[SOLVED] Ece-210-b homework #5 advanced manipulations

This one is a bit longer than the last two – broadcasting and vectorization are
extremely important, and they merit a bit of meditation.
On The Air No reallocation, no loops, no help, no sanity… a typical day at the
Copper Onion. Solve the following with vectorized operations. Broadcasting and meshgrid will be helpful here!

1. Create the matrix
R =


−9 + 9i −8 + 9i . . . 8 + 9i 9 + 9i
−9 + 8i −8 + 8i . . . 8 + 8i 9 + 8i
.
.
.
.
.
.
.
.
.
.
.
.
−9 − 8i −8 − 8i . . . 8 − 8i 9 − 8i
−9 − 9i −8 − 9i . . . 8 − 9i 9 − 9i


(effectively a matrix of coordinates) in the tidiest way you can. (2
lines, 1 if very clever)

2. Approximate
∫ 1
0
sin(t
n
) dt
for n = 1, 2, . . . , 50. Use trapz (the non-cumulative cousin of
cumtrapz) and at least 10000 sample points. Try creating a vector
of points and a vector of n values, combine them via broadcasting
or meshgrid, and go from there. (3 lines)

3. Let’s do something weird – like plot a sphere.
(a) Create two matrices, theta and phi, as follows:
[theta, phi] = meshgrid(…
linspace(0, 2*pi, 5), …
linspace(0, pi, 5));
(This counts as a one-liner in my book.) These will serve as our
spherical coordinates θ and ϕ.

(b) Find the x, y, and z coordinates they correspond to using the
spherical coordinate conversions
x = sin ϕ cos θ,
y = sin ϕ sin θ,
z = cos ϕ.
2
These should take the form of function calls and elementwise
multiplies.

(c) Plot these points using surf. Call pbaspect with appropriate
arguments to make the axes nice and square. Provide a proper
title. All the usual plotting stuff.
Under The Sea Perform the following operations with overly clever indexing.

All line counts include creation of data vectors/matrices.
1. Find the values in sin(linspace(0, 5, 100).*linspace(-5, 0, 100).’)
which are closest to 1/2 (there are a few ties for closest), along with
their 2d indices. (min, abs, and find may be useful!) (3 lines)

2. Approximate the volume contained above by the surface z = e
−(1−xy)
2
and below by the surface z =
1
4
√
x
2 + y
2
. You’ll have to use logical
indexing to selectively sum up volume where one curve is actually
over the other. (Serious bonus points if you 3d-plot the enclosed
volume!) (3 lines, no plot)

I Need a Vacation And now, the weather! Create the following matrices, using
figure and imshow to visualize them. All these matrices are 256 × 256,
and either contain floating-point values or logicals. Add a little comment
describing each one (the meaning of “describe” is up to you, beyond basic
technical information).

Note: the notation aij means the element of A in
row i, column j – i.e., A(i_index, j_index).
1. A where aij is true iff √
(i − 99)
2 + (j − 99)
2 < 29.
2. B where bij is true iff √
(i − 62)
2 + (j − 62)
2 < 58.
3. C where cij is true iff i − 4 sin(
j/10) > 200.

4. S = rand(256, 256, 3) with its two “lower” layers (3rd index 1
and 2) zeroed out.
5. M = A ∩ B
C
(
C denoting logical complement – in other words, ~).
6. Z = (C · S) + M (· denoting elementwise multiply).

A Meshes & Broadcasting
The concepts of broadcasting & meshgrid are often tricky to understand, so
some illustrated examples are likely in order. I’ll start with meshgrid, because
it comes naturally out of evaluating functions using matrices of coordinates –
however, broadcasting (in my opinion) is a more versatile technique (and is
often faster in practice), so keep reading to the end!

A.1 The Grid
Suppose you want to find the area under the surface z = x + y over the unit
square (0 ⩽ x ⩽ 1, 0 ⩽ y ⩽ 1) . In order to do this in matlab, we could do
a discrete Riemann sum: evaluate the function at points on a grid, transform
each of those points into a rectangular prism, and sum the areas of those prisms.

So… how do we create the grid of points? One way might be as follows:
x = [ 0 .2 .4 .6 .8 1; …
0 .2 .4 .6 .8 1; …
0 .2 .4 .6 .8 1; …
0 .2 .4 .6 .8 1; …
0 .2 .4 .6 .8 1; …
0 .2 .4 .6 .8 1];
y = [ 0 0 0 0 0 0; …
.2 .2 .2 .2 .2 .2; …
.4 .4 .4 .4 .4 .4; …
.6 .6 .6 .6 .6 .6; …
.8 .8 .8 .8 .8 .8; …
1 1 1 1 1 1]
z = x + y;

Note the pattern in x and y here: each element of x contains its own x-coordinate,
and each element of y contains its own y-coordinate (flipped vertically). We’re
treating these matrices as gridded data: each element of x and y contains information about its corresponding point in the plane (namely, one of its coordinates).

Evaluating the expression for z elementwise thus inserts the correct x &
y coordinates at the correct positions in the matrix: the upper left corner has
x = 0 and y = 0, so z = 0; the lower right corner has x = 1 and y = 1, so z = 2
– in general, each of the coordinates winds up in the right place, and z becomes
4
z =
0 .2 .4 .6 .8 1.0
.2 .4 .6 .8 1.0 1.2
.4 .6 .8 1.0 1.2 1.4
.6 .8 1.0 1.2 1.4 1.6
.8 1.0 1.2 1.4 1.6 1.8
1.0 1.2 1.4 1.6 1.8 2.0
which is a grid of evaluations of z = x + y. This can be used for any function
(the notes show this technique used to evaluate z = exp(−x
2−y
2
), which translates to matlab code as z = exp(-x.^2 – y.^2)), and, as such, is a handy
technique.

So much so that it has a name: those x & y matrices we created form
a mesh, two matrices that form a gridded set of coordinates, and such meshes
are typically generated not by hand (which is a pain) but by meshgrid. The
following code generates identical matrices to those handcrafted above:
[x, y] = meshgrid(0:.2:1); % same as before
z = x + y;
meshgrid takes in a vector (or two vectors, for non-square matrices) of coordinates and creates matrices which function as x, y coordinate inputs to vectorized
functions. Try it out and see what it does!

A.2 How To Broadcast
meshgrid is usually only used where meshes are really necessary – places where
being able to traverse the space of evaluation in two or three dimensions is necessary. Surface plots, like the sphere in this assignment, are one of those places:
each z-coordinate needs an easy association with an x and y coordinate so that
the plotting engine knows where the hell to put all the points.

If, however, we just want the results of the evaluation (or want another technique to construct matrices really quickly), broadcasting is an alternative. You
met broadcasting in the very first assignment when you performed C + A –
an operation notably not valid in mathematics, because C and A were different
shapes! They were, however, broadcastable shapes – C was a 4×4 matrix and A
was a 4 × 1 column vector, so matlab simply repeated A four times and added

it to C:
C + A →


c11 c12 c13 c14
c21 c22 c23 c24
c31 c32 c33 c34
c41 c42 c43 c44


+


w
x
y
z


→


c11 c12 c13 c14
c21 c22 c23 c24
c31 c32 c33 c34
c41 c42 c43 c44


+






w
x
y
z




w
x
y
z




w
x
y
z




w
x
y
z






→


c11 + w c12 + w c13 + w c14 + w
c21 + x c22 + x c23 + x c24 + x
c31 + y c32 + y c33 + y c34 + y
c41 + z c42 + z c43 + z c44 + z

 .

Any two matrices with dimensions that can be made to match by this sort of
repetition are broadcastable, and any elementwise operation may cause broadcasting to occur. For example, we could compute an elementary multiplication
table very quickly using .*:
first_factor = 0:9; % row vector, 1×10
second_factor = (0:9).’; % column vector (transposed), 10×1
mult_table = first_factor .* second_factor; % becomes 10×10

To perform this calculation, matlab recognizes that if first_factor is repeated 10 times vertically and second_factor is repeated 10 times horizontally, both will be the same size, so it (effectively) does that, producing a 10×10
result.

These examples are, of course, simple, to facilitate understanding. There’s
more that can be done… but that’s up to you!