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In general, the necessary condition equations, Del f(X) = 0, may be difficult to solve numerically. The Newton-Raphson method is an iterative procedure for solving simultaneous nonlinear equations.

**The Newton-Raphson Method**

In general, the necessary condition equations, *Ñ** f(X)* = **0,** may be difficult to solve numerically. The Newton-Raphson method is an iterative procedure for solving simultaneous nonlinear equations.

Consider the simultaneous equations

The idea of the method is to start from an initial point X o and then use the equation above to determine a new point. The process continues until two successive points, X k and X k +1 are approximately equal.

A geometric interpretation of the method is illustrated by a single-variable function in Figure 18.3. The relationship between *xk* and *xk+1* for a single-variable function *f(x) *reduces to

FIGURE 18.3

Illustration of the iterative process in the Newton-Raphson method

The figure shows that *xk+1* is determined from the slope of *f(x)* at *xk* where tan q = *f'* *(xk).*

One difficulty with the method is that convergence is not always guaranteed unless the function *f* is well behaved. In Figure 18.3, if the initial point is *a,* the method will diverge. In general, trial and error is used to locate a "good" initial point.

**Example 18.1-3**

To demonstrate the use of the Newton-Raphson method, consider determining the stationary points of the function

To determine the stationary points, we need to solve

The method converges to *x* = 1.5. Actually, *f(x)* has three stationary points at *x *=* *2/3,* x *=* *13/12,* *and* x *=* *3/2.* *The remaining two points can be found* *by* *selecting different values for* *initial *xo.* In fact, *xo* = .5 and *xo* = 1 should yield the missing stationary points.

**Excel Moment**

Template excelNR.xls can be used to solve any single-variable equation. It requires entering *f(x)/f' (x)* in cell *C3.* For Example 18.1-3, we enter

The variable *x* is replaced with A3. The template allows setting a tolerance limit Δ, which specifies the allowable difference between *xk* and *xk+1* that signals the termination of the iterations. You are encouraged to use different initial *xo* to get a feel of how the method works.

In general, the Newton-Raphson method requires making several attempts before "all" the solutions can be found. In Example 18.1-3, we know beforehand that the equa-tion has three roots. This will not be the case with complex or multi-variable functions, however.

**PROBLEM **SET 18.1B

1. Use NewtonRaphson.xls to solve Problem I(c), Set I8.la.

2. Solve Problem 2(b), Set I8.la by the Newton-Raphson method.

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