Is gradient vector of pole zero carry usful information?
Does gradient vector of pole zero carry useful information?


Yes. The partial derivatives can be used in creating iterative search algorithms for fitting IIR filters to arbitrary targets. Examples of algorithms that use the derivatives are Steepest Descent or Conjugate Gradient. 

It's not a trivial process though and there are lot of details to be worked out. 


Poles, zeros and transfer function targets are all complex, so you either need to work with complex derivatives are use a suitable real-valued representation (real/imag, magnitude/phase, etc.). 
It's important to formulate a suitable error criteria that properly
reflects the requirements and trade-offs for your specific
application,
The search algorithms often get stuck in local minima
Log spaced frequency grids may have to be pre-warped to straighten out the error surface a bit. 
Many search algorithm have "heuristic" parameters that are hard to get right