Question

A. Implement the False-Position (FP) method for solving nonlinear equations in one dimension. The program should be started from a script M-file. -Prompt the user to enter lower and upper guesses for the root. .Use an error tolerance of 107. Allow at most 1000 iterations The code should be fully commented and clear " 1. Use your FP and NR codes and appropriate initial guesses to find the root of the following equation between 0 and 5. Plot the root and the approximate error as a function of the iteration number for both methods, compare the performances of the methods and discuss the results o 3x3 +4x2 - 5x-9 0 2. a) Use your NR code to find the positive root of the equation given below using the following points as initial guesses: xo = 4, 0 and-1, Plot the root and the approximate error as a function of the iteration number. Report the estimated root for each case, describe the differences in the behavior depending on the initial guess and discuss the results o 8000x - 500x3+ 4000 0 b) Verify your NR code using the graphical approach.

Answer 1

The Matlab code is given below. There is newtonRaphson method , false position method and another file to tes them.

If we compare the graphs of errors and roots against the iteration number we find that Newton method converges to the final root very fast as compared to final position

*************************false_position.m*************************

function [ E,R,r ] = false_position( f,a,b,max_iters )
%This is the function for calculation the root
%It takes in the polynomial f and calculates final root r
%Also it calculates the intermediate roots R
%and the errors in between

False position False position 15 1.5 10 0.5 50 teration Newton raphson 100 50 teration Newton raphson 100 6 1.6 4 1.5 1.4 1.3 4 4 teration teration

tol=1e-7;
c=a;

E=[];
R=[];
i=0;
while i<max_iters
%here this is the false position formula
c = (a f(b) - b f(a))/ (f(b) - f(a));
err=abs(f(c)-0);
%add error to vector E
E=[E err];
R=[R c]
%This loop will continue to refine values of c
%until err gets less than tol i.e. 10^-7
if err<tol
break;
end
%update c as per false position rule
if (f(c) * f(a)) < 0
b = c ;
else
a = c ;
end
i=i+1;
end

r=c;
end

***********************************NewtonRaphson.m*************************

function [E,R,r] = NewtonRaphson( F,dFdx,xi,imax )
%Newton raphson method also needs derivative of the function
%xi is the initial guess
%imax is the maximum iterations
%F is the function
%dFdx is the derivative
X=xi;
n=0;
tol=1e-7;
%Fill Errors in E and all intermediate roots in R
E=[];
R=[];
while n<imax
n=n+1;
%below is the newton raphson formula
% to update the approximate root
X=X-F(X)/dFdx(X);
R=[R X];
%calculate the error
%F(x) is the calculated value at x
% while as 0 is the true value
err=abs(F(X)-0);
E=[E err]
if err<tol
break
end

end

r=X;
end

****************************TestBoth.m******************************

%This script file tests the false position and Newton raphson
%functions

%create the polynomial as anonymous function as
f=@(x)3*x^3+4*x^2-5*x-9;

%derivative of above function
dfdx=@(x)9*x^2+8*x-5;
lower_bound=input('Enter lower bound for False position:');
upper_bound=input('Enter upper bound for False position:');
initial_guess=input('Enter initial guess for Netwon Raphson:');
max_iters=1000;

%gather the roots and errors from false position
[E,R,r]=false_position(f,lower_bound,upper_bound,max_iters);
len=length(E);
i=1:1:len;
%gather the roots and errors from Newtonraphson
[E2,R2,r2]=NewtonRaphson(f,dfdx,initial_guess,max_iters);
len2=length(E2);
j=1:1:len2;

%plot errors and roots against iteration for both methods
subplot(221)
plot(i,E);
title('False position')
xlabel('íteration')
ylabel('Error')

subplot(222)
plot(i,R);
title('False position')
xlabel('íteration')
ylabel('Root')

subplot(223)
plot(j,E2);
title('Newton raphson')
xlabel('íteration')
ylabel('Error')

subplot(224)
plot(j,R2);
title('Newton raphson')
xlabel('íteration')
ylabel('Root')

fprintf('Root by false position=%f\n',r);
fprintf('Root by Newton Raphson=%f',r2);