File:Newton iteration.png

 
 
% illustration of Newton's method for finding a zero of a function
 
function main ()
 
a=-1; b=1;   % interval endpoints
fs=20;       % text font size
 
% arrows settings
thickness1=2; thickness2=1.5; arrowsize=0.1; arrow_type=1;
angle=20; % in degrees
 
h=0.1;  % grid size
X=a:h:b; % points on the x axis
f=inline('exp(x)/1.5-0.5');   % function to plot
g=inline('exp(x)/1.5');       % derivative of f
x0=0.7; y0=f(x0);             % point at which to draw the tangent line 
m=g(x0);
Y=f(X);                       % points on the function to plot
XT=-0.1:h:b; YT=y0+(XT-x0)*m; % tangent line
 
% prepare the screen
clf; hold on; axis equal; axis off
 
% plot the graph and the tangent lines
plot(X, Y, 'linewidth', thickness1)
plot(XT, YT, 'r', 'linewidth', thickness1)
plot([x0 x0], [0, y0], '--', 'linewidth', thickness2)
 
% axes
small=0.2;
arrow([a 0], [b, 0], thickness2, arrowsize, angle, arrow_type, [0, 0, 0])
arrow([a+small, -0.1], [a+small, 1.4], thickness2, arrowsize, angle, arrow_type, [0, 0, 0])
 
% text
H=text(-0.29, -0.06,  'x'); set(H, 'fontsize', fs)
H=text(0.1, -0.1,  'x_{n+1}'); set(H, 'fontsize', fs)
H=text(0.7, -0.1,  'x_{n}'); set(H, 'fontsize', fs)
 
% save to disk
saveas(gcf, 'newton_iteration.eps', 'psc2')
 
function arrow(start, stop, thickness, arrow_size, sharpness, arrow_type, colour)
 
% 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 colour, 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, 'Colour', colour);
 
      % fill the arrow
      H=fill(real([stop, point1, point2]), imag([stop, point1, point2]), colour);
      set(H, 'EdgeColor', 'none')
 
   else % two-segment arrow
      plot(real([start, stop]), imag([start, stop]),   'LineWidth', thickness, 'Colour', colour);
      plot(real([stop, point1]), imag([stop, point1]), 'LineWidth', thickness, 'Colour', colour);
      plot(real([stop, point2]), imag([stop, point2]), 'LineWidth', thickness, 'Colour', colour);
   end