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Math 104A Computation Homework
创建日期：2024-08-15 15:45:10
所属分类：数学mathematics
标签： Math 104A
Math 104A Computation Homework

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Math 104A Computation Homework 


1 Main problem

Obtain a fifigure following the instructions. I recommend setting n = 3 to begin with. After you fifinish

programming, you may change it and enjoy other fifigures. The equation x

n

= 1 has n solutions in the

complex plane zk = e

2kπ

n i

= cos(

2kπ

n ) + isin(

2kπ

n ), (k = 0, 1, · · · , n n 1). Newton’s method can be used fifind

roots of complex functions and this homework does so. But you don’t need to do anything new. Just plug

complex numbers into the method when you evaluate f(xn) and f

0

(xn), where xn is a complex number.

That will fifind you a root of f(x) as a complex function. Newton’s method can spot a solution, but we don’t

know which one. That depends on the initial guess. You are going to color the complex plane according

to which solution an initial value leads to using Newton’s method.

Choose a square domain around the origin, and determine a grid resolution so that a matrix stores the

grid points of your domain for initial guesses.

Run Newton’s method to approximate a solution to the equation above with the initial guess x0 being

each of the grid point of your domain. If it turns out to converge to z1, color a small area near the grid

point of the initial guess, say, red. If the initial guess leads to z2, color a small area near the point with a

difffferent color, say, green. If the initial guess leads to z3, color a small area near the point with another

color, say, blue. You may end up getting something weird, say NaN. To treat such abnormalities all at

once, color a small area near your initial guess, say black, for all other cases.

You will have to make various decisions along the way. I have deliberately minimized the instructions

so you can enjoy the process of creating your own program. So, give it shot right away if you believe

you have enough programming experience. But if you are not very confifident with programming, feel

free to consult the tips below to avoid headache.

(Summary)

data

– n ≥ 3 (power of the equation; 3 is recommended to begin with)

output

– img.pdf: a color plot of a domain around the origin. Each difffferent color indicate that if Newton’s

method starts nearby, the method converges to the same zero, or something weird has occurred.

recommended intermediate data (You don’t have to follow these, but they can make things easier. In

particular, the plotting tips assume that you have generated the matrix CC below.)

– ZZ (a complex matrix/array of a grid domain around 0. Each entry serves as an initial guess for the

Newton’s method)

1– CC: a matrix for color plot. If (i, j)–entry of ZZ leads to a convergence to zk , (i, j)–entry of CC is k

(k = 0, 1, · · · , n n 1). If (i, j)–entry of ZZ leads to any other result, (i, j)–entry of CC is n).

– The names of variables ZZ and CC can be arbitrary. But it will enhance the readability of the code.

? Submission: submit the following on GauchoSpace (click on ‘Computation HW1’).

– Your source code (.m, .py, or .ipynb)

– A fractal image fifile (.pdf)

– (optional) A text fifile describing reflflection (.txt). For example, you can share what you struggled

with, how you overcame it, what you newly learned, etc. This may bump up the grade if the result

is not ideal.

2 Tips for plotting

Below, the variable names are assumed to be as follows:

- CC: the matrix as explained above (see the “recommended intermediate data”)

- xmin, xmax: the minimum and maximum of real part of your domain respectively

- ymin, ymax: the minimum and maximum of imaginary part of your domain respectively

(Matlab) Add the following block of code at the end of your main body of code.

figure;

map = [1 0 0; 0 1 0; 0 0 1; 0 0 0]; % [red; green; blue; black]

colormap(map);

image([xmin,xmax], [ymin,ymax], CC);

axis equal; axis tight;

saveas(gcf,‘img.pdf’);

(Python) Add the following block of code at the top of your code:

import matplotlib.pyplot as plt

from matplotlib.colors import ListedColormap

And add the following block of code at the end of your main body of code:

map = [’r’, ’g’, ’b’, ’k’]

cmap = ListedColormap(map)

plt.imshow(CC, cmap=cmap, extent=(xmin, xmax, ymin, ymax))

plt.savefig("img.pdf", format="pdf", bbox inches="tight")

plt.show()

(Python in colab) You can google ’colab’ and use the Jupyter lab environment offffered. However,

to download the fractal image fifile, you will need to add the following block of code at the end in

addition to the above.

2from google.colab import files

files.download(’img.pdf’)

3 Tips for programming

If you haven’t developed your own programming practice, you may want to consider the following.

Do ‘unit test,’ meaning, test each independent functionality separately so that you better know

where bugs are. For example, to make sure your plotting block has no issue, create a small matrix

CC manually and run the plot part before you really compute and plot the fractal. Another example

is to test whether your function of interest f(x) and f

0

(x) (you must determine these) give the

correct values for some known inputs.

(Anonymous function) To construct Newton’s method for a zero of f(x), you will need to specify

f(x) and f

0

(x). For example, if you want to evaluate the function f(x) = 2x

5

+ 3x many times, you

may want one of the the following lines:

– (Matlab) f = @(x) 2*x?5 + 3*x;

– (Matlab vectorized version) f = @(x) 2*x.?5 + 3*x;

– (Python - vectorized by default) f = lambda x: 2*x**5 - 3*x

Vectorize your code. For example, for Matlab, use pointwise operation .*, ./, .? etc whenever

possible to conduct computation in a parallel way rather than use for loop. (The parallelization is

done internally in Matlab.) For Python, I believe the pointwise operation is usually done by default,

say for numpy objects. If this does not make much sense, you may skip this part for the moment.

But your computation can be slow.

If you use Python, you will need to import numpy package to deal with matrices.

Google it if you don’t know something. It is also my teacher. You may feel like you can do

whatever you want.

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