CMPSC/Mathematics 455
Assignement Five
Due 24 October 2022
Fall 2022
You are to write a MATLAB script or function based upon problem 16, p.
62 of the Ascher and Grief textbook.
In that problem, it is pointed out that
f(x;α) = α cosh(x/4)− x = 0 (1)
has two roots for α = 2 and no roots for α = 10. The question asks whethe
there is a value of α for which there is only one root. The answer is yes, and
that root is a double root. That is, there is a value α∗ where there is a root
x∗ such that
f(x∗;α) = 0, f ′(x∗;α∗) = 0.
The differentiation is with respect to x not α.
We can solve for x∗ and α∗ by setting up a nonlinear equation in two
variables z1 = x and z2 = α so we let
z =
(
z1
z2
)
=
(
x
α
)
F(z) =
(
F1(z)
F2(z)
)
=
(
f(z1; z2)
f ′(z1; z2)
)
and then solve
F(z) = 0 (2)
using Newton’s method.
You will try two separate initial guesses z(0) for the root z =
(
x∗
α∗
)
of
equation (2). They are
z(0) =
(
2.5006
6
)
z(0) =
(
4.3944
3
)
You should get approximately the same answer for both initial guesses.
1
With tol = 10−14 = 1e − 14, use Newton’s method to produce the
sequence of iterates {z(k)} until eithe
∥z(k−1) − z(k)∥∞ ≤ tol ∗max{∥z(k)∥∞, tol}
or k = 20. You should only have to store the two most recent iterates. Fo
each of the two initial guesses produce your final values of x∗, α∗, and the
number of iteations k. Of course, here z(k) =
(
x∗
α∗
)
.
I strongly recommend producing a function (I will call it prob16 fun)
that has the first line
function [F, J ] = prob16 fun(z)
where F is the 2-vector F(z), the function value at the point z and J is
the 2 × 2 Jacobian matrix for F evaluated at z. (Yes, part of the problem
is figuring out what the Jacobian is.) You can then build the script around
this function. Your program with all scripts, functions, and output should
e turned in on Canvas by midnight on the due date.
2