Higher-order perturbation
==========================

You can solve for higher-order perturbation if you are in one of the following cases:

	#. you are using *symbolic* derivatives and those derivatives have been precomputed while "rising" the model, through option
	   **max_deriv_order**
	#. you are using *automatic* derivatives
	#. you are using *numerical* derivatives
	#. you are using *matlab* derivatives

And you are able to solve the generalized system of Sylvester equations.	

Choosing the derivatives
----------------------------

To change the type of derivatives, use the option **solve_derivatives_type**. RISE implements the following 4 types of derivatives:

- symbolic
- numerical
- automatic
- matlab

Solving the system of generalized Sylvester equations
---------------------------------------------------------

The complicated linear problem is solved using a solver that can be called using the syntax ::

	[X,flag,relres,iter,resvec] = linsolver(A,b,...
        TolFun,Maxiter,...
        M1,M2,x0,vargs{:});

Where the input arguments are given by

	- *A* : either a square matrix of a function handle that return the result of the operation :math:`A\times x`
	  without explicitly forming and storing :math:`A`.
	- *b* : the right-hand side of the operation :math:`A\times x = b`.
	- *TolFun* : the tolerance criterion 
	- *Maxiter* : the maximum number of iterations 
	- *M1* = [] : not set 
	- *M2* = [] : not set
	- *x0* : initial guess for the solution 
	- *vargs* : potential additional input arguments for the solver utilized.

And the output arguments are :	

	- *X* : the final solution 
	- *flag* : = 0 if the problem solves, =1 if the maximum number of iterations is reached, = some other number otherwise 
	- *relres* : the relative residual norm(b-A*x)/norm(b)
	- *iter* : the iteration number iter at the final solution X was computed
	- *resvec* : a vector of the residual norm at each half iteration, including the first residual norm(b-A*x0)

Change the default setting using option **solve_linsyst_user_algo**.

These are some of the algorithms that can be used in Matlab. 

	- *'tfqmr'* (default)
	- *'bicg'*
	- *'bicgstab'*
	- *'bicgstabl'*
	- *'cgs'*
	- *'gmres'*