Nonlinear modified Newton methods

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... 

Since these underdimensioned nonlinear systems will also be solved using a modified Newton method, it is essential to tackle the subproblem of the solution of underdimensioned linear systems. 

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. ... 

Hartley, H.O. The modified Gauss-Newton method for the fitting of nonlinear regression functions. Technometrics 1961 3 269-280. 

The quantities x, and yj are the iterants, whereas gi and g2 are formed exactly the way Equation 9.5 was developed. Two common methods for finding roots to nonlinear systems are (1) Newton-Raphson and (2) the modified Newton-Raphson. Both approaches are briefly discussed in the subsections below. 

This last equation is a nonlinear algebraic equation in D, it can be solved using the Newton-Raphson method [if the differentiation is very lengthy and cumbersome, which is the case here, then in the Newton-Raphson method, you can use the modified Newton-Raphson by approximating BF/BDaf by - T")/ (Z)" — )") or, easier and sure to converge, use the bisectional... 

The result (10.25) is a nonlinear equation. We can introduce a modified Newton-Raphson method (Owen and Hinton 1980) for solving (10.25). Then we rewrite (10.25), and (10.26)-( 10.29)by inttoducing variables with superscripts k and k — l, which implies the values of the variables at each iteration step, and we have... 

The simplest numerical method for a detailed geometrically and material nonlinear (GMN) analysis is the Newton-Raphson scheme (Crisfield 1979 Bathe 1995), which can be found in three forms (i) the full Newton-Raphson, which is the most accurate, but also the most time consuming, since the tangent stiffness of the structure has to be calculated and factorized within each iteration in the solution procedure (ii) the modified Newton-Raphson, which differs from the full Newton-Raphson in that the calculation and the factorization of the tangent stiffness matrix take place only in some iterations within each step, thus requiring in most cases a larger number of iterations per step but... 

This modified density Is a more slowly varying function of x than the density. The domain of Interest, 0 < x < h, Is discretized uniformly and the trapezoidal rule Is used to evaluate the Integrals In Equations 8 and 9. This results In a system of nonlinear, coupled, algebraic equations for the nodal values of n and n. Newton s method Is used to solve for n and n simultaneously. The domain Is discretized finely enough so that the solution changes negligibly with further refinement. A mesh size of 0.05a was adopted In our calculations. 

In the present chapter, two different but related methods for the optimization of the MCSCF wave function are treated. The first method, which we discuss in this section, is a straightforward application of Newton s method for the optimization of nonlinear functions, modified so as to ensure global convergence. The second method, discussed in Section 12.4, is based on the solution of an eigenvalue equation which gives steps similar to those in the present section and which reduces to the standard eigenvalue problem of Cl theory when no orbital optimization is included. 

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