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 at parse time via the
   ``max_deriv_order`` option to the model factory.
#. 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 four types:

* ``symbolic``
* ``numerical``
* ``automatic``
* ``matlab``


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

The complicated linear problem is solved using a solver with the
syntax::

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

Inputs:

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

Outputs:

* ``X`` -- the final solution.
* ``flag`` -- ``0`` if the problem solves; ``1`` if the maximum
  number of iterations is reached; another number otherwise.
* ``relres`` -- the relative residual ``norm(b - A*x) /
  norm(b)``.
* ``iter`` -- the iteration number at which the final solution
  was computed.
* ``resvec`` -- vector of residual norms at each half iteration,
  including the first ``norm(b - A*x0)``.

Change the default with the option ``solve_linsyst_user_algo``.
Some of the algorithms usable in MATLAB:

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