Nonlinear function root-finding

If the process model is nonlinear, then advanced stability theory can be used (Khalil, 2001), or an approximate stability analysis can be performed based on a linearized transfer function model. If the transfer function model includes time delays, then an exact stability analysis can be performed using root-finding or, preferably, the frequency response methods of Chapter 14. A less desirable alternative is to approximate the terms and apply the Routh stability criterion. 

Newton s method is a classic iterative scheme for solving a nonlinear system /(x) = 0 or for minimizing the multivariate function /(x). These root-finding and minimization problems are closely related since obtaining the minimum of a function f(x) can be formulated as solving for the zeros of / (x) for which f (x) > 0. (See historical note below on the method s name.)... 

Example 1.1 Solution of the Colebrook Equation by Successive Substitution, Linear Interpolation, and Newton Raph on Methods. Develop MATLAB functions to solve nonlinear equations by the successive substitution method, the linear interpolation, and the Newton-Raphson root-finding techniques. Use these functions to calculate the friction factor from the Colebrook equation [Eq. (1.4)] for flow of a fluid in a pipe with e/Z> =10 and Njf, = 10. Compare these methods with each other. 

A dichotomy arises in attempting to minimize function (h). You can either (1) minimize the cost function (h) directly or (2) find the roots of Equation (i). Which is the best procedure In general it is easier to minimize C directly by a numerical method rather than take the derivative of C, equate it to zero, and solve the resulting nonlinear equation. This guideline also applies to functions of several variables. 

Each of the partial derivatives when equated to zero may well yield a nonlinear equation. Hence, the minimization of /(x) is converted into a problem of solving a set of nonlinear equations in n variables, a problem that can be just as difficult to solve as the original problem. Thus, most engineers prefer to attack the minimization problem directly by one of the numerical methods described in Chapter 6, rather than to use an indirect method. Even when minimizing a function of one variable by an indirect method, using the necessary conditions can lead to having to find the real roots of a nonlinear equation. 

With the basis functions cp. (A + G, E, r), a variational solution is sought to the Kohn-Sham equation, equation (B3.2.4). Since the Hamiltonian matrix elements now depend nonlinearly upon the energy due to the energy-dependent basis functions, the resulting secular equation is solved by finding the roots of the determinant of the E) - E E) matrix. (The problem cannot be treated by the eigenvalue routines of linear algebra.)... 

Recall from Chapter 4 that finding equilibriinn points involves solving for the roots to the rate function expressions, which is often a system of nonlinear equations, this may be a difficult task in itself. 

There are many problems which require the finding of the root of a single nonlinear equation in one unknown. For example, the heat capacity of carbon dioxide is given as a function of temperature as... 

Program Description Three MATLAB functions called XGX.m, Ll.m, and NR.m are developed to find the root of a general nonlinear equation using successive substitution [the... 

In the next section, the function builds the coefficients of the Lagrange polynomial and finds its roots, Zj- The vector of Xj is then calculated from Eq. (5.141). The function applies N e wton s method for s olution of the set of nonlinear equations (5.160). Therefore, the starting values for this technique are generated by the second-order Runge-Kutta method, using the guessed initial conditions. The function continues with building the matrices Q,C,A, A , and vectors Y and F. 

Equations 2.52 and 2.54 represent two nonlinear algebraic equations for finding the wire velocity and radius. Find R and V for HDPE at 200 °C using the IMSL subroutine NEQNF, which is described in Appendix D.4 on the accompanying website. Also use Excel and the Solve function or Goal Seek to find the roots of these two equations. 

A fairly thorough look has been taken at Newton s method for solving a single nonlinear (or linear) equation for a root (or roots) of a function. If a function has multiple roots, Newton s method must be started close to each root in order to find more than one solution. It is also seen that for some formulation of a physical problem, it is essentially impossible to get Newton s method to converge to a solution while a rearrangement of the basic equation can rapidly lead to valid solutions with Newton s method. The more one knows about a particular physical problem... 

Either Eq. (3.34) or Eq. (3.37) can form the basis for an iterative solution to a nonlinear equation. Eq. (3.37) would appear to be a more exact expression and thus perhaps the favored expression. However, it is relatively easy to find example equations where the quantity within the square root expression is negative and the equation fails even though there are real roots. One such simple expression is /(x) = exp(x) - A when x is large such that the exponential term dominates. Also the faet that a square root must be ealeulated means additional computational time. Eq. (3.34) does not have these difficulties and will converge for a wider range of functions. Thus this form of Halley s method will only be considered further. 

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