P03
SOLVING NONLINEAR EQUATIONS
1. The (mostly) human part
1.1. The Bisection Method. In class we discussed
various methods for solving a (possibly) nonlinear
equation, e.g.,
F (x) ≡ x5 − x + 1 = 0.
One of the methods we used in the Bisection Method
(i) Describe in your own words, in one brief para-
graph, what the bisection method is and how it
works.
(ii) Consider the following assertion:
The Bisection Method gains one
binary digit in each iteration.
Discuss this assertion. Begin by first explain-
ing in your own words (in at most one brief
paragraph) what is meant by a binary digit.
Then elucidate the assertion in the context of
binary digits.
(iii) Examine the behavior of the bisection method
for the computation of the square root of 2,
as manifested in class and in the Mathematica
notebook provided. Do the computations agree
with the “gain” assertion above, in spirit? Ex-
plain.
1.2. Newton’s Method.
(i) Explain in your own words, in one brief para-
graph, what is meant by Newton’s Method.
(ii) Consider the assertion
When Newton’s method converges
rapidly, the convergence is quadratic
Discuss this assertion briefly (in one modest
paragraph). Be sure to explain what is meant
by quadratic convergence.
(iii) Examine the iterations for Newton’s method
done in class and in the Mathematica note-
book provided. Do the computations affirm the
“quadratic” assertion above? Explain.
2. The computer-centric part
Download the nonlinear equations Mathematica note-
book from Piazza. Work through the notebook and
complete the lines and portions required, mainly fill-
ing in key steps in iteration codes.
The amalgam: putting it together.
Compile the LaTeX writeup for part 1 to generate a
pdf file and export the Mathematica notebook from
part 2 to a pdf file. Concatenate the two pdf files
into a single pdf file and upload it to GradeScope.