Roots Finding: Bracketing Methods

Here are the M-files provided to implement the bisection & false position method. Overall the scripts are the same, the only difference is the iterative function used to evaluate the root (as highlighted in each M-file). 

Bisection M-file

function [root,fx,ea,iter]=bisect(func,xl,xu,es,maxit,varargin)

%bisect:root location zeroes

% [root,fx,ea,iter]=bisect(func,xl,xu,es,maxit,p1,p2,...):

%   uses bisection method to find the root of func

% varargin = variable number of arguments, can input any number of

% arguments in the func

%input:

%  func:name of function

%  xl,xu=lower and upper guesses

%  es=desired relative error(default=0.0001%)

%  maxit=maximum allowable iterations(default=50)

%  p1,p2,...=additional parameters used by func

%output:

%  root=real root

%  fx=function value at root

%  ea=approximate relative error(%)

%  iter=number of iterations 

if nargin<3,error('at least 3 input arguments required'),end

test = func(xl,varargin{:})*func(xu,varargin{:});

if test>0, error('no sign change'),end

if nargin<4||isempty(es),es=0.0001;end

if nargin<5||isempty(maxit),maxit=50;end

iter=0;xr=xl;ea=100;

while(1)

    xrold=xr;

    xr=(xl+xu)/2;

    iter=iter+1;

    if xr~=0,ea=abs((xr-xrold)/xr)*100;end 

    test=func(xl,varargin{:})*func(xr,varargin{:});

    if test<0

        xu=xr;

    elseif test>0

        xl=xr;

    else

        ea=0;

    end

    if ea<=es||iter>=maxit,break,end

end

root=xr;fx=func(xr,varargin{:});

False Position M-file

function [root,fx,ea,iter]=falsepos(func,xl,xu,es,maxit,varargin)

%falsepos:root location zeroes

% [root,fx,ea,iter]=falsepo(func,xl,xu,es,maxit,p1,p2,...):

%   uses falsepos method to find the root of func

% varargin = variable number of arguments, can input any number of

% arguments in the func

%input:

%  func:name of function

%  xl,xu=lower and upper guesses

%  es=desired relative error(default=0.0001%)

%  maxit=maximum allowable iterations(default=50)

%  p1,p2,...=additional parameters used by func

%output:

%  root=real root

%  fx=function value at root

%  ea=approximate relative error(%)

%  iter=number of iterations 

if nargin<3,error('at least 3 input arguments required'),end

% check for number of arg inp, if less than 3, then error message is printed out

test = func(xl,varargin{:})*func(xu,varargin{:});

if test>0, error('no sign change'),end

% if upper and lower boundary same sign, error message is printed out

if nargin<4|isempty(es),es=0.0001;end

% if no input for es, then matlab set default value as 0.0001%

if nargin<5|isempty(maxit),maxit=50;end

% if no input for maxit, then matlab set default value as 50 iterations

iter=0;xr=xl;ea=100;

% in first iteration, if f(xl)f(xr)>0,replace xl with xr, approximate relative error = 100%

while(1)

    xrold=xr;

    % xrold is xr value of previous iteration

    xr=xu-(func(xu)*(xl-xu))/(func(xl)-func(xu));

    % formula to calculate xr

    iter=iter+1;

    % to display next iteration

    if xr~=0,ea=abs((xr-xrold)/xr)*100;end

    % if xr not equal to zero, calculate ea using the formula 

    test=func(xl,varargin{:})*func(xr,varargin{:});

    % evaluate the root by multiplying f(xl) with f(xr)

    if test<0

        xu=xr;

        % if f(xl)f(xr)<0,replace xu with xr

    elseif test>0

        xl=xr;

        % if f(xl)f(xr)>0,replace xl with xr

    else

        ea=0;

        % if f(xl)f(xr)=0,found root and stop iteration

    end

    if ea<=es|iter>=maxit,break,end

    % if ea less than or equal es, then stop iteration

end

root=xr;fx=func(xr,varargin{:});

% put the calculated xr value into the function to check the answer