Image:Cauchy sequence illustration.png

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[edit] MATLAB Source code

% draw an illustration of a Cauchy sequence
function main() 

% prepare the screen and define some parameters   
   figure(1); clf; hold on; axis equal; axis off;

   fontsize=30; thick_line=3; thin_line=2;
   black=[0, 0, 0]; red=[1, 0, 0]; blue=[0, 0, 1];
   arrowsize=0.5; arrow_type=1; arrow_angle=30; % (angle in degrees)
   circrad=0.07; % radius of ball showing up in places


   B=9;
   X=0:0.06:B;

   f=inline('3*exp(-0.4*X)', 'X');
   Y=sin(5*X).*f(X);

   for i=1:length(X)
          ball(X(i), Y(i), circrad, blue);
   end

   X=0:0.05:(B+0.3);
   Z=f(X);
   plot(X, Z, 'r--', 'linewidth', thin_line)
   plot(X, -Z, 'r--', 'linewidth', thin_line)


   % draw the coordinate axes
   shift=-3;
   Kx=1.1; Ky=1.3; 
   L=max(Y); 
   arrow([-1 shift], [Kx*B, shift],  thin_line, arrowsize, arrow_angle, arrow_type, black) 
   arrow([-1, shift], [-1, Ky*L],    thin_line, arrowsize, arrow_angle, arrow_type, black) 


   text(Kx*B+0.6, shift,  '\it{n}', 'fontsize', fontsize, 'HorizontalAlignment', 'c')
   text(-1, Ky*L+0.8,  '\it{x_n}', 'fontsize', fontsize, 'HorizontalAlignment', 'c')

   % save to disk
   saveas(gcf, 'Cauchy_sequence_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, th, arrow_size, sharpness, arrow_type, color)
   
% Function arguments:
% start, stop:  start and end coordinates of arrow, vectors of size 2
% th:           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', th, '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', th, 'Color', color); 
      plot(real([stop, point1]), imag([stop, point1]), 'LineWidth', th, 'Color', color);
      plot(real([stop, point2]), imag([stop, point2]), 'LineWidth', th, 'Color', color);
   end

• Cauchy sequence