Develop a Matlab code to calculate the numerical differentiation at ALL the foll

Question

Develop a Matlab code to calculate the numerical differentiation at ALL the following data points, and produce the following plot. To ensure high accuracy calculations, the equations given below MUST BE USED. First Derivative Error f'(x_i) = f(x_i + 1) - f(x_i)/h f'(x_i) = -f(x_i +2) + 4f(x_i + 1) - 3f(x_j)/2h Error O(h) O(h^2) Backward difference First Derivative f'(x_i) = f(x_i) - f(x_i - 1)/h f'(x_i) = 3f(x_i) - 4f(x_i - 1) + f(x_i - 2)/2h Error O(h) O(h^2) Centered difference First Derivative f'(x_i) = f(x_i + 1) - f(x_i - 1)/2h f'(x_i) = -f(x_i + 2) + 8f(x_i + 1) - 8x_i - 1) + f(x_i - 2)/12 h Error O(h^2) O(h^4)

Explanation / Answer

Numbering equations according to order they are defined in.

Following is the required matlab code :

x = [0,1,2,3,4,5,6,7,8];
fx =[0.5, 4, 5.1, 9.8, 10.3, 9.4, 8.7, 6.6, 5.4 ];

%Equation numbered according to the given order in question
%Equation 1
dfx = zeros(8,1);
for i=1:8
    dfx(i,1) = fx(i+1) - fx(i);
end
figure;
plot( [0,1,2,3,4,5,6,7,8 ], fx, 'r', [0,1,2,3,4,5,6,7 ], dfx , 'black' );
ylabel( 'Amplitude' );
xlabel( 'x' );
legend('f(x)','df(x)');
title('Equation 1');

%Equation 2
dfx = zeros(7,1);
for i=1:7
    dfx(i,1) = (-fx(i+2) + 4*fx(i+1) - 3*fx(i) )/2.0;
end
figure;
plot( [0,1,2,3,4,5,6,7,8 ], fx, 'r', [0,1,2,3,4,5,6 ], dfx , 'black' );
ylabel( 'Amplitude' );
xlabel( 'x' );
legend('f(x)','df(x)');
title('Equation 2');

%Backward Difference
%Equation 3
dfx = zeros(8,1);
for i=2:9
    dfx(i-1,1) = fx(i) - fx(i-1);
end
figure;
plot( [0,1,2,3,4,5,6,7,8 ], fx, 'r', [ 1,2,3,4,5,6,7,8 ], dfx , 'black' );
ylabel( 'Amplitude' );
xlabel( 'x' );
legend('f(x)','df(x)');
title('Equation 3');

%Equation 4
dfx = zeros(7,1);
for i=3:9
    dfx(i-2,1) = ( fx(i-2) - 4*fx(i-1) + 3*fx(i) )/2.0;
end
figure;
plot( [ 0,1,2,3,4,5,6,7,8 ], fx, 'r', [ 2,3,4,5,6,7,8 ], dfx , 'black' );
ylabel( 'Amplitude' );
xlabel( 'x' );
legend('f(x)','df(x)');
title('Equation 4');

%Centerd Difference
%Equation 5
dfx = zeros(7,1);
for i=2:8
    dfx(i-1,1) = (fx(i+1) - fx(i-1))/2;
end
figure;
plot( [0,1,2,3,4,5,6,7,8 ], fx, 'r', [ 1,2,3,4,5,6,7 ], dfx , 'black' );
ylabel( 'Amplitude' );
xlabel( 'x' );
legend('f(x)','df(x)');
title('Equation 5');

%Equation 6
dfx = zeros(5,1);
for i=3:7
    dfx(i-2,1) = ( -fx(i+2) + 8*fx(i+1) - 8*fx(i-1) + fx(i-2) )/12.0;
end
figure;
plot( [ 0,1,2,3,4,5,6,7,8 ], fx, 'r', [ 2,3,4,5,6 ], dfx , 'black' );
ylabel( 'Amplitude' );
xlabel( 'x' );
legend('f(x)','df(x)');
title('Equation 6');

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