Newtons Approximation Technique

The fundamental physical laws governing motion of and transfer to particles immersed in fluids are Newton s second law, the principle of conservation of mass, and the first law of thermodynamics. Application of these laws to an infinitesimal element of material or to an infinitesimal control volume leads to the Navier-Stokes, continuity, and energy equations. Exact analytical solutions to these equations have been derived only under restricted conditions. More usually, it is necessary to solve the equations numerically or to resort to approximate techniques where certain terms are omitted or modified in favor of those which are known to be more important. In other cases, the governing equations can do no more than suggest relevant dimensionless groups with which to correlate experimental data. Boundary conditions must also be specified carefully to solve the equations and these conditions are discussed below together with the equations themselves. 

Equation 7.11 represents a set of two possibly nonlinear algebraic equations in terms of yj(0) and y2(0). This algebraic problem can be solved by trial and error, using for example the Newton-Raphson technique (i pendix A). This will yield y/O) and y2(0), which will form the initial conditions for Eq. 7.9. At this point, we need to assume that the numerical (approximate) solution of Eq. 7.11 will give rise to an initial condition, which will produce a trajectory that is arbitrarily close to the one with the exact initial condition. 

The Newton-Raphson technique Is used since It offers better convergence chan the Gauss-Seldel scheme. There ace, however some limitations to this technique, since the matrix Inversion procedure requires a CPU time approximately proportional to 0(N ) and Che storage Is proportional to N (where N Is Che total number of nodes In the computational region). These factors combine to make Che approach unsuitable for extension to point contact problems where Che number of nodes Is large. 

This equation may be solved for Y with either a financial calculator or an iterative solution process such as Newton s Approximation Technique. 

The kinetic equations such as Equation 7.37, Equation 7.38, Equation 7.40, and Equation 7.41 are known as transcendental equations, whose direct solution cannot be obtained. Such a kinetic equation is generally solved by the use of approximation techniques such as Newton-Raphson iterative method and nonlinear least-squares method. But, these methods have limitations of a different nature. For instance, the nonlinear least-squares method, which is most commonly used in such kinetic studies, tends to provide less reliable values of calculated kinetic parameters with increase in the number of such parameters. 

An iterative loop such as shown in Figure 1.3 may converge to a stable solution set or it may never terminate with a stable solution set. For many simple applications of the Newton-Ralpson technique it is known that a stable solution set can only be obtained if the initial approximation to the solution set is sufficiently close to the final converged solution set. However, no general rules can be given for... 

There Eire other Hessian updates but for minimizations the BFGS update is the most successful. Hessism update techniques are usually combined with line search vide infra) and the resulting minimization algorithms are called quasi-Newton methods. In saddle point optimizations we must allow the approximate Hessian to become indefinite and the PSB update is therefore more appropriate. 

In Equation 44, the intrinsic viscosity is known implicitly and can be approximated by the Newton-Raphson iteration technique (35). The iteration formula is... 

The ij element of the Jacobian represents the partial derivative of equation i with respect to variable j. If analytical derivatives are not available, elements of the Jacobian are obtained by perturbation of the state variable, requiring n + 1 function evaluations for an -equation system of equations. Various quasi-Newton techniques provide approximations to the Jacobian and do not require as many function evaluations, thus reducing computational time. 

Newton s method and quasi-Newton techniques make use of second-order derivative information. Newton s method is computationally expensive because it requires analytical first-and second-order derivative information, as well as matrix inversion. Quasi-Newton methods rely on approximate second-order derivative information (Hessian) or an approximate Hessian inverse. There are a number of variants of these techniques from various researchers most quasi-Newton techniques attempt to find a Hessian matrix that is positive definite and well-conditioned at each iteration. Quasi-Newton methods are recognized as the most powerful unconstrained optimization methods currently available. 

There has been little recent work on methods for differentiable functions which avoid explicit evaluation of derivatives. Powell s conjugate direction method 36 is still used, but the generally accepted approach is now to use standard quasi-Newton methods with finite-difference approximations to the derivatives. On the other hand there has been considerable interest in methods for nondifferentiable functions, as shown by the collection of papers edited by Balinski and Wolfe 37, in which the technique described by Lemarechal is of particular interest. Other contributions in this difficult field are due to Shor 38, ... 

Another technique for handling the mass balances was introduced by Castillo and Grossmann (1). Rather than convert the objective function into an unconstrained form, they implemented the Variable Metric Projection method of Sargent and Murtagh (32) to minimize Gibbs s free energy. This is a quasi-Newton method which uses a rank-one update to the approximation of H l, with the search direction "projected" onto the intersection of hyperplanes defined by linear mass balances. 

Numerical techniques are sometimes required in the solution of thermodynamics problems. Particularly useful is an iteration procedure that generates a sequence of approximations which rapidly converges on the exact solution of an equation. One such procedure is Newton s method, a technique for finding a root X = Xr of the equation... 

The applicability of the topographic technique can be extended by using of an approximated solution of Laplace equation. Here it is necessary to measure the radius n of the fc-th Newton ring of the meniscus surrounding the film, and thickness at which this ring emerges [70]. 

One simplifying approximation can be made for all non-zero concentrations of NaOH, the pH should be basic and we can omit from the charge balance (19.21). The above 7 equations can then be reduced to 1 non-linear equation in 1 unknown, which can be solved by a numerical technique such as Newton-Raphson iteration. Suitable numerical equation solvers are now available as software for personal computers. The range of solutions to these equations for different NaOH concentrations and temperatures is illustrated in Figure 19.1. At very high NaOH concentrations we would also have to consider the doubly deprotonated species H2Si04. ... 

Newton s Method The classical Newton s method is a technique that instead of specifying a step length at each iteration uses the inverse of the Hessian matrix, H(x)" to deflect the direction of steepest descent. The method assumes that /(x) may be approximated locally by a second order Taylor approximation and is derived quite easily by determining the minimum point of this quadratic approximation. Assuming that H(x ) is nonsingular, then the algorithmic process is defined by... 

Approximating the real function by a second-order polynomial forms the basis for the Newton-Raphson optimization techniques described in Section 12.2. 

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