Multistep, Multivalue, and Predictor-Corrector Methods in .NET Develop PDF 417 in .NET Multistep, Multivalue, and Predictor-Corrector Methods

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16.7 Multistep, Multivalue, and Predictor-Corrector Methods using barcode development for vs .net control to generate, create pdf417 image in vs .net applications. USD8 Here r will be a xed vec barcode pdf417 for .NET tor of numbers, in the same way that B is a xed matrix. We x by requiring that the differential equation yn+1 = f(xn+1 , yn+1 ) be satis ed.

The second of the equations in (16.7.9) is hyn+1 = hy n+1 + r2 and this will be consistent with (16.

7.10) provided r2 = 1, = hf(xn+1 , yn+1 ) hy n+1 (16.7.

12) (16.7.11) (16.

7.10). The values of r1 , r3 , a nd r4 are free for the inventor of a given four-value method to choose. Different choices give different orders of method (i.e.

, through what order in h the nal expression 16.7.9 actually approximates the solution), and different stability properties.

An interesting result, not obvious from our presentation, is that multivalue and multistep methods are entirely equivalent. In other words, the value yn+1 given by a multivalue method with given B and r is exactly the same value given by some multistep method with given s in equation (16.7.

2). For example, it turns out that the Adams-Bashforth formula (16.7.

3) corresponds to a four-value method with r1 = 0, r3 = 3/4, and r4 = 1/6. The method is explicit because r1 = 0. The Adams-Moulton method (16.

7.4) corresponds to the implicit four-value method with r1 = 5/12, r3 = 3/4, and r4 = 1/6. Implicit multivalue methods are solved the same way as implicit multistep methods: either by a predictor-corrector approach using an explicit method for the predictor, or by Newton iteration for stiff systems.

Why go to all the trouble of introducing a whole new method that turns out to be equivalent to a method you already knew The reason is that multivalue methods allow an easy solution to the two dif culties we mentioned above in actually implementing multistep methods. Consider rst the question of stepsize adjustment. To change stepsize from h to h at some point xn , simply multiply the components of yn in (16.

7.5) by the appropriate powers of h /h, and you are ready to continue to xn + h . Multivalue methods also allow a relatively easy change in the order of the method: Simply change r.

The usual strategy for this is rst to determine the new stepsize with the current order from the error estimate. Then check what stepsize would be predicted using an order one greater and one smaller than the current order. Choose the order that allows you to take the biggest next step.

Being able to change order also allows an easy solution to the starting problem: Simply start with a rst-order method and let the order automatically increase to the appropriate level. For low accuracy requirements, a Runge-Kutta routine like rkqs is almost always the most ef cient choice. For high accuracy, bsstep is both robust and ef cient.

For very smooth functions, a variable-order PC method can invoke very high orders. If the right-hand side of the equation is relatively complicated, so that the expense of evaluating it outweighs the bookkeeping expense, then the best PC packages can outperform Bulirsch-Stoer on such problems. As you can imagine, however, such a variable-stepsize, variable-order method is not trivial to program.

If. 16. . Integration of Ordinary Differential Equations you suspect that your pro .NET pdf417 2d barcode blem is suitable for this treatment, we recommend use of a canned PC package. For further details consult Gear [1] or Shampine and Gordon [2].

Our prediction, nevertheless, is that, as extrapolation methods like BulirschStoer continue to gain sophistication, they will eventually beat out PC methods in all applications. We are willing, however, to be corrected..

CITED REFERENCES AND FURT HER READING: Gear, C.W. 1971, Numerical Initial Value Problems in Ordinary Differential Equations (Englewood Cliffs, NJ: Prentice-Hall), 9.

[1] Shampine, L.F., and Gordon, M.

K. 1975, Computer Solution of Ordinary Differential Equations. The Initial Value Problem.

(San Francisco: W.H Freeman). [2] Acton, F.

S. 1970, Numerical Methods That Work; 1990, corrected edition (Washington: Mathematical Association of America), 5. Kahaner, D.

, Moler, C., and Nash, S. 1989, Numerical Methods and Software (Englewood Cliffs, NJ: Prentice Hall), 8.

Hamming, R.W. 1962, Numerical Methods for Engineers and Scientists; reprinted 1986 (New York: Dover), s 14 15.

Stoer, J., and Bulirsch, R. 1980, Introduction to Numerical Analysis (New York: Springer-Verlag), 7.

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