Piecewise Functions in Matlab
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This document discusses how to define and plot piecewise functions in Matlab without using loops. It explains that a piecewise function is made up of multiple subfunctions, each applying to an interval of the main function's domain. It provides an example of a 4-interval piecewise function and shows how to define it in Matlab using specially selected indices and the "find" function to assign values based on the interval. Defining the piecewise function this way allows it to be plotted vectorized, without scalar values or iterations.
![Piecewise Functions
First important concept: special indices
In Matlab, if we have vector x = [-2 -1 0 1 2 3 4 5 6 7 8 9], and do this:
i2 = x(0 < x & x <= 3)
we’ll take all the values in vector x that meet the condition 0 < x ≤ 3, that is,
i2 = [1 2 3]
If we do:
i3 = x(3 < x & x <= 8)
we’ll take all the values in vector x that meet the condition 3 < x ≤ 8, thus
i3 = [4 5 6 7 8]](https://image.slidesharecdn.com/piecewise-141004022245-conversion-gate01/85/Piecewise-Functions-in-Matlab-8-320.jpg)
![Piecewise Functions
Second important concept: function find
In Matlab, if we have vector x = [-2 -1 0 1 2 3 4 5 6 7 8 9], and do this:
find(0 < x & x <= 3)
we’ll find the indices (not values) in vector x that meet 0 < x ≤ 3, so we’ll get
the vector [4 5 6].
If we do:
find(3 < x & x <= 8)
we’ll get the vector [7 8 9 10 11]](https://image.slidesharecdn.com/piecewise-141004022245-conversion-gate01/85/Piecewise-Functions-in-Matlab-9-320.jpg)
![Piecewise Functions
Now, if we type this code…
clc, clear all, close all
x = -2 : .01 : 9;
y = piecewise(x);
plot(x, y)
axis([-2 9 -10 25])
grid on
We get this plot…](https://image.slidesharecdn.com/piecewise-141004022245-conversion-gate01/85/Piecewise-Functions-in-Matlab-15-320.jpg)
Piecewise Functions in Matlab
- 1. Piecewise Functions In Matlab
- 2. Piecewise Functions • A piecewise function is a function which is defined by multiple sub functions, each sub function applying to a certain interval of the main function's domain.
- 3. Piecewise Functions • We’ll show one way to define and plot them in Matlab without using loops. • We’ll use a vectorized way: no scalar values or iterations will be used.
- 4. Piecewise Functions Let’s assume a function with 4 intervals ì ï ï í 2 0 6 2 0 3 ( ) 2 ï ï î £ if x x if x + < £ x x if x - + + < £ 6 11 3 8 - < = if x f x 5 8
- 5. Piecewise Functions This is the plot of that 4-interval function x y = f(x)
- 6. Piecewise Functions Ideally, we should type something like this and get the plot shown above x = -2 : .01 : 9; y = piecewise(x); plot(x, y) interval of interest function to be defined call the plot, just as it’s done with any other function
- 7. Piecewise Functions The interesting part is how to define the piecewise function without going number-by-number in the domain, but going instead interval-by-interval. To accomplish this, we’ll use two ideas: • Specially selected indices. • Function find.
- 8. Piecewise Functions First important concept: special indices In Matlab, if we have vector x = [-2 -1 0 1 2 3 4 5 6 7 8 9], and do this: i2 = x(0 < x & x <= 3) we’ll take all the values in vector x that meet the condition 0 < x ≤ 3, that is, i2 = [1 2 3] If we do: i3 = x(3 < x & x <= 8) we’ll take all the values in vector x that meet the condition 3 < x ≤ 8, thus i3 = [4 5 6 7 8]
- 9. Piecewise Functions Second important concept: function find In Matlab, if we have vector x = [-2 -1 0 1 2 3 4 5 6 7 8 9], and do this: find(0 < x & x <= 3) we’ll find the indices (not values) in vector x that meet 0 < x ≤ 3, so we’ll get the vector [4 5 6]. If we do: find(3 < x & x <= 8) we’ll get the vector [7 8 9 10 11]
- 10. Piecewise Functions Putting all together, defining the function: function y = piecewise(x) y(find(x <= -0)) = 2; Name: piecewice Input: vector x Output: vector y First interval Find the indices of values of x that meet criterion x ≤ 0 Assign the corresponding value, according to your f(x)
- 11. Piecewise Functions Putting all together, defining the function Second inteval Take the values of x that meet criterion of 0 < x ≤ 3 Find the corresponding indices Assign the corresponding value, according to your f(x) x2 = x(0<x & x<=3); y(find(0<x & x<=3)) = 6*x2 + 2;
- 12. Piecewise Functions Putting all together, defining the function Third interval Take the values of x that meet criterion of 3 < x ≤ 8 Find the corresponding indices Assign the corresponding value, according to your f(x) x3 = x(3<x & x<=8); y(find(3<x & x<=8)) = -x3.^2 + 6*x3 + 11;
- 13. Piecewise Functions Putting all together, defining the function Fourth inteval Take the corresponding indices for 8 < x We don’t need the values of x now, because they’re not needed for the assignation in this interval y(find(8 < x)) = -5;
- 14. Piecewise Functions Putting all together. This is the final code… function y = piecewise(x) y(find(x <= -0)) = 2; x2 = x(0 < x & x <= 3); y(find(0 < x & x <= 3)) = 6*x2 + 2; x3 = x(3 < x & x <= 8); y(find(3 < x & x <= 8)) = -x3.^2 + 6*x3 + 11; y(find(8 < x)) = -5;
- 15. Piecewise Functions Now, if we type this code… clc, clear all, close all x = -2 : .01 : 9; y = piecewise(x); plot(x, y) axis([-2 9 -10 25]) grid on We get this plot…
- 16. Piecewise Functions For more examples and details, visit: matrixlab-examples.com/piecewise-function.html