Soubor:Newton iteration.png

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[edit] Source code

% 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, 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

[edit] License information

This image has been (or is hereby) released into the public domain by its author, Olegalexandrov at the English Wikipedia project. This applies worldwide. In case this is not legally possible: Olegalexandrov grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

[edit] Original upload log

(All user names refer to en.wikipedia)

• 2004-11-23 19:55 Olegalexandrov 405×340×8 (14290 bytes) Scaled down the picture of Newton's method
• 2004-11-22 21:34 Olegalexandrov 509×406×8 (16510 bytes) I graphed this with Matlab (Illustration of Newton's method) {{PD}}

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