Question

3. Write a code to find 3 roots of the function f(x)-2x3-4x2-22x+24 for the interval -5, 5] considering the following methods a) Bisection Method b) Newton's Method Hint: Plot agraph of f(x) and determine proper intervals and initial guesses for a) and b), respectively

Answer 1

ANSWER:-

MAT LAB CODE:-

. Execute demo-m STDIN 1 f @(x) 2%."3-44x."2-22*x 24; + = 2 df (x) 6*x.2-8*x - 22; 31inspace(-5,5) 4. 5 % plotting Function aท d zero Line to see 3 roots 6 plot (x,f (x)) 7 xlabel("x"); ylabel("f(x)"); 8 grid on g yline(0); % plotting zero Line to identify roots 18 11 %%R XXXX Bisection Method %aaaa:xxxxxx 12 % For interval (-4,-2) 13 x1bis -Bisection(f,-4,-2) 14 % For Interval (0,2) 5 x2 bis Bisection(f,,2) 16 % For Interval (3,5) 17 x3_bis Bisection(f,3,5) 18 19 XXXXXXXX% Newton's Method XXXXXXXXXXX 20 % For starting value (-4) 21 x1_NR NR(f,df, -4); 22 % For starting value (6) 23 x2-NR= NR(f, df ,e) ; 24 % For starting value (3) 25 x3_NR NR(f,df,3); 26 27 fprintf("\n 3 roots by Bisection Method : %.3f \t %.3f \t %.3f",x1-bis,x2-bis,x3-bis); 28 fprintf("\n 3 roots by Newton's Method : %.3f \t %.3f \t %.3f",x1-NR , x2,NR,x3-NR); 39 31 %Function for bisection method

32 function xrBisection(f,xl,xu) 33 % fis function 34 % XL ,xu are initial Guesses 35 error - 1; % Initializing Error 6 tol-10-6; 37 while(error)-tol) 38 xr = (x1+xu)/2; % New Guess calculation 39 if f(x1) *f(xr) < θ % Checking Condition for guess replacement 40 xu Xr; 41 else 42 xl -xr; 43 end 44 45 end 46 end 47 48 % Function For NR 49 function x2 NR(,df,x1) se % f is function 51 % df is derivative function 52 %x1: Initial Guess 53 error - 1; % Initializing Error 54 tol- le-6; % Setting Tolerance 55 while (error - tol) 56 x2 -x1- 57 error - abs(f(x2)); % Defining Error 58 x1 -x2 ; % RepLacing Guesses 59 end 60 end error-abs(f(xr)); % Defining error (f(x1)/df(x1)); % calculation of next guess

OUT PUT SCREEN SHOT:-

I.lı Result

3 roots by Bisection Method: -3.000 .000 4.000 f 3 roots by Newton's Method -3.000 1.000 4.000 1.000 4.000>>

5 4 3 2 1 0 2 3 5 50 0 5 00 50 1 3

MAT LAB CODE:-

f = @(x) 2*x.^3 - 4*x.^2 - 22*x + 24;
df = @(x) 6*x.^2 - 8*x - 22;
x = linspace(-5,5);

% Plotting Function and zero line to see 3 roots
plot(x,f(x))
xlabel("x"); ylabel("f(x)");
grid on
yline(0); % plotting zero line to identify roots

%%%%%%%%% Bisection Method %%%%%%%%%%%
% For interval (-4,-2)
x1_bis = Bisection(f,-4,-2);
% For Interval (0,2)
x2_bis = Bisection(f,0,2);
% For Interval (3,5)
x3_bis = Bisection(f,3,5);

%%%%%%%%% Newton's Method %%%%%%%%%%%
% For starting value (-4)
x1_NR = NR(f,df,-4);
% For starting value (0)
x2_NR = NR(f,df,0);
% For starting value (3)
x3_NR = NR(f,df,3);

fprintf("\n 3 roots by Bisection Method : %.3f \t %.3f \t %.3f",x1_bis,x2_bis,x3_bis);
fprintf("\n 3 roots by Newton's Method : %.3f \t %.3f \t %.3f",x1_NR,x2_NR,x3_NR);

%Function for bisection method
function xr = Bisection(f,xl,xu)
% f is function
% xl ,xu are initial Guesses
error = 1; % Initializing Error
tol = 10^-6;
while(error>=tol)
xr = (xl+xu)/2; % New Guess Calculation
if f(xl)*f(xr) < 0 % Checking Condition for guess replacement
xu = xr;
else
xl = xr;
end
error = abs(f(xr)); % Defining error
end
end

% Function For NR
function x2 = NR(f,df,x1)
% f is function
% df is derivative function
% x1: Initial Guess
error = 1; % Initializing Error
tol = 1e-6; % Setting Tolerance
while (error >= tol)
x2 = x1 - (f(x1)/df(x1)); % Calculation of next guess
error = abs(f(x2)); % Defining Error
x1 = x2; % Replacing Guesses
end
end