Question

. (25 points) The recurrence relation for the Newton's Raphson method is a)0.1.2 f(r.) F(z.) The derivative of the function can be approximately evaluated using finite-difference method. Consider the Forward and Centered finite-difference formulas Forward Finite-Difference Centered Finite-Difference 2h It is worthwhile to mention that modified secant method was derived based on the forward finite- difference formula. Develop a MATLAB functions that has the following syntax function [root,fx,ea,iter]-modnetraph (func,x0,h,es,maxit,sethod, varargin) % modnevtraph: root location zeroes of nonlinear equation f (x) 0 % [root , fx ea, iter) "nodnevtraph(func .xo,b,es,saxit,method,p1.p2,..): % uses modified Newton's Raphson method to find the root of func % inputs: func name of function x0 initial guess h fixed increment to approxinate the derivative ea " desired relative error (default . 0.0001%) % maxit maximum allowable iterations (default-50) method - A string input (F or C) to specify either Forvard or pi,p2, root real root Centered finite-difference formulas (defaultc) aneters used by function % outputs: ea-approximate relative error (%) ter number of iterations 10-6 Test your developed function the following test functions using h Hint: Use avitch command. 0.05 and es

Answer 1

function [root,fx,ea,iter] = modnewtraph(func,x0,h,es,maxit ,method,p)
switch nargin %this is used to set the default values so it required at leat three input , otherwise it will show error
case 3
es = 0.0001 ;
maxit = 50;
method = "C";
p = 0 ; % additional parameter for function
case 4
maxit = 50;
method = "C";
p = 0 ;
case 5
  
method = "C";
p = 0 ;
case 6
p = 0;
  
end
are = 10 ; %%initialise approximate relative error as 10
X0 = x0;
itr = 0; %% initialise iteration variable to keep track of no. of iteration
while(are> es) %%% check if the current ARE is greater then the desired error
itr = itr + 1 ; % increment the itration value to keep track of iteration
if(method =="F")
F_dotX = (func(X0 + h,p) - func(X0,p))/h; %%for Forward finite difference
else
F_dotX = (func(X0 + h,p) - func(X0-h,p))/(2*h); %calculate the derivative of function according to the method,for %centered finite difference
end;
  
  
X1 = X0 - func(X0,p)/F_dotX; %apply Newton_Raphson rule , RECURRENCE formula
  
are = abs(X1-X0)*100/abs(X0) ;%Calculate approximate relative error
  
X0 = X1;
if(maxit ==itr) %%while the iteration is equal to maximum iterarion break the loop
break;
end;
end
fx = func(X1,p);
root =X1;
ea = are;
iter=itr;
  
end

%%%PLEASE UPVOTE !!!!! THANKS