The Successive Substitution Method

The bisection method presented in the previous section can only be used when an interval [a, b] is known. However, if this is not possible, the present and the following sections (Newton-Raphson) will prove to be useful. 

Let us start with the single nonlinear algebraic equation 

The underlying principle of the successive substitution method is to arrange the given equation to the form 

The iteration scheme for the successive substitution method to search for a root is defined as 

Application of the successive method is quite simple. We choose (by guessing) a value for and calculate from the iteration equation (A.5). Repeating the procedure, we obtain x and x and so on. The method may or may not converge, but the method is direct and very easy to apply, despite the uncertainty of convergence. We will demonstrate the method and its uncertainty in the following example. 

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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. 

The equation in cell B1 is copied into cells Cl though El. Then turn on the iteration scheme in the spreadsheet and watch the solution converge. Whether or not convergence is achieved can depend on how you write the equations, so some experimentation may be necessary. Theorems for convergence of the successive substitution method are useful in this regard. 

Convergence proofs are available under certain conditions (Finlayson, 1980), and once the iterate value gets close to the solution, the convergence is very rapid. This method is generally better than the successive substitution method, except for special cases, but sometimes a good initial guess is required. 

The convergence of the successive substitution method is slow, i.e., it may require many iterations for the sequence to converge. The Newton-Raphson method has a faster rate of convergence, which is given as follows ... 

Fig. 3.16 Residual history of the successive substitution method. The convergence depends highly on the relaxation factor D. For D=0.1, oscillations occurred.

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 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. 

In the successive substitution method, sometimes because of a poor initial... 

The Wegstein method converges, even under conditions in which the method of a = (-c) does not. Moreover, it accelerates the convergence when the successive substitution method is... 

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. 

Listing 3.1. Code segment for the successive substitution method applied to Eq. 

The successive substitution method has successfully calculated flie two real roots of flic fourfli order polynomial in this case. However, flic convergence is slow, and follows what is known as linear convergence. This means that as the true solution is approached, the error at each iteration is some linear fraction of the error at flic previous iteration. Let s now study the convergence rate and the condition for convergence of the successive substitution method. Let s let X, =, where x is the true solution value and Sx =x - x is the error in... 

While the first form appears simpler, one finds that the successive substitution method only converges when using the second form. The printed output shows that the method converges to a solution value of x = 0.86033, this time achieved in 41 iterations. Now, let s hasten on to a more general solution method known as Newton s Method. 

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