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Matlab 95 Code

function [c,yc,err,P] = bisect(f,a,b,delta)
%---------------------------------------------------------------------------
%BISECT   The bisection method is used to locate a root.
% Sample calls
%   [c,yc,err] = bisect('f',a,b,delta)
%   [c,yc,err,P] = bisect('f',a,b,delta)
% Inputs
%   f       name of the function
%   a       left endpoint
%   b       right endpoint
%   delta   convergence tolerance
% Return
%   c       solution: the root
%   yc      solution: the function value
%   err     error estimate in c
%   P       History vector of the iterations
%
% NUMERICAL METHODS: MATLAB Programs, (c) John H. Mathews 1995
% To accompany the text:
% NUMERICAL METHODS for Mathematics, Science and Engineering, 2nd Ed, 1992
% Prentice Hall, Englewood Cliffs, New Jersey, 07632, U.S.A.
% Prentice Hall, Inc.; USA, Canada, Mexico ISBN 0-13-624990-6
% Prentice Hall, International Editions:   ISBN 0-13-625047-5
% This free software is compliments of the author.
% E-mail address:      in%"mathews@fullerton.edu"
%
% Algorithm 2.2 (Bisection Method).
% Section    2.2, Bracketing Methods for Locating a Root, Page 61
%---------------------------------------------------------------------------

P = [a b];
ya = feval(f,a);
yb = feval(f,b);
if ya*yb > 0, break, end
max1 = 1 + round((log(b-a)-log(delta))/log(2));
for k=1:max1,
  c  = (a+b)/2;
  yc = feval(f,c);
  if  yc == 0,
    a = c;
    b = c;
  elseif  yb*yc > 0,
    b = c;
    yb = yc;
  else
    a = c;
    ya = yc;
  end
  P = [P;a b];
  if b-a < delta, break, end
end
c  = (a+b)/2;
yc = feval(f,c);
err = abs(b-a)/2;


echo on; clc;
%---------------------------------------------------------------------------
%A2_2   MATLAB script file for implementing
%
% NUMERICAL METHODS: MATLAB Programs, (c) John H. Mathews 1995
% To accompany the text:
% NUMERICAL METHODS for Mathematics, Science and Engineering, 2nd Ed, 1992
% Prentice Hall, Englewood Cliffs, New Jersey, 07632, U.S.A.
% Prentice Hall, Inc.; USA, Canada, Mexico ISBN 0-13-624990-6
% Prentice Hall, International Editions:   ISBN 0-13-625047-5
% This free software is compliments of the author.
% E-mail address:      in%"mathews@fullerton.edu"
%
% Algorithm 2.2 (Bisection Method).
% Section    2.2, Bracketing Methods for Locating a Root, Page 61
%---------------------------------------------------------------------------

clc; clear all; format long;

% - - - - - - - - - - - - - - - - - - - - - - -
%
% This program implements the bisection method.
%
%
% Define and store f(x) in the M-file  f.m
%
% function y = f(x)
% y = x.*sin(x) - 1;

pause % Press any key to continue.

clc;
%.......................................................................
% Begin a section which enters the function(s) necessary for the example
% into M-file(s) by executing the diary command in this script file.
% The preferred programming method is not to use these steps.
% One should enter the function(s) into the M-file(s) with an editor.
delete output
delete f.m
diary  f.m; disp('function y = f(x)');...
            disp('y = x.*sin(x) - 1;');...
diary off;
% Remark. f.m and bisect.m are used for Algorithm 2.2
f(0); % Test for file f.m
pause % Press any key to see the graph y = f(x).

clc;
% ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
% Prepare graphics arrays to plot y = f(x).
% ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
a = 0;
b = 2;
h = (b-a)/150;
X = a:h:b;
Y = f(X);

clc; figure(1); clf;

%~~~~~~~~~~~~~~~~~~~~~~~
% Begin graphics section
%~~~~~~~~~~~~~~~~~~~~~~~
a =  0;
b =  2;
c = -1;
d =  1;
whitebg('w');
plot([a b],[0 0],'b',[0 0],[c d],'b');
axis([a b c d]);
axis(axis);
hold on;
plot(X,Y,'-g');
xlabel('x');
ylabel('y');
title('Graph of y = f(x).');
grid;
hold off;

figure(gcf); pause % Press any key to continue.

clc;

% - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
%
% Example 2.6, page 58.  Use the bisection method to locate
% a zero of the function  f(x) = x sin(x) - 1.
%
% Enter the starting endpoints for [a,b] in  a  and  b
%
% Enter the tolerance in  delta

a = 0;  
b = 2;  
delta = 1e-6;

[p,yp,err,P] = bisect('f',a,b,delta);

pause % Press any key for the list of iterations.

clc;
% ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
% Prepare arrays to graph and print results.
% ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
[m1 m2] = size(P);
n0 = min(7,m1);
Xc = (P(1:n0,1)+P(1:n0,2))'/2;
X0 = [a,Xc,b];
Z0 = zeros(1,n0+2);

clc; figure(2); clf;

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
% Begin graphics section for the results.
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
a =  0;
b =  2;
c = -1;
d =  1;
whitebg('w');
plot([a b],[0 0],'b',[0 0],[c d],'b');
axis([a b c d]);
axis(axis);
hold on;
plot(X,Y,'-g',X0,Z0,'or');
xlabel('x');
ylabel('y');
title('Graphical analysis for the bisection method.');
grid;
hold off;

figure(gcf); pause % Press any key to continue.

clc;
%............................................
% Begin section to print the results.
% Diary commands are included which write all
% the results to the Matlab textfile   output
%............................................
Mx1 = 'Iterations for the bisection method.';
Mx2 = '     a                  b';
Mx3 = 'The approximate root is:';
Mx4 = 'The error estimate for p is  ~ ';
clc,echo off,diary output,...
disp(''),disp(Mx1),disp(''),disp(Mx2),disp(P),...
disp(''),disp(Mx3),disp(''),disp('p = '),disp(p),...
disp('f(p) = '),disp(yp),disp(''),...
disp([Mx4,num2str(err)]),diary off, echo on

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004