Suppose you have a record of coordinates X(t) for every s units of time from 0 to time T.
Suppose in addition you have more data Q(T) that is supposed to be output from some complex numerical computation Q(X) on the X(t) data you have. The dimensions of Q(T) and X(T) are the same.
How does one calculate all partial derivatives of the X coordinates with respect to all the Q coordinates?
A single variable is trivial. Having access to Q(X) is also easy.
The multivariable problem without access to Q(X) seems like it should be simple too, but I don't know under what circumstances (what distributions of the data, etc.) this problem would have a solution, and what algorithms would apply in those circumstances.
It seems like we are we need to solve for a transformation matrix J(T) such that dX(T)=J(T)d*Q(T).
As an example, I could try the algorithm with X(T) being randomly generated rectangular coordinates and Q(T) being spherical coordinates generated from that. I know what the Jacobian should be for this case, and I can compare against a generic algorithm's results.