Homework 3(Bonus): To approximate f' (x) numerically by generating a table of approximations D(j,k) for k s j including final answer in terms of D(n, n). The first column in the matrix could be given as follows:

DCi.07=5 (x+2-9h) -11x-2-91)

2-1+1h

and the elements in the row j could be referred as follows:

DU: 10) = DU: + -1) + DU. k-1) -DU - 1, k-1 for 15k5;

Below program implements basically the Richardson extrapolation. The expression for the elements in row j could be algebraically equivalent to Richardson Extrapolation formula given accordingly:

4*Dk-1 (h) - Dx-1 (2h)

f' (x0) =

+ 0(424+2)

4 -

function [D, err,relerr,n]= diffext(f;,x, delta,toler)

%Input

fis the function input as string f

%

delta is the tolerance for the error

%

toler is the tolerance for the relative error

%Output

D is the matrix of approximate derivatives err is the error bound

%

relerr is the relative error bound

%

n is the coordinate of the best approximation

err=1; relerr=1;

h=1; j=1;

D(1,1)=(feval(f,x+h)-feval(f,x-h))/(2*h);

while relerrtoler & errdelta & i<12

h=h/2;

D(j+1,1)=(feval(f,x+h)-feval(f;,x-h))/(2*h);

for k=1:i

D(j+1,k+1)=D(j+1,k)+(D(j+1,k) - D(j,k))/(4^k)-1);

end

err=abs(D(j+1, j+1) - D(j,j)) ;

relerr=2err/(abs(D(j+1,j+1))+abs(D(j,j))+eps);

]=j+1;

end [n,n]=size(D);

You may use above program to approximate the derivative of function

#(x)=60215-32x3+23385-47x2-77;x=1/V3.

Carry 13 decimal places. You may change the initial values of err, relerr, and h.