calculated step exceeds a threshold, while in the fonuer it is imposed smoothly and is automatically reduced to zero as convergence is approached. 

All of these points are entirely reasonable. What is confusing about Eq. (9.83) is the step length and the undetermined constant K. Fortunately, they can both be eliminated in a single step. If Eq. (9.83) is a solution to Eq. (9.74), then both sides of the latter must be equal when the indicated operations are performed on Eq. (9.83). Carrying out these operations yields the following ... 

It is now necessary to determine the average step length (s) to obtain an expression for (H). The step length is that length moved by the molecule relative to that of the zone center and, while the molecule has move (vta) during time (ta), the zone center... 

For a step train, two neighboring steps collide when the fluctuation Wgq determined in (35) becomes the average step separation i. Therefore, the collision takes place for each step length Lcow 7 /T. To realize a large fluctuation, all the steps meander simultaneously. The step fluctuation for a long step (L > F gon) is reduced to the logarithmic form [3]... 

Anotlrer way of choosing A is to require that the step length be equal to the trust radius R, this is in essence the best step on a hypersphere with radius R. This is known as the Quadratic Approximation (QA) method. ... 

The A for the minimization modes is determined as for the RFO method, eq. (14.8). The equation for Ays is quadratic, and by choosing the solution which is larger than 8ts it is guaranteed that the step component in this direction is along the gradient, i.e. a maximization. As for the RFO step, there is no guarantee that the total step length will be within the trust radius. 

The QA method uses only one shift parameter, requiring that Ats = —A, and restricts the total step length to the trust radius (compare with eq. (14.9)). 

CINT Calculation interval or integration step length in fixed step methods. NOCI Used to specify the communication interval as NOCI CINT. 

RERR Permitted relative error per integration step length. 

Under normal circumstances, the use of a characteristic velocity equation of the type shown above can cause difficulties in computation, owing to the existence of an implicit algebraic loop, which must be solved, at every integration step length. In this the appropriate value of L or G satisfying the value of h generated in the differential mass balance equation, must be found as shown in the information flow diagram of Fig. 3.54. 

CINT The calculation interval. This is the integration step length... 

The classical solution to this problem is by limiting the step-length through the introduction of a stepping parameter, p (0[Pg.140]

If the diamond lattice itself is used for the mapping of the PE chains, each internal bead represents a methylene unit, the step length is the C-C bond length, and CxH2x 2 is represented by x beads. Typical bulk densities for n-alkane melts, which are in the range 0.7-0.8 g/cm3, are achieved with occupancy of 16-19 % of the sites on this lattice. 

A trial move of an atom consisted of a small displacement in its position and momentum simultaneously. Step lengths of 0.9 in velocity and 0.09 in position gave an acceptance rate of about 50%. A cycle consisted of one trial move of all the atoms. 

Fig. 15 Cooling curves of crystallinity (solid line) and demixing parameter of comonomers (dashed line). The latter is defined as the mean fraction of neighboring sites occupied by other comonomers around each comonomer. The cooling program is a stepwise increase of Ep/(k T) from zero with a step length of 0.002 and a step period of 300 Monte Carlo cycles, a The slightly alternating copolymer with a comonomer mole fraction 0.36 b the heterogeneous copolymer with a comonomer mole fraction of 0.36 [52]...

Figure 2. Attraction energy per surface site due to depletion as a function of the plate separation, for two polymer concentrations. The distance is expressed in lattice units (step length 1). r = 1000, x = 0.5, hexagonal lattice.

The walk displacement is represented by the number of steps only. The real distance traversed must be multiplied by the step length a. 

In minimizing a function /(x) of several variables, the general procedure is to (a) calculate a search direction and (b) reduce the value of/(x) by taking one or more steps in that search direction. Chapter 6 describes in detail how to select search directions. Here we explain how to take steps in the search direction as a function of a single variable, the step length a. The process of choosing a is called a unidimensional search or line search. 

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