MATLAB: Losing points near singularities? - matlab

Question

I am trying to plot some bump function supported on the interval (0,3). I've defined the function piecewise so that it is zero outside the interval. Here is the function:
function d = bump2(t)
region1 = (t > 0) & (t < 3);
d(region1) = exp(1./(t(region1).^2 - 3*t(region1)));
region2 = (t <= 0) & (t >= 3);
d(region2) = 0;
end
If I set x = -1:.01:4 and try to run plot(x,bump2(x)) I get an error telling me that my vectors aren't the same length. This is indeed true since length(x) = 501 and length(bump2(x)) = 400 However, when I set x = .01:.01:2.99, then everything checks out fine and the lengths of the vectors are equal.
Also, strangely, I have another, similar function
function b = bump(t)
region1 = abs(t) < 1;
b(region1) = exp(-1./(1 - ((t(region1)).^2)));
region2 = abs(t) >= 1;
b(region2) = 0;
end
with which I've never had this problem. Why are these two functions different? And why does the first function work when I exclude the points 0 and 3, even though I've defined the function to avoid possible singularities there?

Answer 1

This is because the b(region1) is trimming off the ends of your vector x so that it is not the same size.
Consider if you set x = -1:3 and then run the code plot(x, bump3(x)); with:
function d = bump3 (t)
region1 = (t > 0) & (t < 3);
d(region1) = t(region1);
end
Then you get the same problem. The issue is that inside bump3 you have
K>> region1
region1 =
0 0 1 1 0
Now this is the trick. This region1 is of data type logical so MATLAB interprets the next line
d(region1) = t(region1);
as
d(find(region1)) = t(find(region1));
But if you wrote
d = [];
d(5) = 1;
Then you would get a bunch of zeros at the zero, so you do here.
You might consider modifying to say d = t(region1); to get only the good values, but then you still have a sizing problem. A solution is to return the indices you used with the new value as an additional return value in bump like this:
Execute:
x = -1:3;
[y region1] = bump4(x);
plot(x(region1), y);
Where you modify your bump function like this:
function [d region1] = bump4 (t)
region1 = (t > 0) & (t < 3);
d = t(region1);
end
EDIT:
If you really do want those extra zeros on the left and right you can also fix this problem by initializing d before using it in your code like this:
function d = bump2(t)
region1 = (t > 0) & (t < 3);
d = zeros(size(t));
d(region1) = exp(1./(t(region1).^2 - 3*t(region1)));
region2 = (t <= 0) & (t >= 3);
d(region2) = 0;
end
Because I set d = zeros(...) then you have zeros outside the region and your bump function within it. Another way of looking at the problem is that MATLAB is padding on the left side but not on the right because you never set a blank value out there. This fixes that problem by telling MATLAB how big d should be.