Download Practice Finding Roots 1. Consider the following problem: The sum

CS371/AM242
Winter 2016
Practice Finding Roots
1. Consider the following problem:
The sum of two numbers is 20. Add each number to its square root, and multiply
together. The product of the two sums is approximately 155.55.
a) Define a function f(x) such that finding a root of f(x) solves the above problem.
You may assume x is a number between 1 and 9, inclusive.
b) Write Matlab code to find a root of your function f(x)=0 using the NewtonRaphson method. Continue the iterations until abs(f(x)) <= 10-6. Include a table of
the iterate values, when starting from x0=1, x0=9, and a third x0 of your choice.
c) Without implementating the bisection algorithm:
i. Prove that [1,9] is an appropriate initial bracket for a root of f.
ii. Determine the approximate number of iterations required by the bisection
algorithm to find the interval [ak,bk] of length <= 10-6.
2. Consider the function f(x) = x4+2x2-x-3.
a) Show that a fixed point of 𝑔(𝑥) = √
(𝑥+3−𝑥 4 )
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is a root of f.
b) Write Matlab code to calculate the first 20 iterates for a root of f, starting from
x0 = 1, using xk+1 = g(xk). Print out the iterates. What do you observe about the
values?
3𝑥 4 +2𝑥 2 +3
c) Show that a fixed point of ℎ(𝑥) = 4𝑥 3 +4𝑥−1 is a root of f.
d) Write Matlab code to calculate the first 20 iterates for a root of f, starting from
x0 = 1, using xk+1 = h(xk). Print out the iterates. What do you observe about the
values? How does this compare to the performance of g?
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3. Consider the sequence {xk} where 𝑥𝑘+1 = 2 𝑥𝑘 + 𝑥 , used to generate a fixed point of
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𝑘
the function 𝑔(𝑥) = 2 𝑥 + 𝑥.
a) Using the convergence results for fixed point algorithms, show that
lim𝑘→∞ 𝑥𝑘 = √2 when starting with x0>√2. (Verify that the conditions for
convergence as discussed in class apply to g on the interval [√2, ∞). )
b) Determine the order of convergence q and the asymptotic error constant λ for
this sequence, assuming the result from a) is true, where q and λ were defined in
class (see Slide 19 from Module 02 slides, as posted on LEARN). Show your
workings.
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