Piecewise
function  separate ranges in Matlab
In math, a piecewise
function (or piecewisedefined function)
is a function whose definition changes depending on the value of the
independent variable.
A
function f of
a variable x (noted f(x))
is a relationship whose definition is given differently on different
subsets of
its domain.
Piecewise is a term
also used to describe any property of a
piecewise function that is true for each piece but may not be true for
the
whole domain of the function. The function doesn’t need to be
continuous, it
can be defined arbitrarily.


We’re going to experiment
in Matlab with this type of
functions. We’re going to develop three ways to define and graph them.
The first
method involves ifstatements to classify elementbyelement, in a
vector. The
second method uses switchcase statements, and the third method uses
indices to
define different sections of the domain.
Let’s say that this is
our piecewisedefined function
and we’d like to graph it
on the domain of 8 ≤ x ≤ 4
Method 1. Ifelse statements
Notice that this is not the best way to do it in
Matlab, but I mention the idea because this is the fundamental
concept, and in some programming languages it cannot be done in a
dfferent way. It's for study, not for real implementation... So, let's define our
function with ifelse statements for the moment. We
just use different conditions for the different ranges, and assign
appropriate
values.
function y =
piecewise1(x)
if x <=
4
y
= 3;
elseif 4 <
x & x <= 3
y
= 4*x  13;
elseif 3 <
x & x <= 0
y
= x^2 + 6*x + 8;
else
y = 8;
end
Now, we can use scalars
or arrays to call this function, in
a classical way:
%
Define elements
x = [5
4 3 0 3]
%
Submit elementbyelement to the function
for ix = 1
: length(x)
y(ix)
= piecewise1(x(ix))
end
%
Plot discrete values
plot(x,
y, 'ro')
%
Define x and y ranges to display
axis([8
4 2 10])
The result is:
Let’s say that we need
more values. We can call our function
this way
x = 8 :
.01 : 4;
for ix = 1
: length(x)
y(ix)
= piecewise1(x(ix));
end
plot(x,
y)
axis([8
4 2 9])
The new graph (which
obviously gives more information) is:
Method 2. Switchcase statements
Our second method
classifies the elements using switchcase statements.
We separate the different ranges in different cases. If that condition
is true,
then the switchexpression will match the caseexpression, and the
appropriate
statement(s) will be executed.
function y = piecewise2(x)
switch x
% A
scalar switch_expr matches
a case_expr if
%
switch_expr == case_expr
% The
first case must cover
the x = 0 case
case x * (3
< x & x <= 0)
y = x^2 + 6*x + 8;
case x * (x
<= 4)
y = 3;
case x * (4
< x & x <= 3)
y = 4*x  13;
otherwise
y = 8;
end
One important note is that when x = 0 the result of the
caseexpression is also 0, and the first case is executed. That’s why we
need to
place our x = 0 case first in the structure, otherwise we get wrong
that point.
We can test our second
piecewise function definition, like
this:
x
= 8 :
.01 : 4;
for ix = 1
: length(x)
y(ix)
= piecewise2(x(ix));
end
plot(x,
y)
axis([8
4 2 9])
and get the previous
shown graph, too.
Method 3. Vectorized way
The above routines assume
that we’re entering scalars as
input parameters. Now, we’ll assume that we’re submitting whole
vectors, and
we’ll handle indices for that.
This video will show you how to do it without using loops. After the video, another example is given with full code.
The line x2 = x(4
< x & x <= 3);
assigns to x2 the values of vector x that meet the criteria
of 4 < x ≤ 3.
The line y(4
< x & x <= 3) = 4*x2  13; works
with the values in x2, calculates the appropriate function y(x) and
places the values in the correct indices of vector y.
We can combine those two
ideas to work out all the appropriate
values for the whole function.
The following code is
just one way to take full vectors and
pack them into piecewise functions:
function y = piecewise3(x)
%
first range
y(x
<= 4) = 3;
%
second range
x2 =
x(4 < x & x <= 3);
y(4 < x & x <= 3) = 4*x2  13;
%
third range
x3 =
x(3 < x & x <= 0);
y(3 < x & x <= 0) = x3.^2 + 6*x3 + 8;
%
fourth range
y(0
< x) = 8;
Now, we can avoid
forloops before graphing this type of
functions...
We enter the whole array of the independent variable as
parameter
x = 8 :
.01 : 4;
y1 =
piecewise3(x);
plot(x,
y1)
axis([8
4 2 9])
this produces what we
know and
expect...
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