Question

3. Write a code to find 3 roots of the function f(x) 2r3-4x2 -22x +24 for the interval I-5, 5] considering the following methods a) Bisection Method b) Newton's Method Hint: Plot a graph of f(x) and determine proper intervals and initial guesses for a) and b), respectively. 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 a graph of f(x) and determine proper intervals and initial guesses for a) and b), respectively.

Answer 1

Web Browser - Blsection Nr 3roots Blsection Nr 3roots ← → 台 Location file ///C:/Users/sona /Documents MATLAB/html Bisection Nr 3roots.html f = (x) 2*x. ^ 3-4*x.^2-22 *x df-@(x) 6#x.^2- e*x-22; xlinspace (-5, 5) + 24; Plotting Function and zero line to see 3 roots plot(x,(x)) xlabel( ylabel(" grid on yline (0) % plotting zero line to identify roots 388888 Bisection Method 88388888888 % For interval (-4,-2) For Interval (0,2) x2 b1 Bisection(,0,2); % Tor Interval (3,5) x3 bis Bisection (E,3,5): 8888888 Newton' Method 88888888 Eor atarting value {-4) For starting value {0) % For starting value (3) tprintf("\n 3 rooca by Newton ' e Method : .31 \t % .31 \t .31",x1 ITR, x2 11R, x3_11R): %Function for bisection method function xr Bisection (f,xl,xu) f 1 function x1 , xu are initial Gueaae3 Initia1121ng Error while (error>-tol) New Guess Calculation Checking Condition for gue53 replacement e1ae end erro abs ( (xz) % Defining error end end Function For uR function x2 = NRC, dr, x1) % f is function % df is derivative function % xl : Initial Gue 55 Initializing Error t Setting Tolerance rror1: tol 1e-: while (error -tol) Calculation of next gue BB Defining Error errorab((x2)) x1 = x2; % Replacing Guesses end end ubliahed with MATLABO R20186

100 50 -50 -100 -150 200 -250 -5 4 -321 02345

Script TXT:

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