Billede:Simpsons method illustration.png
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Uploaded by Audriusa 16:24, 20 December 2005 (UTC) to use in the non-English Wikipedia versions. Original text follows:
Simpson's method illustration. Done by myself.
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[edit] Source code (carefully documented)
function simpson() % draw an illustration for Simpson's rule
% prepare the scrreen and define some parameters
clf; hold on; axis equal; axis off;
fontsize=25; thick_line=3; thin_line=2; black=[0, 0, 0]; red=[1, 0, 0];
arrowsize=0.1; arrow_type=1; arrow_angle=30; % (angle in degrees)
circrad=0.015; % radius of ball showing up in places
% the function formula and its graph
f=inline('0.45*sin(3.3*(x+0.18))+1'); X=-0.6:0.01:0.8; Y=f(X);
% three points on its graph and the interpolating polynomial going through those points
q=length(X); x1=X(1); y1=Y(1); x2=X(floor(q/2)); y2=Y(floor(q/2)); x3=X(q); y3=Y(q);
Z=y1*(X-x2).*(X-x3)./((x1-x2)*(x1-x3))+y2*(X-x1).*(X-x3)./((x2-x1)*(x2-x3))+y3*(X-x1).*(X-x2)./((x3-x1)*(x3-x2));
% plot the x and y axes
arrow([-0.9 0], [1, 0], thin_line, arrowsize, arrow_angle, arrow_type, black)
arrow([-0.8, -0.1], [-0.8, 1.6], thin_line, arrowsize, arrow_angle, arrow_type, black)
% plot the graph, the interpolating polynomial, some auxiliary lines, and some balls (for beauty)
plot(X, Y, 'linewidth', thick_line)
plot(X, Z, 'linewidth', thick_line, 'color', red)
plot([x1 x1], [0, f(x1)], 'linewidth', thin_line, 'linestyle', '--', 'color', 'black');
plot([x2 x2], [0, f(x2)], 'linewidth', thin_line, 'linestyle', '--', 'color', 'black');
plot([x3 x3], [0, f(x3)], 'linewidth', thin_line, 'linestyle', '--', 'color', 'black');
ball(x1, y1, circrad, red);
ball(x2, y2, circrad, red);
ball(x3, y3, circrad, red);
ball(x1, 0, circrad, black);
ball(x2, 0, circrad, black);
ball(x3, 0, circrad, black);
% place text
tiny=0.1; p0=(x1+x2)/2; q0=(x2+x3)/2;
H=text(x1, -tiny, 'a'); set(H, 'fontsize', fontsize, 'HorizontalAlignment', 'c')
H=text(x2, -tiny, 'm'); set(H, 'fontsize', fontsize, 'HorizontalAlignment', 'c')
H=text(x3, -tiny, 'b'); set(H, 'fontsize', fontsize, 'HorizontalAlignment', 'c')
H=text(p0, 0.43+f(p0), 'P(x)'); set(H, 'fontsize', fontsize, 'HorizontalAlignment', 'c', 'color', 'red')
H=text(q0, 0.15+f(q0), 'f(x)'); set(H, 'fontsize', fontsize, 'HorizontalAlignment', 'c', 'color', 'blue')
saveas(gcf, 'Simpsons_method_illustration.eps', 'psc2') % export to eps
function ball(x, y, r, color)
Theta=0:0.1:2*pi;
X=r*cos(Theta)+x;
Y=r*sin(Theta)+y;
H=fill(X, Y, color);
set(H, 'EdgeColor', 'none');
function arrow(start, stop, thickness, arrow_size, sharpness, arrow_type, color)
% Function arguments:
% start, stop: start and end coordinates of arrow, vectors of size 2
% thickness: thickness of arrow stick
% arrow_size: the size of the two sides of the angle in this picture ->
% sharpness: angle between the arrow stick and arrow side, in degrees
% arrow_type: 1 for filled arrow, otherwise the arrow will be just two segments
% color: arrow color, a vector of length three with values in [0, 1]
% convert to complex numbers
i=sqrt(-1);
start=start(1)+i*start(2); stop=stop(1)+i*stop(2);
rotate_angle=exp(i*pi*sharpness/180);
% points making up the arrow tip (besides the "stop" point)
point1 = stop - (arrow_size*rotate_angle)*(stop-start)/abs(stop-start);
point2 = stop - (arrow_size/rotate_angle)*(stop-start)/abs(stop-start);
if arrow_type==1 % filled arrow
% plot the stick, but not till the end, looks bad
t=0.5*arrow_size*cos(pi*sharpness/180)/abs(stop-start); stop1=t*start+(1-t)*stop;
plot(real([start, stop1]), imag([start, stop1]), 'LineWidth', thickness, 'Color', color);
% fill the arrow
H=fill(real([stop, point1, point2]), imag([stop, point1, point2]), color);
set(H, 'EdgeColor', 'none')
else % two-segment arrow
plot(real([start, stop]), imag([start, stop]), 'LineWidth', thickness, 'Color', color);
plot(real([stop, point1]), imag([stop, point1]), 'LineWidth', thickness, 'Color', color);
plot(real([stop, point2]), imag([stop, point2]), 'LineWidth', thickness, 'Color', color);
end
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