Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

58 CHAPTER 4. FINDING ROOTS
Exercises
(4.1) Consider the bisection method applied to find the zero of the function f(x) = x 2 − 5x + 3,
with a 0 = 0, b 0 = 1. What are a 1 , b 1 ? What are a 2 , b 2 ?
(4.2) Approximate √ 10 by using two steps of Newton’s method, with an initial estimate of x 0 = 3.
(cf. Example Problem 4.11) Your answer should be correct to 5 decimal places.
(4.3) Consider bisection for finding the root to cos x = 0. Let the initial interval I 0 be [0, 2]. What
is the next interval considered, call it I 1 ? What is I 2 ? I 6 ?
(4.4) What does the sequence defined by
x 0 = 1,
x k+1 = 1 2 x k + 1 x k
converge to?
(4.5) Devise a subroutine using only simple operations that finds, via Newton’s Method, the cubed
root of some input number z.
(4.6) Use Newton’s Method to approximate 3√ 9. Start with x 0 = 2. Find x 2 .
(4.7) Use Newton’s Method to devise a sequence x 0 , x 1 , . . . such that x k → ln 10. Is this a reasonable
way to write a subroutine that, given z, computes ln z? (Hint: such a subroutine would require
computation of e x k. Is this possible for rational x k without using a logarithm? Is it practical?)
(4.8) Give an example (graphical or analytic) of a function, f(x) for which Newton’s Method:
(a) Does not find a root for some choices of x 0 .
(b) Finds a root for every choice of x 0 .
(c) Falls into a cycle for some choice of x 0 .
(d) Converges slowly to the zero of f(x).
(4.9) How will Newton’s Method perform for finding the root to f(x) = √ |x| = 0?
(4.10) Implement the inverse finder described in Example Problem 4.10. Your m-file should have
header line like:
function c = invs(z)
where z is the number to be inverted. You may wish to use the builtin function sign. As
an extra termination condition, you should have the subroutine return the current iterate if
zx is sufficiently close to 1, say within the interval (0.9999, 0.0001). Can you find some z for
which the algorithm performs poorly?
(4.11) Implement the bisection method in octave/Matlab. Your m-file should have header line like:
function c = run_bisection(f, a, b, tol)
where f is the name of the function. Recall that feval(f,x) when f is a string with the
name of some function evaluates that function at x. This works for builtin functions and
m-files.
(a) Run your code on the function f(x) = cos x, with a 0 = 0, b 0 = 2. In this case you can
set f = "cos".
(b) Run your code on the function f(x) = x 2 − 5x + 3, with a 0 = 0, b 0 = 1. In this case you
will have to set f to the name of an m-file (without the “.m”) which will evaluate the
given f(x).
(c) Run your code on the function f(x) = x − cos x, with a 0 = 0, b 0 = 1.
(4.12) Implement Newton’s Method. Your m-file should have header line like:
function x = run_newton(f, fp, x0, N, tol)
where f is the name of the function, and fp is its derivative. Run your code to find zeroes of
the following functions:
(a) f(x) = tan x − x.