[SOLVED] Ece-210-b homework #2 vectors & matrices

We’ve now seen some more powerful vector and matrix operations, so let’s
use ‘em! Thinking and writing vectorized code is a powerful way of improving
operation efficiency (as you will see). For each computation, save the result to
a variable or print it to the screen (omit the semicolon). Bonus points if you
use disp or fprintf to label the output and make really pretty charts (FF loves
really pretty charts).

Same submission format as the first assignment, and will be the same going
forward. Have fun…

Read the rest of this document… as before. Some of it may be review, but hey
– lecture notes for this class aren’t super formal.
Spatial Awareness Perform the following operations without the use of for.
1. Use reshape and the : operator to create the matrix
A =


0 1 . . . 7
8 9 . . . 15
.
.
.
.
.
.
.
.
.
.
.
.
56 57 . . . 63


(should be 8 by 8),
then dot-exponentiate 2 by it to create the matrix
B =


2
0
. . . 2
7
.
.
.
.
.
.
.
.
.
2
56
. . . 2
63

 .

2. Flatten B into a vector v (row or column – should be quick either
way!) and extract the prime-numbered components of v (in other
words, v2, v3, . . . , v61). Save these in their own variable, and name
it well!
Note: matlab has a lot of funny little helper functions…

3. Find the geometric mean of those prime components (for some
x1, x2, . . . , xn, the geometric mean is √n x1x2…xn – see the documentation for prod and nthroot for how to compute this concisely).
4. Flip one row of A end for end (see the course notes for how to do
this).

5. Delete one column of A.
Smallest Addition We’re going to try out a couple variations of numerical integration and differentiation. Specifically, let’s see how accurately we can
evaluate
erf(z) = 2
√
π
∫ z
0
exp(−t
2
) dt
and
d
dt [exp(−t
2
)] .
2
Matlab has a function for doing the former (called, unsurprisingly, erf),
as this is a rather useful but nonelementary integral, but because we enjoy
pain we’ll do it by hand.

1. Evaluate exp(−t
2
) at 100 evenly spaced points from 0 to 6.66.
2. Approximate the derivative of this function using diff. Find the
mean squared error (using mean) between this derivative and samples of the analytical derivative over this range. (Note: you will likely
have to truncate or pad these.)

3. Approximate the integral of the original function using cumsum and
cumtrapz. This will result in two estimations of the original. Find
the mean squared error against matlab’s erf over the same range
for each of them. (Sorry to make you do this before we hit functions,
but…)
Comment on the integral estimations – which is better? How close
did they get to erf?

A Indexing & How MATLAB Thinks
Vectors and matrices are indexed in multiple ways. Each element in a vector
or a matrix has an index, which uniquely identifies it (as is probably painfully
evident by now). Vectors are indexed by only this number. Matrix indexing can
be more complex, so I’ll give a refresher here.

All indices in matlab start from 1. If this irks your programmer self to no
end, Python provides some excellent matlab alternatives with zero-indexing
(not to mention the expressive power of a real programming language!). Each
element of a matrix has two (or more!) numbers which represent it. As in mathematics, matrices are indexed first by row, then by column, then by… whatever
other dimensions you have. For a 3 × 3 matrix, you’d have something like this:
A =


A(1, 1) A(1, 2) A(1, 3)
A(2, 1) A(2, 2) A(2, 3)
A(3, 1) A(3, 2) A(3, 3)

 .
Each element of a matrix also has a single scalar associated with it, for an alternate method of indexing. Note the row-first order:
A =


A(1) A(4) A(7)
A(2) A(5) A(8)
A(3) A(6) A(9)

 .

B Notes on Numerical Calculus
The lecture notes on numerical calculus walk you through most of the code
you need to know here. One important thing to note, though, is padding: the
diff function returns the difference between subsequent elements of a vector,
and thus will return a vector one element shorter than the one passed to it. For
example:
x = [0 1 3 5 9]; % length 5
y = diff(x); % y becomes [1 2 2 4], length 4

If you’re computing a derivative via diff, you might want to pad the resulting
vector (depending on what you’re doing with it) so that its length matches with
other data you’re using:
4
% x, y defined above…
dydx = diff(y)./diff(x);
dydx_pad_start = dydx([1 1:end]); % duplicates first element
dydx_pad_end = dydx([1:end end]); % duplicates last element
The functions cumsum and cumtrapz do not need padding and are a little
easier to use – cumtrapz makes doing numerical integrals really easy if you pass
it the right arguments. See its documentation for more details!