Matlab ﬁles used

1.
problem1.m
2.
problem2.m
3.
ssolve.m
4.
problem_4_driver.m

## Math 513 HW1 spring 2014, University of Wisconsin, Madison

January 25, 2020

1.
problems as0.pdf

### 1 Problem 1

Question: Generate a plot of the function $$\sin$$ on the interval [−1, 1] with an appropriate title. Turn in the ﬁgure together with the commands you used to generate the ﬁgure.

%HW1, problem 1, MATH 513, Univ. Of Wisconsin, Madison. SPring 2014
%by Nasser M. Abbasi

close all; clear all;

x    = linspace(-1,1); %default 100 points
fig1 = figure(1);clf;

set(fig1,'defaulttextinterpreter','latex');
plot(x,sin(x))
xlabel('$x$');
ylabel('$\sin(x)$');
title('plot of $\sin(x)$ vs. $x$');
grid on;

### 2 Problem 2

Question: Find the solution to the following 20-by-20 system of equations. Turn in your code and the output $\begin{pmatrix} 1 & & \cdots & 1 & 0 \\ & 1 & & & \\ \vdots & & \ddots & & \vdots \\ & & & 1 & 0 \\ 1 & \cdots & & & 1 \end{pmatrix} x = \begin{pmatrix} 17 \\ 0 \\ \vdots \\ \\ 0 \end{pmatrix}$

[Note: All entries in the matrix not explicitly mentioned are meant to be zero]

%HW1, problem 2, MATH 513, Univ. Of Wisconsin, Madison. SPring 2014
%by Nasser M. Abbasi
close all; clear all;

N      = 20;
A      = zeros(N);
A(:,1) = 1;
A(logical(eye(size(A)))) = 1;
A(1,1:end-1)   = 1;
A(end,:)       = 1;

b    = zeros(N,1);
b(1) = 17;

x = A\b;

fprintf('solution is \n');  x(:)
%-------------------------------
-1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
-17.0000

### 3 Problem 3

Question: Create a Matlab function called ssolve that:

1.
Accepts three parameters, $$a$$, $$b$$, and $$n$$, in that order.
2.
Returns the solution to the following $$n$$ by $$n$$ system

$\begin{pmatrix} 1 & a & & & \\ & 1 & & & \\ & & \ddots & & \\ & & & 1 & \\ b & b & \cdots & b & 1 \end{pmatrix} x = \begin{pmatrix} 1 \\ 2 \\ \vdots \\ n-2 \\ 0 \\ n \end{pmatrix}$

[Note: All entries in the matrix not explicitly mentioned are meant to be zero]

function x = ssolve( a,b,n )
%ssolve(a,b,n)
%HW1, problem 3, MATH 513, Univ. Of Wisconsin, Madison. Spring 2014
%by Nasser M. Abbasi

A              = zeros(n);
A(1,2)         = a;
A(end,1:end-1) = b;

A(logical(eye(size(A)))) = 1;

v        = 1:n;
v(end-1) = 0;

x = A\v(:);
end

### 4 Problem 4

Question: Turn in a printout of the function ssolve and plots of the vectors

1.
ssolve(10, 0.1, 30)
2.
ssolve(2, -0.05, 50)

with the appropriate titles. Also turn in the code that generated these ﬁgures.

EDU>> x = ssolve(10,.1,30)
x =
-19.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
8.0000
9.0000
10.0000
11.0000
12.0000
13.0000
14.0000
15.0000
16.0000
17.0000
18.0000
19.0000
20.0000
21.0000
22.0000
23.0000
24.0000
25.0000
26.0000
27.0000
28.0000
0
-8.6000
EDU>> x = ssolve(2,-0.05,50)
x =

-3.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
8.0000
9.0000
10.0000
11.0000
12.0000
13.0000
14.0000
15.0000
16.0000
17.0000
18.0000
19.0000
20.0000
21.0000
22.0000
23.0000
24.0000
25.0000
26.0000
27.0000
28.0000
29.0000
30.0000
31.0000
32.0000
33.0000
34.0000
35.0000
36.0000
37.0000
38.0000
39.0000
40.0000
41.0000
42.0000
43.0000
44.0000
45.0000
46.0000
47.0000
48.0000
0
108.6000

Now the two solutions are plotted.

%script to plot the solutions. filename: problem_4_driver.m
%HW1, problem 4, MATH 513, Univ. Of Wisconsin, Madison. Spring 2014
%by Nasser M. Abbasi

close all; clear all;
x1 = ssolve(10,.1,30);
x2 = ssolve(2,-0.05,50);

fig1 = figure(1);clf;
set(0,'DefaultAxesFontName', 'Times New Roman')
set(0,'DefaultAxesFontSize', 10)
subplot(1,2,1);

set(fig1,'defaulttextinterpreter','latex');
plot(x1,'-');
xlabel('$x$');
ylabel('$y(x)$');
title('solution of $ssolve(10,.1,30)$');
grid on;

subplot(1,2,2);

plot(x2,'-');
xlabel('$x$');
ylabel('$y(x)$');
title('solution of $ssolve(2,-0.05,50)$');

grid on;