File:Simpsons method illustration.png

DescriptionSimpsons method illustration.png	Simpson's method illustration. Done by myself (Oleg Alexandrov 23:17, 12 August 2007 (UTC)).
Date	23 November 2005 (original upload date)
Source	Transferred from en.wikipedia to Commons.
Author	Oleg Alexandrov at English Wikipedia
Other versions	File:Simpsons method illustration.svg is a vector version of this file. It should be used in place of this PNG file when not inferior. File:Simpsons method illustration.png → File:Simpsons method illustration.svg For more information, see Help:SVG. In other languages Alemannisch ∙ Bahasa Indonesia ∙ Bahasa Melayu ∙ British English ∙ català ∙ čeština ∙ dansk ∙ Deutsch ∙ eesti ∙ English ∙ español ∙ Esperanto ∙ euskara ∙ français ∙ Frysk ∙ galego ∙ hrvatski ∙ Ido ∙ italiano ∙ lietuvių ∙ magyar ∙ Nederlands ∙ norsk bokmål ∙ norsk nynorsk ∙ occitan ∙ Plattdüütsch ∙ polski ∙ português ∙ português do Brasil ∙ română ∙ Scots ∙ sicilianu ∙ slovenčina ∙ slovenščina ∙ suomi ∙ svenska ∙ Tiếng Việt ∙ Türkçe ∙ vèneto ∙ Ελληνικά ∙ беларуская (тарашкевіца) ∙ български ∙ македонски ∙ нохчийн ∙ русский ∙ српски / srpski ∙ татарча/tatarça ∙ українська ∙ ქართული ∙ հայերեն ∙ বাংলা ∙ தமிழ் ∙ മലയാളം ∙ ไทย ∙ 한국어 ∙ 日本語 ∙ 简体中文 ∙ 繁體中文 ∙ עברית ∙ العربية ∙ فارسی ∙ +/−

This work has been released into the public domain by its author, Oleg Alexandrov at English Wikipedia. This applies worldwide. In some countries this may not be legally possible; if so: Oleg Alexandrov grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.Public domainPublic domainfalsefalse

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; kjjkjkjkjkjklhoijthin_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,yuguihguih arrowsize, arrow_angle, arrow_type, black) 
arrow([-0.8, -0.1], [-0.8, 1.6], tipokpkkhin_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,  'x0');          set(H, 'fontsize', fontsize, 'HorizontalAlignment', 'c')
H=text(x2, -tiny,  'x1');          set(H, 'fontsize', fontsize, 'HorizontalAlignment', 'c')
H=text(x3, -tiny,  'x2');          set(H, 'fontsize', fontsize, 'HorizontalAlignment', 'c')
H=text(p0, 0.43+f(p0),  'P2(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

• 2005-11-23 03:12 Oleg Alexandrov 1186×1072×8 (26565 bytes)

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