Solutions to Matlab exercises 1, 2, 3
Last update: October 14, 2008
Exercise 1
In this exercise, we define C to be the plane curve
y^{2} = x^{2} - x^{4},
- Is the origin a point of C?
Ans: Yes. To check this, substitute
(x,y) = (0,0) into the equation of C.
- Use the Matlab script aPlot to plot C
in the region -1.2 < x < 1.2,
-1 < y < 1, in each of the following ways:
- with a 100 by 100 grid;
- with a 51 by 51 grid;
- with a 500 by 500 grid. (omitted)
- Which of the plots are more accurate than the others? What features
could be contributing to this situation?
Ans: Most parts of the plot with the 100 by 100 grid are
somewhat smoother than the corresponding parts of the plot with the 51 by 51
grid, because of the improved resolution. On the other hand, the plot with
the 100 by 100 grid misses the origin (0,0)
¡because of the even number of grid points!
A plot with a 500 by 500 grid would be still smoother. It also
would miss the origin, but this phenomenon would be shown at an extremely small
scale. Accordingly, you might not see it without zooming in on the origin.
Exercise 2
- Use the Matlab script aPlot to plot the plane curve
y^{2} = x^{4} + y^{4},
in the region -1 < x < 1,
-1.2 < y < 1.2, in each of the following ways:
- with a 51 by 51 grid;
- with a 501 by 501 grid;
- with a 1000 by 1000 grid.
- Answer the same questions as in Part c of Exercise 1.
Ans: While the 51 by 51 grid gives a plot that looks
somewhat "shaky", it is at least accurate enough to include the origin.
The 501 by 501 grid gives a smoother plot and also includes the origin.
While the 1000 by 1000 grid gives a very smooth plot, it not only misses
the origin, but also omits very visible piece of the curve near the origin.
- (Challenge question) Can you identify what feature makes
this curve "worse" than the curve of Exercise 1?
Ans: Despite the higher resolution than in the 100 by
100 grid in exercise 1, the present plot with the 1000 by 1000 grid actually
seems to leave a larger gap. This seems to be related to the
shape of the curve near near the origin. In exercise 1 the curve crosses
itself at the origin, with an angle that is fairly close to a right angle.
But in exercise 2, the portion of the curve near the origin looks like
two parabolas that are pasted together. And thus, each point on the
upper half of the curve is very close to a point on
the lower half of the curve. {You could express this in a
quantitative way by using the equation of the curve to express y as a
power series in x.}
Well, that's as much as I can explain clearly. We could give a more
penetrating explanation if we knew what algorithm Matlab uses to connect
"consecutive" points on the curve. Also, as one member of the class mentioned
to me, round-off error also could be contributing.
Exercise 3
Consider the following family of plane cubic curves:
y^{2} = x^{3} - 3x + c
Plot 6 curves in this family, corresponding to the following parame
ter
values:
- c = 2
- c = -2
- A value of c slightly above c = 2
and a
value of c slightly below c = 2.
- A value of c slightly above c = -2
and a
value of c slightly below c = -2.
Print a picture that includes all 6 curves, with distinct colors or patterns,
and suitable labels.
Alternatively if this is too cluttered, print 2 pictures, each
including 3 curves.
Comments: In this first figure the curve that
corresponds to c = 2 crosses itself at the origin,
while the curve that corresponds to c = 1.5 is
disconnected.
Comments: In this second figure, the curve that
corresponds to c = -1.5 still has two components,
but at c = -2 it has shrunk to a single point.
- There is a feature of the curve corresponding to c = -2
that Matlab won't show you,
but I may not divulge particulars about
this before the due date of the assignment.
Comment: The point (-1,0) is far enough
away from any other points of the curve so that Matlab can't connect it to a
nearby point, and therefore has completely omitted it from the sketch. It's
shown here because I inserted it separately.
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