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Simpson's Rule Illustration - How to create those parabolas?

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Dominic
Dominic on 27 Mar 2013
Commented: Richard Treuren on 17 Apr 2014
I managed to create the rectangle and trapezium strips, but stuck on the parabola strips for Simpson's Rule like the one shown below.
In this code, the user has to input the strip width, function, limits,
Here's my code for the RECTANGLE STRIPS
% 7. Display figure
clf, hold on;
plot([a b], [0 0], 'k' ), plot([0 0], [min( y_x(x) ) max( y_x(x))], 'k') % This shows a black line for the x-axis (y=0) and the y-axis (x=0).
xlabel('x', 'FontWeight', 'bold'), ylabel('y(x)', 'FontWeight', 'bold')
title(['Numeric integration through Rectangle Rule of y(x)=' , y_xs , ' with ', num2str(n), ' slices ||| Result is ', num2str(S_r) '.'] , 'FontWeight', 'bold')
% 8. To create the rectangular strips
for x=a:dx:(b-dx);
y_x(x);
left = x; right = x+dx; bottom = 0; top = y_x(x);
X = [left left right right]; Y = [bottom top top bottom]; %to create the shape
fill(X,Y, 'b', 'FaceAlpha', 0.3)
end
% 9. Display a smooth line in the graph
x = a:dx/100:b;
plot(x, y_x(x), 'k')
Here's my code for the TRAPEZIUM STRIPS:
% 7. Display figure
clf, hold on;
plot(x, y_x(x), 'k--')
plot([a b], [0 0], 'k'), plot([0 0], [min( y_x(x) ) max( y_x(x))], 'k')
xlabel('x', 'FontWeight', 'bold'), ylabel('y(x)', 'FontWeight', 'bold')
title(['Numeric integration through Trapezium Rule of y(x)=' , y_xs , ' with ', num2str(n), ' slices ||| Result is ', num2str(S_t) '.'] , 'FontWeight', 'bold')
% 8. Display Trapezium strips
for x=a:dx:(b-dx);
y_x(x);
left = x; right = x+dx; bottom = 0; top1 = y_x(x); top2 = y_x(x+dx);
X = [left left right right]; Y = [bottom top1 top2 bottom]; %to create the shape
fill(X,Y, 'b', 'FaceAlpha', 0.3)
end
% 9. Display a smooth line in the graph
x = a:dx/100:b;
plot(x, y_x(x), 'b')
Comments on how to optimise and improve brevity this code would also be appreciated! Cheers

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Dominic
Dominic on 29 Mar 2013
Let me know if you want to have a look at the other parts of this code.

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Answers (3)

bym
bym on 31 Mar 2013
here is what I have done
clc;clear; close all
x = linspace(0,4*pi); % create data
f = sin(x);
xs = linspace(0,4*pi,11); % sample points
fs = sin(xs); % sample function
plot(x,f,'g') % plot function
hold on
plot(xs,fs,'bo') % plot sample points
xp = reshape(x,[],5)'; % prepare for plotting
xp(5,21)=x(end); % duplicate end point
xp(1:4,21)=xp(2:5,1); % duplicate end points
c(5,3)=0; %preallocate
for k = 1:5
c(k,:) = polyfit(xs(2*k-1:2*k+1),fs(2*k-1:2*k+1),2); %fit coefficients
fill([xp(k,1) xp(k,:) xp(k,end)],[0 polyval(c(k,:),xp(k,:)) 0] ...
,'c', 'FaceAlpha',.1)
end
ylim([-1.25,1.25])

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Dominic
Dominic on 2 Apr 2013
This is the figure that is showing up with mine...
Here is the code, I am not sure what to replace the numerical values I am asking above, so I just copied what you gave. Surprise, surprise, it didn't show as the way I want it to.
Here is my try
% 4. Create the strips (with parabolic arcs)
x_s = linspace(a,b); % create x-data
y_s = y_x(x_s); % create y-data
x_arcs = linspace(a,b,n+1); % sample points
y_arcs = y_x(x_arcs); % sample function
plot(x_arcs, y_arcs, 'bo-') % plot sample points
xp = reshape(x_s, [], 5)'; % !!! prepare for plotting
xp(5,21)=x(end); % !!! duplicate end point
xp(1:4,21)=xp(2:5,1); % !!! duplicate end points
c(5, 3)=0; % !!! preallocate
for k = 1:5
c(k,:)=polyfit(x_arcs(2*k-1:2*k+1), y_arcs(2*k-1:2*k+1), 2); %fit coefficients
fill([xp(k,1) xp(k,:) xp(k,end)], [0 polyval(c(k,:), xp(k,:)) 0], 'c', 'FaceAlpha', 0.1)
end

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Richard Treuren
Richard Treuren on 17 Apr 2014
Edited: Richard Treuren on 17 Apr 2014
I changed the script of proecsm a bit so that it does work for different step sizes and some other changes in the area plot
clc;clear; close all
steps = 5; % number of steps
x = linspace(0,2*pi,steps*12); % create data
xs = linspace(0,2*pi,2*steps+1); % sample points
f = sin(x);
fs = sin(xs); % sample function
c(steps,3)=0;
for k = 1:steps
c(k,:) = polyfit(xs(2*k-1:2*k+1),fs(2*k-1:2*k+1),2); %fit coefficients
hold on
z = linspace(xs(2*k-1),xs(2*k+1),12);
y = c(k,1).*z.^2+c(k,2).*z+c(k,3);
area(z,y,'FaceColor',[0.6,1,1])
end
hold on
plot(x,f,'r','LineWidth',2) % plot function
hold on
plot(xs,fs,'bo','LineWidth',2) % plot sample points

  1 Comment

Richard Treuren
Richard Treuren on 17 Apr 2014
if you want to calculate the area (using simpson's rule) you can add the next lines to the script:
sim=0;
for i=1:steps
sim = sim + (xs(2*i+1)-xs(2*i-1))/6*(fs(2*i-1)+4*fs(2*i)+fs(2*i+1));
simp(i)= sim; %variable for plotting
end
sim

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