Question

Please use Python

Problem 2: 10 pts - Here is a cubic polynomial with three closely spaced real roots: p(x) = 580x4-2320x3-1 160x2 + 6960-1740 What are the exact roots of p? . Plot p(x) for -2 x 3 4. And plot the location of the four roots on the graph. Starting with xo2, what does Newton's method do? . Starting with 0.3 and xi0.9, what does the secant method do? Starting with the interval [0.5, 2.9, what does bisection do?

Answer 1

Matlab code for finding root using Newton, Secant and Bisection method clear all close all Function for which root have to find fun-e (x) 580.*x.*4-2320. *x. *3-1160.*x. *2+6960.*x-1740 %plotting of the function xx-linspace (-2,4,1001); yy-fun (xx); plot (xx, yy) xlabel(x) ylabel'Y) title( f (x) vs x plot) %displaying the function fprintf( For the function\n) disp (fun) %Finding roots p = [580-2320-1160 6960-1740); r = roots(p); fprintf( The roots are occured atn) disp(r) hold on plot(r,fun(r), 'r*) Root using Newton method X0-2; %Initial guess maxit-1000; %maximum iteration root]-newton_method (fun, x0, maxit); fprintf("Root using Newton method for initial guess %f is %2.15f. n',x0,root); Root using Secant method :0-0 . 3 ; x1-0 . 9 ; %Initial guess maxit-1000; %maximum iteration [root]-secant method fun, x0, x1,maxit); fprint f ("Root using Secant method for initial guess[%f,%f] n',x0,x1, root); IS %2.15f. Root using Bisection method x0=0 . 5 ; x1-2 . 9 ; %Initial guess maxit-1000; %maximum iteration root]-bisection_method (fun, x0, xl,maxit); fprint f ("Root using Bisection method for initial guess[%f,%f] is 2.15f.\n', x0,x1,root); 용음음응용음음응용음음응용음음응용음음응용음음용용음음용용음음용용음음용용음음용욤음음용욤음음各욤음응용욤음응용욤음응용욤응응용욤응응

%Matlab function for Bisection Method function [root]-bisection _method (fun, x0, xl,maxit) if fun (x0)<=0 x0 x1; xl-t; end %f(x1) should be positive (XO) should be negative k-10 count-0 while k>eps count-count+l; xx (count) (x0+x1)/2; mm-double (fun(xx (count))) if mm>-0 :0#xx( count) ; else x1=xx( count) ; end err count)-abs (fun (x1)); k-abs (fun (x1)) if count-maxit break end root-xx(end); end 8888888888888888888888888888888888888888응%8888888888888888888888% %Matlab function for Newton Method function [root ]=newton-method (fun,x0,maxit ) syms x g1(x) =d if f (fu n , x ); %1st Derivative of this xx-x0; %Loop for all intial guesses function sinitial guess] n-eps; %error limit for close itteration for i-1: maxit x2-double (xx-(fun(xx)-/gl (xx))); cc=abs ( fun ( x2 ) ) ; %Newton Raphson Formula Error xx-x2; if cc<-n break end end root xx; end 용음음응용음음응용음음응용음음응용음음용용음음용용음음응용음음용용음응용용음응용욤음응용욤응응용욤응응용욤응응용욤응응욤욤응응욤욤응응 %Matlab function for Secant Method function [root ]=secantmethod (fun,x0 , x 1 , maxit) -

f(x1) should be positive %f(x0) should be negative k-10 count 0; while k>eps count-count+1; xx-double(xl-(fun(xl) *abs ( (x1-x0)/(fun (x1)-fun (x0)))) x0=x1; xl-xx; k-abs (fun (xx)); if count-maxit break end root-xx; end 88 88888888 End of Code 8888888888888888 For the function (x)580. *x. *4-2320. *x. *3-1160. *x. *2+6960. *x-1740 The roots are occured at 3. 73205080756888 -1. 73205080756888 1.73205080756888 0.267949192431123 Root using Newton method for initial guess 2.000000 is 1.732050807568878. Root using Secant method for initial guess [0.300000,0.900000] is 0.267949192431123. Root using Bisection method for initial guess[ 0.500000,2.900000] is 1.732050807568878.

f(x) vs x plot 8000 6000 4000 2000 2000 -6000 8000 Published with MATLAB R2018a

%%Matlab code for finding root using Newton, Secant and Bisection method
clear all
close all

%Function for which root have to find
fun=@(x) 580.*x.^4-2320.*x.^3-1160.*x.^2+6960.*x-1740;

%plotting of the function
xx=linspace(-2,4,1001);
yy=fun(xx);
plot(xx,yy)
xlabel('x')
ylabel('y')
title('f(x) vs x plot')

%displaying the function
fprintf('For the function\n')
disp(fun)

%Finding roots
p = [580 -2320 -1160 6960 -1740];
r = roots(p);
fprintf('The roots are occured at\n')
disp(r)
hold on
plot(r,fun(r),'r*')

%Root using Newton method
x0=2; %Initial guess
maxit=1000; %maximum iteration
[root]=newton_method(fun,x0,maxit);
fprintf('Root using Newton method for initial guess %f is %2.15f.\n',x0,root);

%Root using Secant method
x0=0.3; x1=0.9; %Initial guess
maxit=1000; %maximum iteration
[root]=secant_method(fun,x0,x1,maxit);
fprintf('Root using Secant method for initial guess[%f,%f] is %2.15f.\n',x0,x1,root);

%Root using Bisection method
x0=0.5; x1=2.9; %Initial guess
maxit=1000; %maximum iteration
[root]=bisection_method(fun,x0,x1,maxit);
fprintf('Root using Bisection method for initial guess[%f,%f] is %2.15f.\n',x0,x1,root);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Matlab function for Bisection Method
function [root]=bisection_method(fun,x0,x1,maxit)
if fun(x0)<=0
    t=x0;
    x0=x1;
    x1=t;
end
%f(x1) should be positive
%f(x0) should be negative
k=10; count=0;
while k>eps
    count=count+1;
    xx(count)=(x0+x1)/2;
    mm=double(fun(xx(count)));
    if mm>=0
        x0=xx(count);
    else
        x1=xx(count);
    end
    err(count)=abs(fun(x1));
    k=abs(fun(x1));
    if count>=maxit
        break
    end
end
root=xx(end);
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Matlab function for Newton Method
function [root]=newton_method(fun,x0,maxit)
syms x
g1(x) =diff(fun,x);   %1st Derivative of this function
xx=x0;            %initial guess]
%Loop for all intial guesses
    n=eps; %error limit for close itteration
    for i=1:maxit
        x2=double(xx-(fun(xx)./g1(xx))); %Newton Raphson Formula
        cc=abs(fun(x2));                 %Error
        err(i)=cc;
        xx=x2;
        if cc<=n
            break
        end
      
    end
    root=xx;
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Matlab function for Secant Method
function [root]=secant_method(fun,x0,x1,maxit)
%f(x1) should be positive
%f(x0) should be negative
k=10; count=0;
while k>eps
    count=count+1;
    xx=double(x1-(fun(x1)*abs((x1-x0)/(fun(x1)-fun(x0)))));
    x0=x1;
    x1=xx;
    k=abs(fun(xx));
    if count>=maxit
            break
    end
end
root=xx;
end
  
%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%