Modified Newton

Techniques used to find global and local energy minima include sequential simplex, steepest descents, conjugate gradient and variants (BFGS), and the Newton and modified Newton methods (Newton-Raphson). 

Usually, modified Newton-Raphson methods with relaxation are applied. Additional iteration loops are necessary for the determination of the dynamic pressure losses in ducts and duct fittings. 

The modified Newton method [12] offers one way of dealing with multiple roots. If a new function is defined... 

The Gill-Murray modified Newton s method uses a Cholesky factorization of the Hessian matrix (Gill and Murray, 1974). The method is described in detail by Scales (1985). 

Modified Newton methods require calculation of second derivatives. There might be cases where these derivatives are not available analytically. One may then calculate them by finite differences (Edgar and Himmelblau, 1988 Gill et al. 1981 Press et al. 1992). The latter, however, requires a considerable number of... 

It is noted that the Rosenbrock function given by the next equation has been used to test the performance of various algorithms including modified Newton s and conjugate gradient methods (Scales, 1986)... 

In the lumped parameter model, the transient temperature of a single droplet during flight in a high speed atomization gas is calculated using the modified Newton s law of cooling, 1561 considering the frictional heat produced by the violent gas-droplet interactions due... 

In a supersonic gas flow, the convective heat transfer coefficient is not only a function of the Reynolds and Prandtl numbers, but also depends on the droplet surface temperature and the Mach number (compressibility of gas). 154 156 However, the effects of the surface temperature and the Mach number may be substantially eliminated if all properties are evaluated at a film temperature defined in Ref. 623. Thus, the convective heat transfer coefficient may still be estimated using the experimental correlation proposed by Ranz and Marshall 505 with appropriate modifications to account for various effects such as turbulence,[587] droplet oscillation and distortion,[5851 and droplet vaporization and mass transfer. 555 It has been demonstrated 1561 that using the modified Newton s law of cooling and evaluating the heat transfer coefficient at the film temperature allow numerical calculations of droplet cooling and solidification histories in both subsonic and supersonic gas flows in the spray. 

MMI/MMPI incorporates a modified Newton-Raphson energy minimization algorithm that moves atoms one by one and is quite efficient. The force field is parameterized not only for saturated hydrocarbons including cyclopropane, but also for nonconjugated olefins (17c),... 

Molecule Starting coordinates Pattern Search (BIGSTRN) Modified Newton-Raphson (MMI)... 

When f is nonlinear, as it nearly always is, then an iteration is required to determine y +i. For stiff problems, the iterative solution is usually accomplished with a modified Newton method. We seek the solution yn+i to a nonlinear system that may be stated in residual form as... 

The modified Newton algorithm for solving such a system is discussed in Section 15.5.2. The correction vector for the mth iteration, defined as Ay= y " "J"1 — yl +1 may be found by solving the following linear system of equations ... 

Fig. 15.5 Illustration of the full Newton and modified Newton algorithm on a scalar problem, F(y) = 0. The curve represents a nonlinear function F(y), and the solution is the value of y at which the function is zero.

The modified Newton iteration, and the reason that damping is effective, can be explained in physical terms. Chemical-kinetics problems often have an enormous range of characteristic scales—this is the source of stiffness, as discussed earlier. These problems are also highly nonlinear. 

In solving the underlying model problem, the Jacobian matrix is an iteration matrix used in a modified Newton iteration. Thus it usually doesn t need to be computed too accurately or updated frequently. The Jacobian s role in sensitivity analysis is quite different. Here it is a coefficient in the definition of the sensitivity equations, as is 3f/9a matrix. Thus accurate computation of the sensitivity coefficients depends on accurate evaluation of these coefficient matrices. In general, for chemically reacting flow problems, it is usually difficult and often impractical to derive and program analytic expressions for the derivative matrices. However, advances in automatic-differentiation software are proving valuable for this task [36]. 

Program a modified Newton method to solve the problem, seeking the solution near x 0.5. Explore the performance of the algorithm (including failure to converge) beginning with initial iterates of xo = 1 and xo = 3. 

Twopnt, solves systems of nonlinear, stiff, boundary-value problems [158]. It implements a hybrid, damped, modified Newton algorithm [159]. Local error is controlled using adaptive placement of mesh intervals. 

Once p has been determined we calculate the step from the modified Newton equations Eq. (3.24). Therefore, the RF and RSO steps Eire calculated in the same way. The only difference is the prescription for determining the level shift. In the RSO approach p reflects the trust radius h, in the RF model p reflects the metric S. By varying h and S freely the same steps are obtained in the two models. 

First, when Hk is not positive-definite, the search direction may not exist or may not be a descent direction. Strategies to produce a related positive-definite matrix Hk, or alternative search directions, become necessary. Second, far away from x, the quadratic approximation of expression [34] may be poor, and the Newton direction must be adjusted. A line search, for example, can dampen (scale) the Newton direction when it exists, ensuring sufficient decrease and guaranteeing uniform progress toward a solution. These adjustments lead to the following modified Newton framework (using a line search). 

Newton variants are constructed by combining various strategies for the individual components above. These involve procedures for formulating Hk or Hk, dealing with structures of indefinite Hessians, and solving for the modified Newton search direction. For example, when Hk is approximated by finite differences, the discrete Newton subclass emerges.5 91-94 When Hk, or its inverse, is approximated by some modification of the previously constructed matrix (see later), QN methods are formed.95-110 When is nonzero, TN methods result,111-123 because the solution of the Newton system is truncated before completion. 

Many variations of the correction method we have proposed can be used. Among these are the use of the same Jacobian for several iterations and the use of modified Newton methods such as Marquardt s method (8). We have tried Marquardt s method on some of these problems without observing any significant improvement, but this is only a tentative evaluation. Improved methods for generating starting conditions would be helpful. 

Numerical calculation has been carried out using a software interface which is based on the so-called "Method of lines" (14). Gear s backward difference formulas (15) are used for the time integration. A modified Newton s method with the internally generated Jacobian matrix is utilized to solve the nonlinear equations. ... 

A number of iterative methods exist, as described in Appendix L. TK Solver uses a modified Newton-Raphson iterative procedure (see Sec. L.2), which is satis-fectory for a wide variety of problems. 

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