First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 74
to get
c = f(p 2 )
b = (p 0 − p 2 )(f(p 1 ) − f(p 2 ))
(p 1 − p 2 )(p 0 − p 1 )
− (p 1 − p 2 )(f(p 0 ) − f(p 2 ))
(p 0 − p 2 )(p 0 − p 1 )
a = f(p 0) − f(p 2 )
(p 0 − p 2 )(p 0 − p 1 ) − f(p 1) − f(p 2 )
(p 1 − p 2 )(p 0 − p 1 ) .
(2.14)
Now that we have determined P (x), the next step is to solve P (x) =0, and set the next
iterate p 3 to its solution. To this end, put w = x − p 2 in (2.13) to rewrite the quadratic
equation as
aw 2 + bw + c =0.
From the quadratic formula, we obtain the roots
ŵ =ˆx − p 2 =
−2c
b ± √ b 2 − 4ac . (2.15)
Let Δ=b 2 − 4ac. We have two roots (which could be complex numbers), −2c/(b + √ Δ)
and −2c/(b − √ Δ), and we need to pick one of them. We will pick the root that is closer to
p 2 , in other words, the root that makes |ˆx − p 2 | the smallest. (If the numbers are complex,
the absolute value means the norm of the complex number.) Therefore we have
⎧
⎨ −2c
b+
ˆx − p 2 =
√ if |b + √ Δ| > |b − √ Δ|
Δ
⎩ −2c
b− √ if |b + √ Δ|≤|b − √ . (2.16)
Δ|
Δ
The next iterate of Muller’s method, p 3 , is set to the value of ˆx obtained from the above
calculation, that is,
⎧
⎨p 2 −
p 3 =ˆx =
⎩p 2 −
2c
b+ √ if |b + √ Δ| > |b − √ Δ|
Δ
2c
b− √ if |b + √ Δ|≤|b − √ Δ|
Δ
.
Remark 36.
1. Muller’s method can find real as well as complex roots.
2. The convergence of Muller’s method is superlinear, that is,
∣
|p − p n+1 | ∣∣∣
lim
n→∞ |p − p n | = f (3) (p)
α 6f ′ (p)
where α ≈ 1.84, provided f ∈ C 3 [a, b],p∈ (a, b), andf ′ (p) ≠0.
∣
α−1
2