I don't understand why you can't invert a hessian matrix using gauss jordan method. Can't you integrate or differentiate an entire row (because they are linear operators) and then subtract/add/swap?
To clarify, I mean inverting the generic hessian formulation before it is applied to any function. In this way, I can get the hessian inverse so I can apply it to other functions. otherwise, I'd need to calculate hessian of a function and then invert, which takes a larger time complexity.