Newton’s methods

Another option is a q,p) = p and b q,p) = VU q). This guarantees that we are discretizing a pure index-2 DAE for which A is well-defined. But for this choice we observed severe difficulties with Newton s method, where a step-size smaller even than what is required by explicit methods is needed to obtain convergence. In fact, it can be shown that when the linear harmonic oscillator is cast into such a projected DAE, the linearized problem can easily become unstable for k > . Another way is to check the conditions of the Newton-Kantorovich Theorem, which guarantees convergence of the Newton method. These conditions are also found to be satisfied only for a very small step size k, if is small. 

A difficulty with the energy conserving method (6), in general, is the solution of the corresponding nonlinear equations [6]. Here, however, using the initial iterate (q + A p , p ) for (q +i, p +i), even for large values of a we did not observe any difficulties with the convergence of Newton s method. 

The steepest descent method is quite old and utilizes the intuitive concept of moving in the direction where the objective function changes the most. However, it is clearly not as efficient as the other three. Conjugate gradient utilizes only first-derivative information, as does steepest descent, but generates improved search directions. Newton s method requires second derivative information but is veiy efficient, while quasi-Newton retains most of the benefits of Newton s method but utilizes only first derivative information. All of these techniques are also used with constrained optimization. 

If q is an initial approximate solution, then a better approximation is given by Newton s method according to... 

One application of Newton s method gives to three significant figures e = 0.00748,... 

Figure 14.1 Newton s method. Dotted line is tangent at j = 0.5...

Newton s method can be easily re-written for the problem of finding stationary points, where (d//dx) = 0 rather than j = 0. The formula 14.9 becomes... 

Newton, R., 517, 536,539,560 Newton-Raphson method, 86 Newton s method, 79,80 Newton, T. 0., 555 Neyman-Pearson lemma, 306 Nodal point, 326 Node, 326 proper, 326 Noise... 

Refer to equation 5, which relates Ny to the parameters in the reactor. For the continuous reactor these parameters are evaluated at t = t. However, the solution to equation 5 is complicated by the fact that Ny is not only on the left hand side, but Ny also appears in the expression for Rj p as a first power. Newton s method of convergence is used to solve equation 5 for the continuous reactor. 

Use Newton s method to solve the algebraic equations in Example 4.2. Note that the first two equations can be solved independently of the second two, so that only a two-dimensional version of Newton s method is required. 

This example found the reactor throughput that would give the required annual capacity. For prescribed values of the design variables T and V, there is only one answer. The program uses a binary search to find that answer, but another root-finder could have been used instead. Newton s method (see Appendix 4) will save about a factor of 4 in computation time. 

The next example treats isothermal and adiabatic PFRs. Newton s method is used to determine the throughput, and Runge-Kutta integration is used in the Reactor subroutine. (The analytical solution could have been used for the isothermal case as it was for the CSTR.) The optimization technique remains the random one. 

WBout = bout Q MwB LOOP End of Newton s method CALL Cost(WAin, V, aout, bout, cout, total)... 

A double trial-and-error procedure is needed to determine uq and Tq. If done only once, this is probably best done by hand. This is the approach used in the sample program. Simultaneous satisfaction of the boundary conditions for concentration and temperature was aided by using an output response that combined the two errors. If repeated evaluations are necessary, a two-dimensional Newton s method can be used. Dehne... 

If the mixture has gelled, the program proceeds to calculate P(Fa° ) and P(Fg° ) using a binary search method (lines 2510-2770). This method is more convenient that the earlier approach of Bauer and Budde (10) who used Newton s method, since derivatives of the functions are not required. The program also calculates the probability generating functions used to calculate sol fractions and the two crosslink densities (lines 2800-3150). Finally, the sol fraction and crosslink densities are calculated and printed out (lines 3160-3340). The program then asks for a new percents of reaction for the A and B groups. To quit enter a percent reaction for A of >100. 

In this chapter the new difference schemes are constructed for the quasilin-ear heat conduction equation and equations of gas dynamics with placing a special emphasis on iterative methods available for solving nonlinear difference equations. Among other things, the convergence of Newton s method is established for implicit schemes of gas dynamics. 

In this regard, Newton s method suits us perfectly in connection with solving the nonlinear difference equation (2). It is worth recalling here its algorithm ... 

A difference scheme. Newton s method. We now proceed to constructing difference schemes for the quasilinear heat conduction equation. 

In practical implementations Newton s method converges with any prescribed accuracy e only if... 

It is worth noting here that Newton s method is quite applicable for solving problem (35) in addition to the well-established method of iterations. 

Numerical solution of difference equations by Newton s method. As... 

After that, applying Newton s method yields... 

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