Nonlinear equations successive substitution

Some formulas, such as equation 98 or the van der Waals equation, are not readily linearized. In these cases a nonlinear regression technique, usually computational in nature, must be appHed. For such nonlinear equations it is necessary to use an iterative or trial-and-error computational procedure to obtain roots to the set of resultant equations (96). Most of these techniques are well developed and include methods such as successive substitution (97,98), variations of Newton s rule (99—101), and continuation methods (96,102). 

Successive Substitutions Let/(x) = 0 be the nonlinear equation to be solved. If this is rewritten as x = F x), then an iterative scheme can be set up in the form Xi + = F xi). To start the iteration an initial guess must be obtained graphically or otherwise. The convergence or divergence of the procedure depends upon the method of writings = F x), of which there will usually be several forms. However, if 7 is a root of/(x) = 0, and if IF ( 7)I < I, then for any initial approximation sufficiently close to a, the method converges to a. This process is called first order because the error in xi + is proportional to the first power of the error in xi for large k. 

Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

Successive Substitutions Let fix) = 0 be the nonlinear equation to be solved. If this is rewritten as x = Fix), then an iterative scheme can be set up in the form x,. Fixf). To start the iteration an initial... 

A variety of names are used in the literature to describe calculational procedures for solving sets of linear and nonlinear algebraic equations such as iteration, successive iteration, and successive substitution. 

Discuss the methods of interval halving, successive substitution, and New-ton-Raphson for solving nonlinear algebraic equations. What are their relative advantages and disadvantages ... 

In the previous subsection, the successive substitution and Wegstein methods were introduced as the two methods most commonly implemented in recycle convergence units. Other methods, such as the Newton-Raphson method, Broyden s quasi-Newton method, and the dominant-eigenvalue method, are candidates as well, especially when the equations being solved are highly nonlinear and interdependent. In this subsection, the principal features of all five methods are compared. 

Equation (7.18) represents a set of N nonlinear algebraic equations, which must be solved by a trial and error method such as Newton-Raphson or successive substitution for y . j (Appendix A). This is the characteristic difference between the implicit and explicit types of solution the easier explicit method allows sequential solution one at a time, while the implicit method requires simultaneous solutions of sets of equations hence, an iterative solution at a given time... 

The common factor in the implicit Euler, the trapezoidal (Crank-Nicolson), and the Adams-Moulton methods is simply their recursive nature, which are nonlinear algebraic equations with respect to y +j and hence must be solved numerically this is done in practice by using some variant of the Newton-Raphson method or the successive substitution technique (Appendix A). 

Divide the equation by a and add x,- to the LHS and RHS (a similar procedure is used in Appendix A for the successive substitution method for solving nonlinear equations) to yield the equation... 

The main advantage of the implicit techniques is their stability for any given value of the step size. This advantage, however, requires the solution of a set of nonlinear equations via an iterative approach. For this purpose, methods such as successive substitution or Newton-Raphson can be used [1]. 

Other methods The method of successive substitution (SS), and the full Newton method for solving the system of nonlinear algebraic equations of flash have certain limitations. They have also certain desirable... 

The classical Newton method has quadratic convergence properties whereas the successive substitution method has linear rate of convergence. However, because of the overshoot, the Newton iteration may fail to converge when the initial estimate is not a good estimate of the solution of the system of nonlinear equations. 

Equation (1), with the associated boundary conditions, is a nonlinear second-order boundary-value ODE. This was solved by the method of collocation with piecewise cubic Hermite polynomial basis functions for spatial discretization, while simple successive substitution was adequate for the solution of the resulting nonlinear algebraic equations. The method has been extensively described before [9], and no problems were found in this application. 

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. 

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... 

Successive Substitution method to find one root of a nonlinear equation. 

The method of successive substitutions (also called fixed point iteration) is perhaps the simplest method of obtaining a solution to a nonlinear equation. This technique begins by rearranging the basic F(x) = 0 equation so that the variable x is given as some new function of the same variable x. The original equation is thus converted into an equation of the form ... 

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