MATLAB 5 Solution.pdf
-
Upload
muni-sankar-matam -
Category
Documents
-
view
13 -
download
1
description
Transcript of MATLAB 5 Solution.pdf
Name: Foster, Antony (XXXX)
Course: Math 20300 XX
Instructor: Prof A Foster
Due Date: June XX, XXXX
MATLAB ASSIGNMENT 5
Purpose: Using Matlab to create 2-D labeled/unlabeled contour plots of surfaces
defined over closed bounded rectangular regions where we
can identify and classify potential critical points (relative extrema) that the
function may have on as guaranteed by the Extreme Value Theorem.
New Commands Used: plot (x,y, options)
text(x,y,'Your Phrase',options}
contour(x,y,f(x,y),n,options)
[C h] = contour(x,y,f(x,y))
clabel(C,h)
u = a:b:c
Exercise 5.2 Suppose that a surface S is defined by
A) Write a script M-file that will produce a labeled contour plot for over the (closed and
bounded) region, S, given by
B) Based on the contour plot you found in A) estimate the coordinates of 2 saddle points of
in the region defined in A). Mark these points using the Data Cursor.
MATLAB CODE
clear all clc; clf reset
figure(2)
f = @(x,y)sin(3*y - x.^2 + 1) + cos(2*y.^2 - 2*x);
u = linspace(-2,2,45); v = linspace(-1,1,45); %Levels = [-2:.2:2]; [x,y] = meshgrid(u,v);
[C,h] = contour(x,y,f(x,y),'k'); hold on clabel(C,h);
plot(-1.1818,0.63636,'--rs','LineWidth',2,'MarkerEdgeColor','k','MarkerFaceColor',
'r','MarkerSize',6)
plot(1,0.77273,'--rs','LineWidth',2,'MarkerEdgeColor','k','MarkerFaceColor',
'r','MarkerSize',6) plot(0.5,-0.72727,'--rs','LineWidth',2,'MarkerEdgeColor','k','MarkerFaceColor',
'r','MarkerSize',6)
text(-1.09,0.63636,'\leftarrow This is a saddlepoint','HorizontalAlignment','left',
'FontWeight','bold')
text(1.1,0.77273,'\leftarrow This is a saddle point','HorizontalAlignment','left',
'FontWeight','bold')
text(0.6,-0.72727,'\leftarrow This is a saddle point','HorizontalAlignment','left',
'FontWeight','bold')
set(gca,'XTick',[-2:0.5:2],'XMinorTick','on','FontName','times','FontWeight',
'bold')
set(gca,'YTick',[-1:0.5:1],'YMinorTick','on','FontName','times','FontWeight',
'bold')
title({' ','20 labeled Contour curves of the surface f(x,y) = sin(3y - x^2 + 1) +
cos(2y^2 - 2x)','over the rectangle R = \{(x,y) | -2\,\leq\,x\,\leq 2,\, -
1\,\leq\,y\,\leq\,1\}','by Antony Foster'})
xlabel('X-Axis','Color','red','FontName','mathematica','FontWeight','bold',
'FontSize',12)
ylabel('Y-Axis','Color','red','FontName','mathematica','FontWeight','bold',
'FontSize',12)
axis square axis([-2 2 -1 1]); grid on %%
MATLAB OUTPUT
-1.5
-1.5
-1.5
-1.5
-1.5
-1.5
-1.5
-1.5 -1
.5-1
-1
-1
-1
-1
-1-1
-1
-1
-1
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
0
0
0
0
0
0
0
0
0
0
0
0 00
0.5
0.5 0.5
0.5
0.5
0.5
0.5
0.5
0.5
1
1
1
1
1 1
1
11.5
1.5
1.5
1.5
1.5
This is a saddle point
This is a saddle point
This is a saddle point
10 labeled Contour curves of the surface f(x,y) = sin(3y - x2 + 1) + cos(2y2 - 2x)
over the rectangle S = {(x,y) | -2 x 2, -1 y 1 }
by Antony Foster
X-Axis
Y-A
xis
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-1
-0.5
0
0.5
1
NOTE: THIS IS NOT THE ANSWER TO PART A) or B). I just wanted to see more
contours curves (40 unlabeled ones) of so that I could better identify the
saddle points (I can see about 4 such points).
MATLAB CODE
clear all clc; clf reset
figure(1)
f = @(x,y)sin(3*y - x.^2 + 1) + cos(2*y.^2 - 2*x);
u = linspace(-2,2,25); v = linspace(-1,1,25); [x,y] = meshgrid(u,v);
n = 40; % number of contour curves. contour(x,y,f(x,y),n,'k'); % 2-Dimensional contour plots of z = f(x,y)
set(gca,'XTick',[-2:0.5:2],'XMinorTick','on','FontName','times','FontWeight',
'bold')
set(gca,'YTick',[-1:0.25:1],'YMinorTick','on','FontName','times','FontWeight',
'bold')
title({' ','40 unlabeled contour curves of f(x,y) = sin(3y - x^2 + 1) + cos(2y^2 -
2x)','over the rectangle R = \{(x,y) | -2\,\leq\,x\,\leq 2,\, -1\,\leq\,y\,\leq\,1
\}','by Antony Foster'})
xlabel('X-Axis','Color','red','FontName','mathematica','FontWeight','bold',
'FontSize',12)
ylabel('Y-Axis','Color','red','FontName','mathematica','FontWeight','bold',
'FontSize',12)
axis equal square axis([-2 2 -1 1]); grid on %%
MATLAB OUTPUT
40 unlabeled contour curves of f(x,y) = sin(3y - x2 + 1) + cos(2y2 - 2x)
over the rectangle R = {(x,y) | -2 x 2, -1 y 1 }
by Antony Foster
X-Axis
Y-A
xis
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
Exercise 5.3 Consider the surface given by .
A) Write a script M-file that will produce labeled contour curves for over the (closed and bounded)
square region given by
.
B) Based on the contour plot you found in A) determine whether has any critical points in
the square defined in A). If there are any such points, provide estimates from the graph for their
and coordinates and provide a justification from the graph as to whether these points occur at a
relative maxima, minima or saddle points. Indicate your reasons as comments in the script M-file
and publish the contour plot and the script M-file.
MATLAB CODE clear all clc; clf reset figure(4)
f = @(x,y)sin(3*x + y) - 2*cos(x - y);
u = linspace(-2,2,45); v = linspace(-2,2,45);
Levels = -3:0.4:3; [x,y] = meshgrid(u,v);
plot(-1.1818,-1.1818,'--rs','LineWidth',2,'MarkerEdgeColor','k','MarkerFaceColor',
'r','MarkerSize',10)
hold on plot(-0.36364,-0.36364,'--
rs','LineWidth',2,'MarkerEdgeColor','k','MarkerFaceColor','g','MarkerSize',10)
hold on plot(1.1818,-2,'--
rs','LineWidth',2,'MarkerEdgeColor','k','MarkerFaceColor','b','MarkerSize',10)
hold on [C,h] = contour(x,y,f(x,y),Levels,'k');
clabel(C,h,Levels);
plot(0.36364,0.36364,'--rs','LineWidth',2,'MarkerEdgeColor','k','MarkerFaceColor', 'r','MarkerSize',10)
plot(1.1818,1.1818,'--rs','LineWidth',2,'MarkerEdgeColor','k','MarkerFaceColor',
'g','MarkerSize',10)
plot(-2,-2,'--rs','LineWidth',2,'MarkerEdgeColor','k','MarkerFaceColor',
'g','MarkerSize',10)
plot(-2,1.1818,'--rs','LineWidth',2,'MarkerEdgeColor','k','MarkerFaceColor',
'b','MarkerSize',10)
text(0.4,0.36364,'(0.36, 0.36)','VerticalAlignment','top','HorizontalAlignment',
'center','FontWeight','bold')
text(-1.09,-1.1818, '(-1.18, -1.18)','VerticalAlignment','top',
'HorizontalAlignment','center','FontWeight','bold)
text(-0.35,-0.36364,'(-0.36,-36)','VerticalAlignment','top','HorizontalAlignment',
'center','FontWeight','bold')
text(1.2,-2,'(1.1818, -2)','VerticalAlignment','bottom','HorizontalAlignment',
'center','FontWeight','bold)
text(1.2,1.1818,'(1.1818,1.1818)','VerticalAlignment','top','HorizontalAlignment','
center','FontWeight','bold')
text(-1.9,1.1818,'(-2,1.1818)\leftarrow','HorizontalAlignment','left','FontWeight',
'bold')
text(-1.9,-2,'(-2,-2)\leftarrow','HorizontalAlignment','left','FontWeight','bold')
set(gca,'XTick',[-2:0.5:2],'XMinorTick','on','FontName','times','FontWeight',
'bold')
set(gca,'YTick',[-2:0.5:2],'YMinorTick','on','FontName','times','FontWeight',
'bold')
title({' ','15 labeled Contour curves f(x,y) = sin(3x + y) - 2cos(x - y)','over the
rectangle R = \{(x,y) | -2 \leq x \leq 2, -2 \leq y \leq 2\}','by Antony Foster'})
legend('saddle point','local min','local max','Location','NorthEastOutside')
xlabel('X-Axis','Color','red','FontName','mathematica','FontWeight','bold',
'FontSize',12)
ylabel('Y-Axis','Color','red','FontName','mathematica','FontWeight','bold',
'FontSize',12)
axis square axis([-2 2 -2 2]);
grid on
%%
MATLAB OUTPUT
Comments: (THOUGH MY PICTURE IS A BIT CLUTTERED (TOO MANY LABELS!)) Points at the
blue markers are points where has its largest value locally. We see this in
the contour plot when we move closer and closer towards the blue markers the z
values (from the data cursor) increase and as you move away the z (data cursor)
decrease.
Points at the green markers are points where has its smallest value locally.
We see this in the contour plot when we move closer and closer towards the green
markers the z values (from the data cursor) decrease and as you move away the z
(data cursor) increase.
Points at the red markers are points where has neither a largest nor smallest
value locally. We see this in the contour plot where contour curves seem to
intersect.
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2.6
-2.6
-2.6
-2.6
-2.6
-2.2
-2.2
-2.2 -2
.2
-2.2
-2.2
-2.2
-1.8
-1.8
-1.8-1
.8
-1.8
-1.8-1
.8-1.4
-1.4
-1.4
-1.4
-1.4
-1.4
-1.4
-1.4
-1
-1
-1
-1
-1-1-1
-1
-1
-0.6
-0.6
-0.6
-0.6
-0.6
-0.6
-0.6
-0.2
-0.2
-0.2
-0.2
-0.2
-0.2
-0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.6
0.6
0.6
0.6
0.6
1
1
1 1
11
1
1.4
1.4
1.4
1.4
1.4
1.8
1.8
1.8
1.8
1.8
2.2 2
.2
2.2
2.2
2.6
2.6
(0.36, 0.36)
(-1.18, -1.18)
(-0.36,-36)
(1.1818, -2)
(1.1818,1.1818)(-2,1.1818)
(-2,-2)
15 labeled Contour curves f(x,y) = sin(3x + y) - 2cos(x - y)
over the rectangle R = {(x,y) | -2 x 2, -2 y 2}
by Antony Foster
X-Axis
Y-A
xis
saddle point
local min
local max
MATLAB ASSIGNMENT ENDS HERE
NOTE: THIS IS NOT THE ANSWER TO PART A) or B). I just wanted to see more
contours curves (40 unlabeled ones) of so that I could better identify the
saddle points (I can see about 4 such points).
MATLAB CODE
clear all clc; clf reset figure(3)
f = @(x,y)sin(3*x + y) - 2*cos(x - y);
u = linspace(-2,2,45); v = linspace(-2,2,45); [x,y] = meshgrid(u,v);
contour(x,y,f(x,y),40,'k')
set(gca,'XTick',[-2:0.5:2],'XMinorTick','on','FontName','times','FontWeight',
'bold')
set(gca,'YTick',[-2:0.5:2],'YMinorTick','on','FontName','times','FontWeight',
'bold')
title({' ','40 unlabeled contour curves of f(x,y) = sin(3x + y) - 2cos(x -
y)','over the rectangle R = \{(x,y) | -2 < x < 2, -2 < y < 2\}','by Antony
Foster'})
xlabel('X-Axis','Color','red','FontName','mathematica','FontWeight','bold',
'FontSize',12)
ylabel('Y-Axis','Color','red','FontName','mathematica','FontWeight','bold',
'FontSize',12)
axis equal square axis([-2 2 -2 2]); grid on %%
MATLAB OUTPUT
40 unlabeled contour curves of f(x,y) = sin(3x + y) - 2cos(x - y)
over the rectangle R = {(x,y) | -2 x 2, -2 y 2}
by Antony Foster
X-Axis
Y-A
xis
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2