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The Initial Guess

One good way to estimate an initial guess for starting the N-R method is to linearize the equation being solved. For example, the equation f(x) = x*sin(pi x)-exp(-x) may be rewritten as a sum of two series:

(8)

 

Linearization implies that we approximate this equation with one in which we neglect all terms containing x raised to a power m>1 (x², x³, etc.) (c.f. Introduction). In this example, linearization reduces the equation to:

f(x) = 1 - x  (9)

 

The root of the linearized equation f(x)=1-x=0 is 1, which provides a reasonable initial guess for the actual roots on the nonlinear equation, in the absence of a plot such as Figure 1.
 

1998-10-01