MAT 423 Numerical Analysis I, Dr. Espa˜nol, Fall 2022
Homework Set 1
Each problem is worth 10 points.
Due date: Monday September 5th
The purpose of these problems is to acquaint you with features of MATLAB that we
use extensively in this class. Review the MATLAB Overview, paying particular attention
to Sections 3, 4, and 5. For each problem, submit both the code and the output by copying
them into a Word document. Use as little paper as necessary.
1. Dot Operators. All variables in MATLAB are considered to be matrices, so x*y,
for example, is treated as matrix multiplication. There are plenty of calculations,
however, where x and y are vectors that we think of as sets of scalars, and we want
to perform scalar operations on the individual elements. MATLAB has a set of ‘dot
operators’ to do this, for operations other than addition and subtraction (which are
the same for vectors and scalars). Run this code to see how these operators work.
Line 3 gives an error message – learn to recognize this so you can fix your code quickly.
First, type these lines in the command window to see what happens. Then put the
commands in a script file with line 3 omitted or commented. Submit the script file
and output for the problem.
x = [1 2 3 4]
y = [2 1 -2 3]
x*y
x.*y
x./y
x.^y
x+y
x-y
2. Plotting Functions by the Build-A-Vector Method. Sometimes, we want a quick plot
of a function with no intention of using the function later. The plot command requires
a vector of x values and a vector of y values. It is convenient in this case to create
the y vectors directly without using MATLAB’s data structures for functions. As an
example,
h = 1;
x = 0:h:10;
y = x.^2;
figure(1)
plot(x,y)
h = .01;
x = 0:h:10;
y = x.^2; % note that when you change the x vector,
% you need to recompute the y vector
figure(2)
plot(x,y)
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Figure 1 is in a window underneath Figure 2’s window. See the MATLAB Overview
Section 2, for the description of what’s happening with x. I call h the resolution; it
is important because the plot is a set of straight line segments connecting the dots in
the input vectors, clearly seen in the first plot. This is called a linear spline and we
will discuss it later in this course. Your assignment is to plot f(x) = √
x
3 + 1 sin x on
the interval [5, 10] with poor resolution (h = 1) and good resolution (h = 0.01). To
print the plots, you have 2 choices. You can print from the plot window (full page for
each plot, don’t do this for assignments) or you can save the plot as a jpg file (click
on File in the plot window, Save As and use File Type JPEG image). Then you can
insert the figure in your Word document and resize it so you don’t waste paper.
3. Sometimes, you want multiple graphs on one plot. Do one of following 2 options for
the function in problem 2, again on [5, 10]. Use dots for the poor resolution graph.
helpwin plot
% First approach
x1 = 0:10; y1 = x1.^2;
x2 = 0:.01:10; y2 = x2.^2;
plot(x1,y1,’o’,x2,y2) % the ’o’ is an optional third parameter
% that formats the graph of x1,y1
% Second approach
x = 0:10; y = x.^2;
figure(1)
hold on
plot(x,y,’o’)
x = 0:.01:10; y = x.^2;
plot(x,y)
4. Sometimes we want to define a function to use several times, or in different ways, in a
code. MATLAB allows us to define ‘anonymous functions’; the name of the function
is called the handle. There is a specific syntax that must be used in the definition:
f = @(x) x.^2+1;
g = @(x,y) x ./ (y.^3+x+1);
Here, x and y are dummy variables that have no specific values. Here are some
examples of how to use these functions:
f = @(x) x.^2+1;
f(3)
x=[1,2,3]
f(x)
x=0:.01:5; y = f(x);
figure(1)
plot(x,y)
2
figure(2)
plot(x,f(x))
g = @(x,y) x ./ (y.^3+x+1);
g(1,2)
g(5,6)
g([1 2],[5 6])
Your assignment is to define f(x) = e
x
sin x
x
2 + 1
as an anonymous function and plot it on
[3, 7] using poor resolution and good resolution (same h values as before) in one plot,
with dots for the poor resolution, as in the previous problem.
5. In the previous example, the values f(3) and [f(1), f(2), f(3)] were displayed in the
command window using the numeric format we specified in preferences. Sometimes
we want more control over the format, so we use the fprintf command (Section 1 of
the Overview). As an example,
f = @(x) x.^2+1;
x = .6221;
fprintf(’f(%6.5f) = %12.4e \n’,x,f(x))
fprintf(’f(%6.5f) = %12.4e \n’,3.706,f(3.706))
Note that changing the output display does not change the value stored in MATLAB. Your assignment is to print the values of f(x) = sin(x
3
) at the points x =
5.201, −8323.6, 0.0003 in floating point (f format) with 8 digits after the decimal and
in scientific notation (e format) with 10 digits after the decimal. Check the help page
(type helpwin fprintf) for details. Notice that the floating point format is not appropriate for the third x; get in the habit of printing in scientific notation to see the
magnitude of a number.
6. A third way to define functions is to create a separate file for the function. This is
used for more complicated functions, or algorithms, that require more than one line
of code. The function uses only dummy variables and cannot be run directly. Instead
you create a script file in which the inputs are defined and then the function is called.
As an example, one way to approximate the derivative of a function at a point x is
called the first order forward difference method (FD1):
f
0
(x) = f(x + h) − f(x)
h
To implement this, we create a function file called FD1.m, shown below. To signify
that this file defines a function, the first line must have the proper syntax: function
[outputs] = name(inputs).
function d = FD1(f,x,h)
d = (f(x+h)-f(x))/h;
A sample script that uses this simple function is
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f = @(x) x.^2;
x = 3;
exactderiv = 2*x;
h1 = .1; h2 = .01;
d1 = FD1(f,x,h1); d2 = FD1(f,x,h2);
fprintf(’The approximate derivative at %f using h=%f is %e\n’,x,h1,d1)
fprintf(’The approximate derivative at %f using h=%f is %e\n’,x,h2,d2)
Your assignment follows the same structure. Write a function file that computes
the roots of a quadratic using the form for x
+ and x
−1 that are least susceptible to
cancellation error. Also write a script file to find the roots of
ax2 + bx + c = 0
For reference, your function file will have the structure
function [xp,xm] = quadform(a,b,c)
if (b<0)
xp =
xm =
else
xm =
xp =
end
Calculate the roots for the following cases:
a) a = 1; b = −105
; c = 1.
b) a = 6 ∗ 1030; b = 5 ∗ 1030; c = −4 ∗ 1030
.
c) a = 10−30; b = −1030; c = 1030
.
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