Convergence factor

Therefore a convergence factor p can be used to speed up or slow down the rate at which X( , is permitted to change from iteration to iteration. 

The integral is indeterminate at the upper limit but the introduction of a convergence factor reduces the value at the upper limit to zero. As Fermi points out, this is a reasonable procedure as the beam intensity will decay at its edges. We thus obtain... 

Thus, it becomes apparent the output and the impulse response are one-sided in the time domain and this property can be exploited in such studies. Solving linear system problems by Fourier transform is a convenient method. Unfortunately, there are many instances of input/ output functions for which the Fourier transform does not exist. This necessitates developing a general transform procedure that would apply to a wider class of functions than the Fourier transform does. This is the subject area of one-sided Laplace transform that is being discussed here as well. The idea used here is to multiply the function by an exponentially convergent factor and then using Fourier transform technique on this altered function. For causal functions that are zero for t < 0, an appropriate factor turns out to be where a > 0. This is how Laplace transform is constructed and is discussed. However, there is another reason for which we use another variant of Laplace transform, namely the bi-lateral Laplace transform. 

In the special case of a Coulomb potential we perform the integration ill (3.40) by introducing a convergence factor and taking the limit ju 0 after the integration. We write (3.40) in the form... 

In random walk simulations with configurational temperature, the calculations are started with a convergence factor / = exp(O.l). When / > exp(10 ), the density of states calculated from the temperature is used as the initial density of states for the next stage, the convergence factor is reduced by and the temperature accumulators are reset to zero... 

Fig. 1. Statistical errors in the density of states as a function of convergence factor [17]...

Another approach to tackle the conditionally convergent Coulomb sum is used by Strebel and Sperb [15] and called the MMM method. Instead of defining the summation order, the sum is made convergent by means of a convergence factor ... 

The limit E exists and has as its value that of the Ewald method minus the dipole term [16]. Starting from this convergence factor approach, Strebel and Sperb constructed a method of computational order 0(A / ) or, with a more clever algorithm, (P(AllogAl), MMM [15]. Unlike the particle mesh Ewald methods, no mesh is introduced, so that no interpolation errors occur, and it is comparatively easy to find error estimates for the method. Consequently, this method allows much higher precisions compared to P M or (S)PME. 

Due to the convergence factor, it is possible to calculate the total energy by a pairwise potential 4> such that... 

As described above, we now introduce the two formulas used to calculate (j> as in (15), however only for two periodic coordinates x and y. Moreover a cubic simulation box is often inappropriate for partially periodic systems, therefore we use a general simulation box of dimensions Xx x Xy x Xz- Here, the convergence factor energy E is equal to the energy obtained from the standard spherical summation, and no dipole term occurs. The MMM2D far formula is... 

Here, f/ is a positive infinitesimal which ensures that the integrals in Eq. (2) are convergent (see Section III for a discussion of the physical reason for the appearance of this convergence factor). The summation in Eq. (2) extends over all states m> different from the reference state 0>. The real values of the (first-order) pole of P 6 are... 

The exponential convergence factor minimizes series termination errors. To the extent that 1//(s) )/(j) /(/b - /app) in the case of Eq. (7a) and FiFjfFkF s in the case of Eq. (7b) are nearly constant, and 117,(5) -17,(5), smin and k are equal to zero, the individual peaks of the radial distribution curves will have nearly Gaussian shapes. In these cases D(r) will be given to good approximation by... 

For the Hermitian Hamiltonian ftnmm the norm of the vectors defined by recursion relations, Eq. (27), does not decay with the number n, and therefore the series in Eq. (26) does not converge in the usual sense. Nevertheless, it does converge in the following more general sense. One of the ways to sum such a series is to make an analytic continuation in tp by introducing a constant convergence factor ze ... 

Since the nuclear magnetic moments are small in magnitude, the corrections arising from the commutator relations can be neglected. However, it is convenient to retain k0 as a convergence factor so that we arrive at the orbital hyperfine interaction in the form... 

To overcome this problem we may define the Fourier transform to include a convergence factor exp( — t /K where >/ is a small, real, positive quantity. After the integration is performed, we can then take the limit / - 0+ (Mat-tuck, 1967). The Fourier transform of the GF may then be expressed as... 

The Fourier transform of Eq. (6.7) then becomes [the definition of the Fourier transform of the GF always contains the exp(-i f ) convergence factor, although henceforth we do not explicitly express this fact]... 

Such integrals can be made well defined by contour integration in the complex plane and with appropriate convergence factors and limiting procedures, as discussed in the previous chapter. 

Fig. 2 Convergence factor, y = r(P,u)— T[P/2,u)] as a function of number of beads P. Symbols alternate filled or open with each temperature and indicate classical-beads (CB) approach (circles) SCB-QFH (squares) SCB-TI (diamonds). Temperatures are T = 2.5 K (black open symbols connected by solid lines) T = 10.0 K (redfilled symbols connected by dotted lines) T = 50.0 K (green open symbols connected by dashed lines) T = 500.0 K (blue filled symbols connected by dash-dot lines). Confidence limits (68 %) are smaller than the symbol sizes except where shown...

The parameter e(e > 0) is a convergence factor which is set equal to zero at the end of the calculation. Hint First multiply both sides of the differential equation obeyed by G(x, x t) by and integrate over f,... 

Yet another approach to tackle the conditionally convergent Coulomb stun is used for MMM. Instead of defining the summation order, one can also multiply each summand by a continuous factor c, nj, Ukim) such that the sum is absolutely convergent for > 0, but c(0,., .)= . The energy is then defined as the limit 0 of the sum, i.e., is an artificial convergence parameter. For a convergence factor of the limit is the same as the spherical limit, and... 

Starting from this convergence factor approach along the lines of the Lekner sum, R. Strebel and R. Sperb constructed a method of computational order O (N log N), MMM [54]. The favorable scaling is obtained, very much as in the Ewald case, by technical tricks in the calculation of the far formula. The far formula has a product decomposition and can be evaluated hierarchically, similar in spirit to the fast multipole methods. 

Equation 51 is derived using the same convergence factor approach as used for Eq. 49, and consequently the same singularity in /3 is obtained (and omitted). This is important, since otherwise the charge neutrality argument does not hold and the limit /3 0 cannot be performed. 

Recently we proposed a new method called MMM2D [24,58], which has an 0(N / ) complexity and full error control that is based on a convergence factor approach similar to MMM [12]. In two dimensions the convergence factor based methods and the Ewald sum methods yield exactly the same results. However, this will still only allow simulations including up to a few thousand charges due to the power law scaling. 

---