First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 61
by
This is called Newton’s method.
p n = p n−1 − f(p n−1)
,n≥ 1. (2.6)
f ′ (p n−1 )
Graphical interpretation:
Start with p 0 . Draw the tangent line at (p 0 ,f(p 0 )) and approximate p by the intercept p 1
of the line:
f ′ (p 0 )= 0 − f(p 0)
p 1 − p 0
⇒ p 1 − p 0 = − f(p 0)
f ′ (p 0 ) ⇒ p 1 = p 0 − f(p 0)
f ′ (p 0 ) .
Now draw the tangent at (p 1 ,f(p 1 )) and continue.
Remark 31. 1. Clearly Newton’s method will fail if f ′ (p n )=0for some n. Graphically
this means the tangent line is parallel to the x-axis so we cannot get the x-intercept.
2. Newton’s method may fail to converge if the initial guess p 0 is not close to p. In Figure
(2.3), either choice for p 0 results in a sequence that oscillates between two points.
1
p0
p0
0
Figure 2.3: Non-converging behavior for Newton’s method
3. Newton’s method requires f ′ (x) is known explicitly.
Exercise 2.3-1:
f(x) =0?
Sketch the graph for f(x) =x 2 −1. What are the roots of the equation