A practical advantage is that the UT can be easier to implement because it avoids the need to derive and implement a linearizing Jacobian matrix.
32.
It acts between tangent spaces, and is in coordinates represented by the Jacobian matrix of the " coordinate representation " of the function.
33.
AutoChem symbolically differentiates the time derivatives to give the Jacobian matrix, and symbolically differentiates the Jacobian matrix to give the Hessian matrix and the adjoint.
34.
AutoChem symbolically differentiates the time derivatives to give the Jacobian matrix, and symbolically differentiates the Jacobian matrix to give the Hessian matrix and the adjoint.
35.
If is a differentiable function, a " critical point " of is a point where the rank of the Jacobian matrix is not maximal.
36.
Just as for = 1, the case of the largest absolute value being 1 needs to be investigated further the Jacobian matrix test is inconclusive.
37.
The linearization of this map, i . e . the Jacobian matrix ~ J _ n describes the change of an infinitesimal perturbation h _ n.
38.
If a function is differentiable at, then all of the partial derivatives exist at, in which case the linear map is given by the Jacobian matrix.
39.
However, in order to use this class of methods, we would need to formulate a Jacobian matrix, and we would need to solve it often.
40.
Broyden suggests using the current estimate of the Jacobian matrix and improving upon it by taking the solution to the secant equation that is a minimal modification to:
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