Question

in matlab

-Consider the equation f(x) = x-2-sin x = 0 on the interval x E [0.1,4 π] Use a plot to approximately locate the roots of f. To which roots do the fol- owing initial guesses converge when using Function 4.3.1? Is the root obtained the one that is closest to that guess? )xo = 1.5, (b) x0 = 2, (c) x.-3.2, (d) xo = 4, (e) xo = 5, (f) xo = 27. Function 4.3.1 (newton) Newton's method for a scalar rootfinding problem. I function x = newton (f,dfdx.x1) 2 %NEWTON Newton's method for a scalar equation. 3 % Input: objective function derivative function initial root approximation 5 % dfdx 6%x1 7 % Output vector of root approximations (last one is best) 10 % Operating parameters 12 14 funto1=100*eps ; xto1= 100 * eps; maxiter=40; 15 dx inf; % for initial pass below k=1; 16 17 s while (abs (dx) > xtol) && Cabs (y) >funtol) && Ck

Answer 1

Matlab code for plotting the function clear all close all %function for which root have to find f-ê (x) x.*-2-sin(x); %function for which root have to find fprintf(function for which root have to find f(x) disp(f) %interval of x for which root have to find xx-linspace (1, 4*pi,500) %Plotting of the function plot (xx, yy) xlabel('x) ylabel(f(x) title( Plotting of x vs. f(x)') grid orn fprintf( From the plot we can say roots are obtained approximately at fprint f ("\t x=1.07 and function value is %f\n",f(1.07)) fprintf( "\t x-3.04 and function value is %f\n',f(3.04)) fprint f ("\t x-6.308 and function value is Af\n',f(6.308)) fprint f ("At x-9.414 and function value is %f\n', f (9.414)) %Finding root for initial guess using Newton method 4AII initial guess xx0-[1.5 2 3.2 4 5 2*pil; for i-l: length ( xx0) fprintf("\n\t For initial guess x0-% 2 .2 f \n" , x0) %root using Newton method [root] newton_method (E,x0,1000); fprint f("Root using Newton method is %f\n' ,root) end 용음음응용음음응용음음응용음음응용음음응용음음용용음음용용음음용용음음용용음응용욤음응용욤음음各욤음응용욤음응용욤음응용욤응응용욤

%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 function sinitial guess ] %Loop for all intial guesses ïimit for close it teration n-eps; %error for ¡= 1: maxit x2-double(xx-(fun(xx)-/g1(xx))); cc=abs (fun ( x2 )); err(i)-cci xx x2; if cc<-n %Newton Raphson Formula Error break end end if i==maxit warning( 'Maximum number of iteration reached.In') end root =xx ; end 8888옹옹옹옹옹옹88%옹88888888888888888888옹%88옹%88응%888888옹옹88888888888888 function for which root have to find f(x)e(x)x. *-2-sin(x) From the plot we can say roots are obtained approximately at x=1.07 and function value is -0.003762 x=3.04 and function value is 0.006788 x-6. 308 and function value is 0.000319 x-9.414 and function value is 0.000506 For initial guess x0-1.50 Root using Newton method is 1.068224 For initial guess x0=2.00 Root using Newton method is 6.308317 For initial guess x0-3.20 Root using Newton method is 3.032645 For initial guess x0-4.00 Root using Newton method is 3.032645 For initial guess x0#5.00 Warning: Maximum number of iteration reached. In Root using Newton method is 9.4134913 For initial guess x0=6.28 Root using Newton method is 6.308317

Plotting of x vs. (x) 1.5 0.5 -0.5 0 10 12 14 Published with MATLAB R2018a

%%Matlab code for plotting the function

clear all
close all

%function for which root have to find

f=@(x) x.^-2-sin(x);

%function for which root have to find

fprintf('function for which root have to find f(x)=')
disp(f)

%Interval of x for which root have to find

xx=linspace(1,4*pi,500);
yy=f(xx);

%Plotting of the function
plot(xx,yy)
xlabel('x')
ylabel('f(x)')
title('Plotting of x vs. f(x)')
grid on

  
fprintf('From the plot we can say roots are obtained approximately at\n')
fprintf('\t x=1.07 and function value is %f\n',f(1.07))
fprintf('\t x=3.04 and function value is %f\n',f(3.04))
fprintf('\t x=6.308 and function value is %f\n',f(6.308))
fprintf('\t x=9.414 and function value is %f\n',f(9.414))

%Finding root for initial guess using Newton method

%All initial guess

xx0=[1.5 2 3.2 4 5 2*pi];

for i=1: length(xx0)
  
    x0=xx0(i);
    fprintf('\n\t For initial guess x0=%2.2f\n',x0)
    %root using Newton method
    [root]=newton_method(f,x0,1000);
    fprintf('Root using Newton method is %f\n',root)
  
end

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%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
    if i==maxit
        warning('Maximum number of iteration reached.\n')
    end
    root=xx;
end

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