Accelerating Convergence

Any sequence that is linear convergent can be accelerated by a method called the Aitken method. 

Let a be a linearly convergent sequence with a limit a. By definition, we 

Given a sequence a we now vnsh to construct a new sequence 5 , which will converge at a rate faster than the original sequence (a ). 

For n sufficiently large, and if we assume that all components in the sequence e have the same sign, then 

By applying Aitken s method to a linearly convergent sequence obtained from fixed point (successive substitution) iteration, we can accelerate the convergence to quadratie order. This procedure is known as the Steffenson s method, which leads to Steffenson s algorithm as follows. 

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Thus, HyperChem occasionally uses a three-point interpolation of the density matrix to accelerate the convergence of quantum mechanics calculations when the number of iterations is exactly divisible by three and certain criteria are met by the density matrices. The interpolated density matrix is then used to form the Fock matrix used by the next iteration. This method usually accelerates convergent calculations. However, interpolation with the MINDO/3, MNDO, AMI, and PM3 methods can fail on systems that have a significant charge buildup. 

Method ofWegstein This is a variant of the method of successive substitutions which forces and/or accelerates convergence. The iterative procedure Xi + = F xC) is revised by setting x + i = F xi) and then taking Xi + = qxi -i- (1 — q)xi + i, where is a suitably chosen number which may be taken as constant throughout or may be adjusted at each step. Wegstein found that suitable q s are ... 

Wegstein JH (1958) Accelerating Convergence of Iterative Processes, Commun Assoc Comp Mach, 1 9. 

Enhanced sampling in conformational space is not only relevant to sampling classical degrees of freedom. An additional reason to illustrate this particular method is that the delocalization feature of the underlying distribution in Tsallis statistics is useful to accelerate convergence of calculations in quantum thermodynamics [34], We focus on a related method that enhances sampling for quantum free energies in Sect. 8.4.2. 

In continuing efforts, the same authors constructed tris(galactoside)-modified DAB dendrimers by the accelerated convergent strategy. In this particular approach, the saccharides were connected first to a small TRIS branching derivative to form a cluster which served as a building block, once suitably functionalized.417 N-Succini-midyl-activated ester 561 was then coupled onto G(l) and G(2)-PPI dendrimers with... 

The search algorithm employs a successive approximation and accelerated convergence technique on the Independent variable in eqn (6), then approximates the dependent variable from the simultaneous solution of the equations for Mn and Mw moments of the polydisperse standard distribution. Convergence to within 0.1% of true and values of a broad MWD standard is usually achieved in six to nine iterations. 

Yanai, T., Kurashige, Y., Ghosh, D., Chan, G.K-L. Accelerating convergence in iterative solutions of large active-space self-consistent field calculations. Int. J. Quantum Chem., 2009, 109, 2178-90. 

The Accelerated Convergence Method and the Electrostatic Potential in a Point-Charge Crystal... 

For the point-charge contributions, the accelerated convergence expression, Eq. (9.21), is used with the substitution, Eq. (9.22). The explicit expression for the point-charge contribution is then (Bertaut 1952)... 

With K in the convergence function of Eq. (9.18) equal to 0.4 A-1, the accelerated convergence method reaches an accuracy of nine significant figures with inclusion of only 1000 unit cells. By comparison, about 50,000 unit cells would be required for a summation performed exclusively in direct space. 

Moore et al.[104 106a reported a double exponential dendrimer growth , which is an accelerated convergent process for the synthesis of monodendrons by means of a bidirectional procedure, where the degree of polymerization follows Equation (5.1). 

Roussel, M. R. Fraser, S. J. Geometry of the steady-state approximation. Perturbation and Accelerated Convergence Methods 1990,93 1072-1081. 

D. E. Williams, Accelerated convergence of crystal-lattice potential sums, Acta Cryst. A27 452-455 (1971). 

Performance of the CG method is generally very sensitive to roundoff in the computations that may destroy the mutual conjugacy property. The method was actually neglected for many years until it was realized that a preconditioning technique can be used to accelerate convergence significantly.14 80 81... 

Quasi-Newton methods can be viewed as extensions of nonlinear CG methods, in which additional curvature information is used to accelerate convergence. Thus, the required analytic Hessian information, memory, and computational requirements are kept as low as possible, and the main strength of Newton methods—employing curvature information to detect and move away from saddle points efficiently—is retained. 

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