Convergence with order

This gives a good estimate provided At is small enough that the method is truly convergent with order p. This process can also be repeated in the same way Romberg s method was used for quadrature. 

It is well known that for sufficiently differentiable functions, the Euler method converges with order At (0(At)). This means that the total error after n integration steps can be made as small as wanted provided the interval At is small enough. Unfortunately this says nothing about the At to be used in order to... 

In short, the error in the approximation obtained on [0, r] is reduced in direct proportion to the number of steps taken to cover this interval. Another way to say this is that e och, or, using the order notation, e = 0(h). Because the global error is of order h , where r= 1, we say that Euler s method is a first order method, or that it converges with order r = 1. 

A numerical solution converges with order p with respect to x and with order... 

If the multistep discretization defined by (J. 1.12) is consistent of order p and zero stable, then the method (5.2.6) converges with order p, i.e. the global error is... 

All explicit Runge-Kutta methods which are convergent with order q for explicit ODEs can be applied in this way to index-1 ODEs too. The order of the error in y t) and t) is q. 

The original definition of natural orbitals was in terms of the density matrix from a full Cl wave function, i.e. the best possible for a given basis set. In that case the natural orbitals have the significance that they provide the fastest convergence. In order to obtain the lowest energy for a Cl expansion using only a limited set of orbitals, the natural orbitals with the largest occupation numbers should be used. 

The pseudopotential density-functional technique is used to calculate total energies, forces on atoms and stress tensors as described in Ref. 13 and implemented in the computer code CASTEP. CASTEP uses a plane-wave basis set to expand wave-functions and a preconditioned conjugate gradient scheme to solve the density-functional theory (DFT) equations iteratively. Brillouin zone integration is carried out via the special points scheme by Monkhorst and Pack. The nonlocal pseudopotentials in Kleynman-Bylander form were optimized in order to achieve the best convergence with respect to the basis set size. 5... 

The existence of very small energy differences ( 0.5 mRy) between the fct and bcc phase in CuZn, forces us to take a suffir intly large k-mesh with 2500 k-points per cell, whereas for the FeaNi system we may use a much smaller k-mesh (between 500 and 900) for good convergence. In order to get this high precision for the CuZn total... 

Convergence in order k was tracked for three different particle numbers, N = 10,30,75, and for two values of the interaction strength, V = 0.8,1.6. Comparisons were made by recording the percentage of the correlation accounted for using second-, third-, and fourth-order approximations, with the results recorded in Table I. 

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