Multistep, Multivalue, and Predictor-Corrector Methods in .NET Drawer barcode pdf417 in .NET Multistep, Multivalue, and Predictor-Corrector Methods

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16.7 Multistep, Multivalue, and Predictor-Corrector Methods using visual .net tocreate pdf417 on web,windows application iOS of the error ter pdf417 for .NET m by a fractional amount. So dubious an improvement is certainly not worth the effort.

Your extra effort would be better spent in taking a smaller stepsize. As described so far, you might think it desirable or necessary to predict several intervals ahead at each step, then to use all these intervals, with various weights, in a Simpson-like corrector step. That is not a good idea.

Extrapolation is the least stable part of the procedure, and it is desirable to minimize its effect. Therefore, the integration steps of a predictor-corrector method are overlapping, each one involving several stepsize intervals h, but extending just one such interval farther than the previous ones. Only that one extended interval is extrapolated by each predictor step.

The most popular predictor-corrector methods are probably the AdamsBashforth-Moulton schemes, which have good stability properties. The AdamsBashforth part is the predictor. For example, the third-order case is predictor: yn+1 = yn + h (23yn 16yn 1 + 5yn 2 ) + O(h4 ) 12 (16.

7.3). Here information at the current point xn , together with the two previous points xn 1 and xn 2 (assumed equally spaced), is used to predict the value yn+1 at the next point, xn+1 . The Adams-Moulton part is the corrector. The third-order case is corrector: yn+1 = yn + h (5y + 8yn yn 1 ) + O(h4 ) 12 n+1 (16.

7.4). Without the tria PDF-417 2d barcode for .NET l value of yn+1 from the predictor step to insert on the right-hand side, the corrector would be a nasty implicit equation for yn+1 . There are actually three separate processes occurring in a predictor-corrector method: the predictor step, which we call P, the evaluation of the derivative yn+1 from the latest value of y, which we call E, and the corrector step, which we call C.

In this notation, iterating m times with the corrector (a practice we inveighed against earlier) would be written P(EC)m . One also has the choice of nishing with a C or an E step. The lore is that a nal E is superior, so the strategy usually recommended is PECE.

Notice that a PC method with a xed number of iterations (say, one) is an explicit method! When we x the number of iterations in advance, then the nal value of yn+1 can be written as some complicated function of known quantities. Thus xed iteration PC methods lose the strong stability properties of implicit methods and should only be used for nonstiff problems. For stiff problems we must use an implicit method if we want to avoid having tiny stepsizes.

(Not all implicit methods are good for stiff problems, but fortunately some good ones such as the Gear formulas are known.) We then appear to have two choices for solving the implicit equations: functional iteration to convergence, or Newton iteration. However, it turns out that for stiff problems functional iteration will not even converge unless we use tiny stepsizes, no matter how close our prediction is! Thus Newton iteration is usually an essential part of a multistep stiff solver.

For convergence, Newton s method doesn t particularly care what the stepsize is, as long as the prediction is accurate enough. Multistep methods, as we have described them so far, suffer from two serious dif culties when one tries to implement them: Since the formulas require results from equally spaced steps, adjusting the stepsize is dif cult..

16. . Integration of Ordinary Differential Equations Starting and s topping present problems. For starting, we need the initial values plus several previous steps to prime the pump. Stopping is a problem because equal steps are unlikely to land directly on the desired termination point.

Older implementations of PC methods have various cumbersome ways of dealing with these problems. For example, they might use Runge-Kutta to start and stop. Changing the stepsize requires considerable bookkeeping to do some kind of interpolation procedure.

Fortunately both these drawbacks disappear with the multivalue approach. For multivalue methods the basic data available to the integrator are the rst few terms of the Taylor series expansion of the solution at the current point xn . The aim is to advance the solution and obtain the expansion coef cients at the next point xn+1 .

This is in contrast to multistep methods, where the data are the values of the solution at xn , xn 1, . . .

. We ll illustrate the idea by considering a four-value method, for which the basic data are yn hy (16.7.

5) yn 2 n (h /2)yn 3 (h /6)yn It is also conventional to scale the derivatives with the powers of h = xn+1 xn as shown. Note that here we use the vector notation y to denote the solution and its rst few derivatives at a point, not the fact that we are solving a system of equations with many components y. In terms of the data in (16.

7.5), we can approximate the value of the solution y at some point x: y(x) = yn + (x xn )yn + (x xn )2 (x xn )3 yn + yn 2 6 (16.7.

6). Set x = xn+1 in visual .net PDF417 equation (16.7.

6) to get an approximation to yn+1 . Differentiate equation (16.7.

6) and set x = xn+1 to get an approximation to yn+1 , and similarly for yn+1 and yn+1 . Call the resulting approximation yn+1 , where the tilde is a reminder that all we have done so far is a polynomial extrapolation of the solution and its derivatives; we have not yet used the differential equation. You can easily verify that yn+1 = B yn where the matrix B is 1 0 B= 0 0 1 1 0 0 1 2 1 0 1 3 3 1 (16.

7.7). (16.7.8).

We now write the .NET PDF417 actual approximation to yn+1 that we will use by adding a correction to yn+1 : yn+1 = yn+1 + r (16.7.

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