Question

************matlab code please*******************

4. Errors in finite difference approximations. Consider the second-order accurate central difference formula for computing the second derivative "h)) + 0(h2) Generate an appropriate error-vs-h log-log plot to verify the second order accuracy of the above expression for computing the second derivative of e*sin(3x) at xo0.4. Comment on how well the plot predicts the behavior of the error.

************matlab code please*******************

Answer 1

Matlab code for log log plot of 2nd derivative clear all close all %function for which derivative have to find f-e(x) exp (X ) . *sin(3*x); %displaying the function fprintf( The function for which derivative have to find f(x)n) disp(f) %function for exact derivative df-ext= %second @(x) 6*cos ( 3 *x ) * exp(x) -84sīn ( 3 *x ) * exp ( x ); derivative of the function fprintf(Second derivative of the function n') disp(df_ext) %All step sizes hh-log space(-1,-10, 100) ; %formula for central derivative f2_cen- (x,h)(f (x+h)-2 f (x)+f (x-h)./(h. 2) here x0=0.4 x0-0.4 ext_val-df ext (x0); fprint f ('Exact value for 2nd derivative at x=0.4 is %f.\n" ,ext-val); %loop for error for different h for i-1:length (hh) dff (i)-f2_cen (x0, h error ( 1 )-abs (ext-val-dff ( í ) ) ; h(i)); plotting error vs. step size loglog (hh,error) xlabel('step size (h) ylabel(Error') title( 'Error vs. step size for central difference) 88888888888888888888응 End of code钐88钐88888888888888 The function for which derivative have to find f(x) @ ( x) exp(x) . *sin( 3#x) Second derivative of the function @(x)6"cos (3tx)"ехр (x)-8tsin(3#x)"exp (x) Exact value for 2nd derivative at x=0.4 is -7.880066.

Error vs. step size for central difference 10 10-10 10 10 10 10 102 10° step size (h) Published with MATLAB R2018a

%%Matlab code for log log plot of 2nd derivative

clear all
close all

%function for which derivative have to find
f=@(x) exp(x).*sin(3*x);
%displaying the function
fprintf('The function for which derivative have to find f(x)\n')
disp(f)

%function for exact derivative
df_ext= @(x) 6*cos(3*x)*exp(x) - 8*sin(3*x)*exp(x);
%second derivative of the function
fprintf('Second derivative of the function \n')
disp(df_ext)

%All step sizes
hh=logspace(-1,-10,100);

%formula for central derivative
f2_cen=@(x,h) (f(x+h)-2*f(x)+f(x-h))./(h.^2);

%here x0=0.4
x0=0.4;
ext_val=df_ext(x0);

fprintf('Exact value for 2nd derivative at x=0.4 is %f.\n',ext_val);

%loop for error for different h
for i=1:length(hh)
  
    dff(i)=f2_cen(x0,hh(i));
    error(i)=abs(ext_val-dff(i));
  
end

%plotting error vs. step size
loglog(hh,error)
xlabel('step size (h)')
ylabel('Error')
title('Error vs. step size for central difference')

%%%%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%%%%