Complex Step Derivative of 3D rotation in exponential coordinates at u = [0 0 0] not working?

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Adri 2025-3-18

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https://ww2.mathworks.cn/matlabcentral/answers/2175340-complex-step-derivative-of-3d-rotation-in-exponential-coordinates-at-u-0-0-0-not-working

评论： Adri 2025-3-19

Applying complex step differentiation to the matrix exponential works fine, except at u = [0 0 0]. Here, the exponential coordinates have a removable singularity. Why does the complex differentiation fail only at this point? Can you think of a complex differentiable matrix exponential that returns the correct derivative at u = [0 0 0]? Similar to the Complex-step-compatible atan2()?

Thank you for your time.

u0 = [0;0;0]; % The problematic point
% Central finite-difference
sh = 1e-5;
for ii = 1:3
    dh = zeros(3,1);
    dh(ii) = sh;
    JFD(:,ii) = (vec(rexpm(u0+dh)) - vec(rexpm(u0-dh))) / (2*sh);
end
% Complex step differentiation
sh = 1e-16;
for ii = 1:3
    dh = zeros(3,1);
    dh(ii) = sh*1i;
    JCS(:,ii) = imag(vec(rexpm(u0+dh))) / (sh);
end
JFD - JCS % should be close to 0!
% Functions
function R = rexpm(u)
Su = [ 0   -u(3) u(2)
       u(3) 0   -u(1)
      -u(2) u(1)   0];
R = expm(Su);
end
function y = vec(x)
y = x(:);
end