Hi, my prob is find the most accurate 1st derivative of f(x)=exp(cos(x)) at x=1, with h=0.5,0.25,..,2^(-16). I calculate the 1st der. using 1st order central diff formula & trying to improve the accuracy using Richardson extrap. got incorrect result

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Umesh Prajapati 2016-3-9

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https://ww2.mathworks.cn/matlabcentral/answers/272403-hi-my-prob-is-find-the-most-accurate-1st-derivative-of-f-x-exp-cos-x-at-x-1-with-h-0-5-0-25

编辑： John D'Errico 2016-3-9

clear; clc; format shortG
f = @(x) exp(cos(x)); df = @(x) -exp(cos(x))*sin(x);
x = 1; Truef1 = df(x); A = [];
h = 1/2;
while (h >= 2^-16)
    f1 = (f(x+h)-f(x-h))/(2*h);
    A = [A; h f1];
    h = h/2;
end
D(:,1) = A(:,2); E(:,1) = abs((Truef1-D(:,1))/Truef1);
for i = 1:16
    for j = 2:i
        D(i,j) = ((4^j)*D(i,j-1)-D(i-1,j-1))/(4^j-1);
        E(i,j) = abs((Truef1-D(i,j))/Truef1);
    end
end
disp(D); disp(E);
Order = (log(E(3,2))-log(E(2,2)))/(log(A(3,1))-log(A(2,1)))
loglog(A(:,1),E,'-');