Convergence rate linear

A rather slow convergence rate (linear convergence) persists. 

In the above equation, the norm is usually the Euclidean norm. We have a linear convergence rate when 0 is equal to 1. Superlinear convergence rate refers to the case where 0=1 and the limit is equal to zero. When 0=2 the convergence rate is called quadratic. In general, the value of 0 depends on the algorithm while the value of the limit depends upon the function that is being minimized. 

On the debit side, the linearization method is quite sensitive to the form of the network element model. Jeppson and Tavallaee (J2) reported that convergence rate was slow when the usual pump and reservoir models were incorporated, but they obtained significant improvements after the models had been suitably transformed. Although the number of iterations required is small using formulations A and B, the dimension of the matrix equation is substantial. Hence, it becomes essential to use sparse computation techniques if the method is to retain its competitive edge in larger problems. 

Steepest descent is simple to implement and requires modest storage, O(k) however, progress toward a minimum may be very slow, especially near a solution. The convergence rate of SD when applied to a convex quadratic function, as in Eq. [22], is only linear. The associated convergence ratio is no greater than [(k - 1)/(k + l)]4 where k, the condition number, is the ratio of largest to smallest eigenvalues of A ... 

As the convergence ratio measures the reduction of the error at every step (llx +i x ll — Pita x ll for a linear rate), the relevant SD value can be arbitrarily close to 1 when k is large (Figure 12). In other words, because the n lengths of the elliptical axes belonging to the contours of the function are proportional to the eigenvalue reciprocals, the convergence rate of SD is slowed as the contours of the objective function become more eccentric. Thus, the SD search vectors may in some cases exhibit very inefficient paths toward a solution (see final section for a numerical example). 

Figure 32 shows a typical microelectrode voltammogram for an electro-chemically reversible system under near steady-state conditions. Of course at very fast scan rates the behaviour returns to that of planar diffusion and a characteristic transient-type cyclic voltammetric response is obtained as the mass transport changes from convergent to linear diffusion. 

For roots of multiplicity 1, Newton s method has quadratic convergence when Xn is close to the solution. If the multiplicity of a root is greater than 1, the convergence rate becomes linear. Knowing the multiplicity k of the root one can obtain quadratic convergence using... 

If we use the Euclidean norm, then (3F/3x) = which is the largest eigenvalue of (dF/dx) in magnitude. From this expression, we can show a linear convergence rate, but the speed of these iterations is related to l)i l. Now by recurring the iterations for k, we can develop the following relationship ... 

For the Poisson problem, the total number of grid points for all the grid levels in three dimensions is only a small constant (greater than one) times the number of fine-grid points (indicated by Ng). It is observed that the convergence rate does not depend on the system size, and the residual decreases exponentially with the number of iterations, i.e., a plot of the log of the residual vs. the number of iterations decreases linearly. Thus, the... 

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