HW2- numerical integration and differentiation

ECE-210-B HW2

This homework will review numerical integration and differentiation using vectorized operations, some plotting operations. Remember the style guidelines
from the previous assignment. For each question, either save the result to a
variable or print out the result to the screen (don’t suppress the result).
1. Now it’s your turn to try out numerical integration and differentiation.
(a) Create two vectors of length 100 and 1000, each containing evenly
spaced samples of the function g(t) = (1 + t)
−1
for 0 ≤ t ≤ 2π.
(b) Approximate the derivatives of the vectors using the difference quotient method (i.e., use diff to take the difference between adjacent
verbs and divide elementwise by ∆t). (This will give you vectors of
length 99 and 999). Call these numerical estimates ˆg
0
(t).
(c) Find the analytical derivative g
0
(t) by hand, and evaluate g
0 over the
same intervals of t. (This will give you vectors of length 100 and
1000; truncate them to their first 99 and 999 elements, respectively).
(d) Say we define the error of the estimate of the derivative to be a
normalized mean-square error:
(g
0
, gˆ
0
) = 1
N
X
t
(g
0
(t) − gˆ
0
(t))2
where N is the number of samples (99 or 999). Calculate the error
for both estimates for the derivative – which is smaller?
(e) Approximate the integrals of your original vectors using cumtrapz
and cumsum. This will give you four approximations of the integral
of g. These will be length 100 and 1000, so you do not have to perform
truncation. Repeat the steps in parts (c) and (d): find the analytical
antiderivative, evaluate it at the same t points, and calculate the
error estimates for each of the four estimates of the integral.
(f) Plot the best estimate for the integral. Title your plot.
(g) (Optional) Explore the plotting API: Give the horizontal and vertical
axes a label. Turn the grid on/off. Change the axis ticks. Subplots!
This is useful for writing reports for your classes, and these plotting
functions are closely mirrored by Python’s matplotlib library.
(h) (Optional) Integrate using Simpson’s rule and compare results.
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2. Perform the following matrix operations (without for loops). Save each
result to a separate variable (i.e., don’t alter A after creating it).
(a) Generate the matrix:
A =





100 101
. . . 104
105 106
. . . 109
.
.
.
.
.
.
.
.
.
.
.
.
1045 1046
. . . 1049





∈ M10×5(R)
(Hint: use logspace and reshape.)
(b) Flip the third row of A left to right.
(c) Create a column vector of the geometric means of each row. (Recall
that the geometric mean of x1, x2, . . . , xn is √n x1x2 . . . xn. The prod
function will probably be helpful.)
(d) Create the submatrix B ∈ M3×3(R) such that bij = a(i+5)(j+1).
(e) Delete rows 5-10 of A. (Do this in at most five operations.)
3. Create a matrix C ∈ M1000×1000(R) such that cij = 2i
4/(3j + 2), using
each of the following methods. Time each method using tic/toc and
write a comment on whether you think the times make sense.
(a) Using for loops and no pre-allocation.
(b) Using for loops and pre-allocation.
(c) Using (meshgrid or repmat) and elementwise matrix operations.
(d) Using broadcasting.
(Caution! Make sure you clear the variables before running this, especially
the matrices in (a) and (b) to show the effect of allocation. E.g., if the four
matrices created in this section are called C1, C2, C3, and C4, make sure
to do something like clear C* before running your code for this question.
If you don’t clear your variables and you re-run your code, then you will
have pre-allocated the matrix for part (a).)
4. (Optional) Try these problems using Python and numpy.
5. (Optional) How does vectorization work (in MATLAB or in general)? Do
a little research and write a few sentences explaining in your own words
how it achieves speedup over for loops.
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