First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 71
Geometric interpretation: The slope of the secant line through the points (p n−1 ,f(p n−1 ))
and (p n−2 ,f(p n−2 )) is f(p n−1)−f(p n−2 )
p n−1 −p n−2
.Thex-intercept of the secant line, which is set to p n ,is
0 − f(p n−1 )
= f(p n−1) − f(p n−2 )
p n−1 − p n−2
⇒ p n = p n−1 − f(p n−1 )
p n − p n−1 p n−1 − p n−2 f(p n−1 ) − f(p n−2 )
which is the recursion of the secant method.
The following theorem shows that if the initial guesses are "good", the secant method
has superlinear convergence. A proof can be found in Atkinson [3].
Theorem 35. Let f ∈ C 2 [a, b] and assume f(p) =0,f ′ (p) ≠0, for p ∈ (a, b). If the initial
guesses p 0 ,p 1 are sufficiently close to p, then the iterates of the secant method converge to p
with
∣ |p − p n+1 | ∣∣∣
lim
n→∞ |p − p n | = f ′′ r
(p)
1
r 0 2f ′ (p) ∣
where r 0 = √ 5+1
2
≈ 1.62,r 1 = √ 5−1
2
≈ 0.62.
Julia code for the secant method
The following code is based on Equation (2.12); the recursion for the secant method. The
initial guesses are called pzero and pone in the code. The same stopping criterion as in
Newton’s method is used. Notice that once a new iterate p is computed, pone is updated as
p, andpzero is updated as pone.
In [1]: function secant(f::Function,pzero,pone,eps,N)
n=1
p=0. # to ensure the value of p carries out of the while loop
while n