Roots of Equations - Numerical Methods in Engineering with MATLAB - page 156

Tangent line

f ( x )

Figure 4.4. Graphical interpretation of the Newton-Raphson
formula. f ( x i )

x

x

x +1

i i

Agraphical depiction of the Newton-Raphson formulais shown in Fig. 4.4. The for-
mula approximates f ( x ) by the straight linethat istangent to the curve at x i . Thus x i + 1
is at the intersection of the x -axis and the tangentline.
The algorithm for the Newton-Raphsonmethodissimple:it repeatedly applies
Eq. (4.3), starting with an initial value x 0 , until the convergence criterion

|

x i + 1 − x 1 | <ε

is reached, being the error tolerance. Only the latest valueof x hastobe stored. Here

is the algorithm:

ε

1. Let x be aguess for the root of f ( x ) = 0.

f ( x ). 2. Compute x =− f ( x ) /

| | <ε

3. Let x ← x + x and repeat steps 2-3until x .

f ( x ) f ( x )

x x x

x

1 0 2

x

x

0

(a) (b)

Figure 4.5. Examples where the Newton-Raphsonmethod

diverges.

Although the Newton-Raphsonmethod converges fast near the root, its global
convergence characteristics are poor. The reasonisthat the tangentline is not al-
ways an acceptable approximation of the function, as illustratedinthe twoexamples
inFig. 4.5. But themethod canbemade nearly fail-safe by combining it with bisection,
as in Brent's method.

newtonRaphson

The following safe version of the Newton-Raphsonmethodassumes that the root
to becomputedis initiallybracketedin (a,b) . The midpointofthebracket is used
as the initial guess of the root. The brackets are updatedafter each iteration. If a