First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 73
next iteration p 2 . If p 1 /∈ (a, b), then we will not accept p 1 as the next iteration: instead
the code will switch to the bisection method, determine which subinterval among
(a, p 0 ), (p 0 ,b) contains the root, updates the interval (a, b) as the subinterval that contains
the root, and sets p 1 to the midpoint of this interval. Once p 1 is obtained, the
code will check if the stopping criterion is satisfied. If it is satisfied, the code will
return p 1 and the iteration number, and terminate. If it is not satisfied, the code will
use Newton’s method, with p 1 as the initial guess, to compute p 2 . Then it will check
whether p 2 ∈ (a, b), and continue in this way. If the code does not terminate after N
iterations, output an error message similar to Newton’s method.
Apply the hybrid method to:
• a polynomial with a known root, and check if the method finds the correct root;
• y =logx with (a, b) =(0, 6), for which Newton’s method failed in part (a).
c) Do you think in general the hybrid method converges to the root, provided the initial
interval (a, b) contains the root, for any starting value p 0 ? Explain.
2.5 Muller’s method
The secant method uses a linear function that passes through (p 0 ,f(p 0 )) and (p 1 ,f(p 1 )) to
find the next iterate p 2 . Muller’s method takes three initial approximations, passes a parabola
(quadratic polynomial) through (p 0 ,f(p 0 )), (p 1 ,f(p 1 )), (p 2 ,f(p 2 )), and uses one of the roots
of the polynomial as the next iterate.
Let the quadratic polynomial written in the following form
P (x) =a(x − p 2 ) 2 + b(x − p 2 )+c. (2.13)
Solve the following equations for a, b, c
P (p 0 )=f(p 0 )=a(p 0 − p 2 ) 2 + b(p 0 − p 2 )+c
P (p 1 )=f(p 1 )=a(p 1 − p 2 ) 2 + b(p 1 − p 2 )+c
P (p 2 )=f(p 2 )=c