_root.py

"""
Unified interfaces to root finding algorithms.

Functions
---------
- root : find a root of a vector function.
"""
__all__ = ['root']

import numpy as np

ROOT_METHODS = ['hybr', 'lm', 'broyden1', 'broyden2', 'anderson',
                'linearmixing', 'diagbroyden', 'excitingmixing', 'krylov',
                'df-sane']

from warnings import warn

from ._optimize import MemoizeJac, OptimizeResult, _check_unknown_options
from ._minpack_py import _root_hybr, leastsq
from ._spectral import _root_df_sane
from . import _nonlin as nonlin


def root(fun, x0, args=(), method='hybr', jac=None, tol=None, callback=None,
         options=None):
    r"""
    Find a root of a vector function.

    Parameters
    ----------
    fun : callable
        A vector function to find a root of.
    x0 : ndarray
        Initial guess.
    args : tuple, optional
        Extra arguments passed to the objective function and its Jacobian.
    method : str, optional
        Type of solver. Should be one of

            - 'hybr'             :ref:`(see here) <optimize.root-hybr>`
            - 'lm'               :ref:`(see here) <optimize.root-lm>`
            - 'broyden1'         :ref:`(see here) <optimize.root-broyden1>`
            - 'broyden2'         :ref:`(see here) <optimize.root-broyden2>`
            - 'anderson'         :ref:`(see here) <optimize.root-anderson>`
            - 'linearmixing'     :ref:`(see here) <optimize.root-linearmixing>`
            - 'diagbroyden'      :ref:`(see here) <optimize.root-diagbroyden>`
            - 'excitingmixing'   :ref:`(see here) <optimize.root-excitingmixing>`
            - 'krylov'           :ref:`(see here) <optimize.root-krylov>`
            - 'df-sane'          :ref:`(see here) <optimize.root-dfsane>`

    jac : bool or callable, optional
        If `jac` is a Boolean and is True, `fun` is assumed to return the
        value of Jacobian along with the objective function. If False, the
        Jacobian will be estimated numerically.
        `jac` can also be a callable returning the Jacobian of `fun`. In
        this case, it must accept the same arguments as `fun`.
    tol : float, optional
        Tolerance for termination. For detailed control, use solver-specific
        options.
    callback : function, optional
        Optional callback function. It is called on every iteration as
        ``callback(x, f)`` where `x` is the current solution and `f`
        the corresponding residual. For all methods but 'hybr' and 'lm'.
    options : dict, optional
        A dictionary of solver options. E.g., `xtol` or `maxiter`, see
        :obj:`show_options()` for details.

    Returns
    -------
    sol : OptimizeResult
        The solution represented as a ``OptimizeResult`` object.
        Important attributes are: ``x`` the solution array, ``success`` a
        Boolean flag indicating if the algorithm exited successfully and
        ``message`` which describes the cause of the termination. See
        `OptimizeResult` for a description of other attributes.

    See also
    --------
    show_options : Additional options accepted by the solvers

    Notes
    -----
    This section describes the available solvers that can be selected by the
    'method' parameter. The default method is *hybr*.

    Method *hybr* uses a modification of the Powell hybrid method as
    implemented in MINPACK [1]_.

    Method *lm* solves the system of nonlinear equations in a least squares
    sense using a modification of the Levenberg-Marquardt algorithm as
    implemented in MINPACK [1]_.

    Method *df-sane* is a derivative-free spectral method. [3]_

    Methods *broyden1*, *broyden2*, *anderson*, *linearmixing*,
    *diagbroyden*, *excitingmixing*, *krylov* are inexact Newton methods,
    with backtracking or full line searches [2]_. Each method corresponds
    to a particular Jacobian approximations.

    - Method *broyden1* uses Broyden's first Jacobian approximation, it is
      known as Broyden's good method.
    - Method *broyden2* uses Broyden's second Jacobian approximation, it
      is known as Broyden's bad method.
    - Method *anderson* uses (extended) Anderson mixing.
    - Method *Krylov* uses Krylov approximation for inverse Jacobian. It
      is suitable for large-scale problem.
    - Method *diagbroyden* uses diagonal Broyden Jacobian approximation.
    - Method *linearmixing* uses a scalar Jacobian approximation.
    - Method *excitingmixing* uses a tuned diagonal Jacobian
      approximation.

    .. warning::

        The algorithms implemented for methods *diagbroyden*,
        *linearmixing* and *excitingmixing* may be useful for specific
        problems, but whether they will work may depend strongly on the
        problem.

    .. versionadded:: 0.11.0

    References
    ----------
    .. [1] More, Jorge J., Burton S. Garbow, and Kenneth E. Hillstrom.
       1980. User Guide for MINPACK-1.
    .. [2] C. T. Kelley. 1995. Iterative Methods for Linear and Nonlinear
       Equations. Society for Industrial and Applied Mathematics.
       <https://archive.siam.org/books/kelley/fr16/>
    .. [3] W. La Cruz, J.M. Martinez, M. Raydan. Math. Comp. 75, 1429 (2006).

    Examples
    --------
    The following functions define a system of nonlinear equations and its
    jacobian.

    >>> def fun(x):
    ...     return [x[0]  + 0.5 * (x[0] - x[1])**3 - 1.0,
    ...             0.5 * (x[1] - x[0])**3 + x[1]]

    >>> def jac(x):
    ...     return np.array([[1 + 1.5 * (x[0] - x[1])**2,
    ...                       -1.5 * (x[0] - x[1])**2],
    ...                      [-1.5 * (x[1] - x[0])**2,
    ...                       1 + 1.5 * (x[1] - x[0])**2]])

    A solution can be obtained as follows.

    >>> from scipy import optimize
    >>> sol = optimize.root(fun, [0, 0], jac=jac, method='hybr')
    >>> sol.x
    array([ 0.8411639,  0.1588361])

    **Large problem**

    Suppose that we needed to solve the following integrodifferential
    equation on the square :math:`[0,1]\times[0,1]`:

    .. math::

       \nabla^2 P = 10 \left(\int_0^1\int_0^1\cosh(P)\,dx\,dy\right)^2

    with :math:`P(x,1) = 1` and :math:`P=0` elsewhere on the boundary of
    the square.

    The solution can be found using the ``method='krylov'`` solver:

    >>> from scipy import optimize
    >>> # parameters
    >>> nx, ny = 75, 75
    >>> hx, hy = 1./(nx-1), 1./(ny-1)

    >>> P_left, P_right = 0, 0
    >>> P_top, P_bottom = 1, 0

    >>> def residual(P):
    ...    d2x = np.zeros_like(P)
    ...    d2y = np.zeros_like(P)
    ...
    ...    d2x[1:-1] = (P[2:]   - 2*P[1:-1] + P[:-2]) / hx/hx
    ...    d2x[0]    = (P[1]    - 2*P[0]    + P_left)/hx/hx
    ...    d2x[-1]   = (P_right - 2*P[-1]   + P[-2])/hx/hx
    ...
    ...    d2y[:,1:-1] = (P[:,2:] - 2*P[:,1:-1] + P[:,:-2])/hy/hy
    ...    d2y[:,0]    = (P[:,1]  - 2*P[:,0]    + P_bottom)/hy/hy
    ...    d2y[:,-1]   = (P_top   - 2*P[:,-1]   + P[:,-2])/hy/hy
    ...
    ...    return d2x + d2y - 10*np.cosh(P).mean()**2

    >>> guess = np.zeros((nx, ny), float)
    >>> sol = optimize.root(residual, guess, method='krylov')
    >>> print('Residual: %g' % abs(residual(sol.x)).max())
    Residual: 5.7972e-06  # may vary

    >>> import matplotlib.pyplot as plt
    >>> x, y = np.mgrid[0:1:(nx*1j), 0:1:(ny*1j)]
    >>> plt.pcolormesh(x, y, sol.x, shading='gouraud')
    >>> plt.colorbar()
    >>> plt.show()

    """
    if not isinstance(args, tuple):
        args = (args,)

    meth = method.lower()
    if options is None:
        options = {}

    if callback is not None and meth in ('hybr', 'lm'):
        warn('Method %s does not accept callback.' % method,
             RuntimeWarning)

    # fun also returns the Jacobian
    if not callable(jac) and meth in ('hybr', 'lm'):
        if bool(jac):
            fun = MemoizeJac(fun)
            jac = fun.derivative
        else:
            jac = None

    # set default tolerances
    if tol is not None:
        options = dict(options)
        if meth in ('hybr', 'lm'):
            options.setdefault('xtol', tol)
        elif meth in ('df-sane',):
            options.setdefault('ftol', tol)
        elif meth in ('broyden1', 'broyden2', 'anderson', 'linearmixing',
                      'diagbroyden', 'excitingmixing', 'krylov'):
            options.setdefault('xtol', tol)
            options.setdefault('xatol', np.inf)
            options.setdefault('ftol', np.inf)
            options.setdefault('fatol', np.inf)

    if meth == 'hybr':
        sol = _root_hybr(fun, x0, args=args, jac=jac, **options)
    elif meth == 'lm':
        sol = _root_leastsq(fun, x0, args=args, jac=jac, **options)
    elif meth == 'df-sane':
        _warn_jac_unused(jac, method)
        sol = _root_df_sane(fun, x0, args=args, callback=callback,
                            **options)
    elif meth in ('broyden1', 'broyden2', 'anderson', 'linearmixing',
                  'diagbroyden', 'excitingmixing', 'krylov'):
        _warn_jac_unused(jac, method)
        sol = _root_nonlin_solve(fun, x0, args=args, jac=jac,
                                 _method=meth, _callback=callback,
                                 **options)
    else:
        raise ValueError('Unknown solver %s' % method)

    return sol


def _warn_jac_unused(jac, method):
    if jac is not None:
        warn('Method %s does not use the jacobian (jac).' % (method,),
             RuntimeWarning)


def _root_leastsq(fun, x0, args=(), jac=None,
                  col_deriv=0, xtol=1.49012e-08, ftol=1.49012e-08,
                  gtol=0.0, maxiter=0, eps=0.0, factor=100, diag=None,
                  **unknown_options):
    """
    Solve for least squares with Levenberg-Marquardt

    Options
    -------
    col_deriv : bool
        non-zero to specify that the Jacobian function computes derivatives
        down the columns (faster, because there is no transpose operation).
    ftol : float
        Relative error desired in the sum of squares.
    xtol : float
        Relative error desired in the approximate solution.
    gtol : float
        Orthogonality desired between the function vector and the columns
        of the Jacobian.
    maxiter : int
        The maximum number of calls to the function. If zero, then
        100*(N+1) is the maximum where N is the number of elements in x0.
    epsfcn : float
        A suitable step length for the forward-difference approximation of
        the Jacobian (for Dfun=None). If epsfcn is less than the machine
        precision, it is assumed that the relative errors in the functions
        are of the order of the machine precision.
    factor : float
        A parameter determining the initial step bound
        (``factor * || diag * x||``). Should be in interval ``(0.1, 100)``.
    diag : sequence
        N positive entries that serve as a scale factors for the variables.
    """

    _check_unknown_options(unknown_options)
    x, cov_x, info, msg, ier = leastsq(fun, x0, args=args, Dfun=jac,
                                       full_output=True,
                                       col_deriv=col_deriv, xtol=xtol,
                                       ftol=ftol, gtol=gtol,
                                       maxfev=maxiter, epsfcn=eps,
                                       factor=factor, diag=diag)
    sol = OptimizeResult(x=x, message=msg, status=ier,
                         success=ier in (1, 2, 3, 4), cov_x=cov_x,
                         fun=info.pop('fvec'))
    sol.update(info)
    return sol


def _root_nonlin_solve(fun, x0, args=(), jac=None,
                       _callback=None, _method=None,
                       nit=None, disp=False, maxiter=None,
                       ftol=None, fatol=None, xtol=None, xatol=None,
                       tol_norm=None, line_search='armijo', jac_options=None,
                       **unknown_options):
    _check_unknown_options(unknown_options)

    f_tol = fatol
    f_rtol = ftol
    x_tol = xatol
    x_rtol = xtol
    verbose = disp
    if jac_options is None:
        jac_options = dict()

    jacobian = {'broyden1': nonlin.BroydenFirst,
                'broyden2': nonlin.BroydenSecond,
                'anderson': nonlin.Anderson,
                'linearmixing': nonlin.LinearMixing,
                'diagbroyden': nonlin.DiagBroyden,
                'excitingmixing': nonlin.ExcitingMixing,
                'krylov': nonlin.KrylovJacobian
                }[_method]

    if args:
        if jac:
            def f(x):
                return fun(x, *args)[0]
        else:
            def f(x):
                return fun(x, *args)
    else:
        f = fun

    x, info = nonlin.nonlin_solve(f, x0, jacobian=jacobian(**jac_options),
                                  iter=nit, verbose=verbose,
                                  maxiter=maxiter, f_tol=f_tol,
                                  f_rtol=f_rtol, x_tol=x_tol,
                                  x_rtol=x_rtol, tol_norm=tol_norm,
                                  line_search=line_search,
                                  callback=_callback, full_output=True,
                                  raise_exception=False)
    sol = OptimizeResult(x=x)
    sol.update(info)
    return sol

def _root_broyden1_doc():
    """
    Options
    -------
    nit : int, optional
        Number of iterations to make. If omitted (default), make as many
        as required to meet tolerances.
    disp : bool, optional
        Print status to stdout on every iteration.
    maxiter : int, optional
        Maximum number of iterations to make. If more are needed to
        meet convergence, `NoConvergence` is raised.
    ftol : float, optional
        Relative tolerance for the residual. If omitted, not used.
    fatol : float, optional
        Absolute tolerance (in max-norm) for the residual.
        If omitted, default is 6e-6.
    xtol : float, optional
        Relative minimum step size. If omitted, not used.
    xatol : float, optional
        Absolute minimum step size, as determined from the Jacobian
        approximation. If the step size is smaller than this, optimization
        is terminated as successful. If omitted, not used.
    tol_norm : function(vector) -> scalar, optional
        Norm to use in convergence check. Default is the maximum norm.
    line_search : {None, 'armijo' (default), 'wolfe'}, optional
        Which type of a line search to use to determine the step size in
        the direction given by the Jacobian approximation. Defaults to
        'armijo'.
    jac_options : dict, optional
        Options for the respective Jacobian approximation.
            alpha : float, optional
                Initial guess for the Jacobian is (-1/alpha).
            reduction_method : str or tuple, optional
                Method used in ensuring that the rank of the Broyden
                matrix stays low. Can either be a string giving the
                name of the method, or a tuple of the form ``(method,
                param1, param2, ...)`` that gives the name of the
                method and values for additional parameters.

                Methods available:

                    - ``restart``
                        Drop all matrix columns. Has no
                        extra parameters.
                    - ``simple``
                        Drop oldest matrix column. Has no
                        extra parameters.
                    - ``svd``
                        Keep only the most significant SVD
                        components.

                        Extra parameters:

                            - ``to_retain``
                                Number of SVD components to
                                retain when rank reduction is done.
                                Default is ``max_rank - 2``.
            max_rank : int, optional
                Maximum rank for the Broyden matrix.
                Default is infinity (i.e., no rank reduction).

    Examples
    --------
    >>> def func(x):
    ...     return np.cos(x) + x[::-1] - [1, 2, 3, 4]
    ...
    >>> from scipy import optimize
    >>> res = optimize.root(func, [1, 1, 1, 1], method='broyden1', tol=1e-14)
    >>> x = res.x
    >>> x
    array([4.04674914, 3.91158389, 2.71791677, 1.61756251])
    >>> np.cos(x) + x[::-1]
    array([1., 2., 3., 4.])

    """
    pass

def _root_broyden2_doc():
    """
    Options
    -------
    nit : int, optional
        Number of iterations to make. If omitted (default), make as many
        as required to meet tolerances.
    disp : bool, optional
        Print status to stdout on every iteration.
    maxiter : int, optional
        Maximum number of iterations to make. If more are needed to
        meet convergence, `NoConvergence` is raised.
    ftol : float, optional
        Relative tolerance for the residual. If omitted, not used.
    fatol : float, optional
        Absolute tolerance (in max-norm) for the residual.
        If omitted, default is 6e-6.
    xtol : float, optional
        Relative minimum step size. If omitted, not used.
    xatol : float, optional
        Absolute minimum step size, as determined from the Jacobian
        approximation. If the step size is smaller than this, optimization
        is terminated as successful. If omitted, not used.
    tol_norm : function(vector) -> scalar, optional
        Norm to use in convergence check. Default is the maximum norm.
    line_search : {None, 'armijo' (default), 'wolfe'}, optional
        Which type of a line search to use to determine the step size in
        the direction given by the Jacobian approximation. Defaults to
        'armijo'.
    jac_options : dict, optional
        Options for the respective Jacobian approximation.

        alpha : float, optional
            Initial guess for the Jacobian is (-1/alpha).
        reduction_method : str or tuple, optional
            Method used in ensuring that the rank of the Broyden
            matrix stays low. Can either be a string giving the
            name of the method, or a tuple of the form ``(method,
            param1, param2, ...)`` that gives the name of the
            method and values for additional parameters.

            Methods available:

                - ``restart``
                    Drop all matrix columns. Has no
                    extra parameters.
                - ``simple``
                    Drop oldest matrix column. Has no
                    extra parameters.
                - ``svd``
                    Keep only the most significant SVD
                    components.

                    Extra parameters:

                        - ``to_retain``
                            Number of SVD components to
                            retain when rank reduction is done.
                            Default is ``max_rank - 2``.
        max_rank : int, optional
            Maximum rank for the Broyden matrix.
            Default is infinity (i.e., no rank reduction).
    """
    pass

def _root_anderson_doc():
    """
    Options
    -------
    nit : int, optional
        Number of iterations to make. If omitted (default), make as many
        as required to meet tolerances.
    disp : bool, optional
        Print status to stdout on every iteration.
    maxiter : int, optional
        Maximum number of iterations to make. If more are needed to
        meet convergence, `NoConvergence` is raised.
    ftol : float, optional
        Relative tolerance for the residual. If omitted, not used.
    fatol : float, optional
        Absolute tolerance (in max-norm) for the residual.
        If omitted, default is 6e-6.
    xtol : float, optional
        Relative minimum step size. If omitted, not used.
    xatol : float, optional
        Absolute minimum step size, as determined from the Jacobian
        approximation. If the step size is smaller than this, optimization
        is terminated as successful. If omitted, not used.
    tol_norm : function(vector) -> scalar, optional
        Norm to use in convergence check. Default is the maximum norm.
    line_search : {None, 'armijo' (default), 'wolfe'}, optional
        Which type of a line search to use to determine the step size in
        the direction given by the Jacobian approximation. Defaults to
        'armijo'.
    jac_options : dict, optional
        Options for the respective Jacobian approximation.

        alpha : float, optional
            Initial guess for the Jacobian is (-1/alpha).
        M : float, optional
            Number of previous vectors to retain. Defaults to 5.
        w0 : float, optional
            Regularization parameter for numerical stability.
            Compared to unity, good values of the order of 0.01.
    """
    pass

def _root_linearmixing_doc():
    """
    Options
    -------
    nit : int, optional
        Number of iterations to make. If omitted (default), make as many
        as required to meet tolerances.
    disp : bool, optional
        Print status to stdout on every iteration.
    maxiter : int, optional
        Maximum number of iterations to make. If more are needed to
        meet convergence, ``NoConvergence`` is raised.
    ftol : float, optional
        Relative tolerance for the residual. If omitted, not used.
    fatol : float, optional
        Absolute tolerance (in max-norm) for the residual.
        If omitted, default is 6e-6.
    xtol : float, optional
        Relative minimum step size. If omitted, not used.
    xatol : float, optional
        Absolute minimum step size, as determined from the Jacobian
        approximation. If the step size is smaller than this, optimization
        is terminated as successful. If omitted, not used.
    tol_norm : function(vector) -> scalar, optional
        Norm to use in convergence check. Default is the maximum norm.
    line_search : {None, 'armijo' (default), 'wolfe'}, optional
        Which type of a line search to use to determine the step size in
        the direction given by the Jacobian approximation. Defaults to
        'armijo'.
    jac_options : dict, optional
        Options for the respective Jacobian approximation.

        alpha : float, optional
            initial guess for the jacobian is (-1/alpha).
    """
    pass

def _root_diagbroyden_doc():
    """
    Options
    -------
    nit : int, optional
        Number of iterations to make. If omitted (default), make as many
        as required to meet tolerances.
    disp : bool, optional
        Print status to stdout on every iteration.
    maxiter : int, optional
        Maximum number of iterations to make. If more are needed to
        meet convergence, `NoConvergence` is raised.
    ftol : float, optional
        Relative tolerance for the residual. If omitted, not used.
    fatol : float, optional
        Absolute tolerance (in max-norm) for the residual.
        If omitted, default is 6e-6.
    xtol : float, optional
        Relative minimum step size. If omitted, not used.
    xatol : float, optional
        Absolute minimum step size, as determined from the Jacobian
        approximation. If the step size is smaller than this, optimization
        is terminated as successful. If omitted, not used.
    tol_norm : function(vector) -> scalar, optional
        Norm to use in convergence check. Default is the maximum norm.
    line_search : {None, 'armijo' (default), 'wolfe'}, optional
        Which type of a line search to use to determine the step size in
        the direction given by the Jacobian approximation. Defaults to
        'armijo'.
    jac_options : dict, optional
        Options for the respective Jacobian approximation.

        alpha : float, optional
            initial guess for the jacobian is (-1/alpha).
    """
    pass

def _root_excitingmixing_doc():
    """
    Options
    -------
    nit : int, optional
        Number of iterations to make. If omitted (default), make as many
        as required to meet tolerances.
    disp : bool, optional
        Print status to stdout on every iteration.
    maxiter : int, optional
        Maximum number of iterations to make. If more are needed to
        meet convergence, `NoConvergence` is raised.
    ftol : float, optional
        Relative tolerance for the residual. If omitted, not used.
    fatol : float, optional
        Absolute tolerance (in max-norm) for the residual.
        If omitted, default is 6e-6.
    xtol : float, optional
        Relative minimum step size. If omitted, not used.
    xatol : float, optional
        Absolute minimum step size, as determined from the Jacobian
        approximation. If the step size is smaller than this, optimization
        is terminated as successful. If omitted, not used.
    tol_norm : function(vector) -> scalar, optional
        Norm to use in convergence check. Default is the maximum norm.
    line_search : {None, 'armijo' (default), 'wolfe'}, optional
        Which type of a line search to use to determine the step size in
        the direction given by the Jacobian approximation. Defaults to
        'armijo'.
    jac_options : dict, optional
        Options for the respective Jacobian approximation.

        alpha : float, optional
            Initial Jacobian approximation is (-1/alpha).
        alphamax : float, optional
            The entries of the diagonal Jacobian are kept in the range
            ``[alpha, alphamax]``.
    """
    pass

def _root_krylov_doc():
    """
    Options
    -------
    nit : int, optional
        Number of iterations to make. If omitted (default), make as many
        as required to meet tolerances.
    disp : bool, optional
        Print status to stdout on every iteration.
    maxiter : int, optional
        Maximum number of iterations to make. If more are needed to
        meet convergence, `NoConvergence` is raised.
    ftol : float, optional
        Relative tolerance for the residual. If omitted, not used.
    fatol : float, optional
        Absolute tolerance (in max-norm) for the residual.
        If omitted, default is 6e-6.
    xtol : float, optional
        Relative minimum step size. If omitted, not used.
    xatol : float, optional
        Absolute minimum step size, as determined from the Jacobian
        approximation. If the step size is smaller than this, optimization
        is terminated as successful. If omitted, not used.
    tol_norm : function(vector) -> scalar, optional
        Norm to use in convergence check. Default is the maximum norm.
    line_search : {None, 'armijo' (default), 'wolfe'}, optional
        Which type of a line search to use to determine the step size in
        the direction given by the Jacobian approximation. Defaults to
        'armijo'.
    jac_options : dict, optional
        Options for the respective Jacobian approximation.

        rdiff : float, optional
            Relative step size to use in numerical differentiation.
        method : {'lgmres', 'gmres', 'bicgstab', 'cgs', 'minres'} or function
            Krylov method to use to approximate the Jacobian.
            Can be a string, or a function implementing the same
            interface as the iterative solvers in
            `scipy.sparse.linalg`.

            The default is `scipy.sparse.linalg.lgmres`.
        inner_M : LinearOperator or InverseJacobian
            Preconditioner for the inner Krylov iteration.
            Note that you can use also inverse Jacobians as (adaptive)
            preconditioners. For example,

            >>> jac = BroydenFirst()
            >>> kjac = KrylovJacobian(inner_M=jac.inverse).

            If the preconditioner has a method named 'update', it will
            be called as ``update(x, f)`` after each nonlinear step,
            with ``x`` giving the current point, and ``f`` the current
            function value.
        inner_tol, inner_maxiter, ...
            Parameters to pass on to the "inner" Krylov solver.
            See `scipy.sparse.linalg.gmres` for details.
        outer_k : int, optional
            Size of the subspace kept across LGMRES nonlinear
            iterations.

            See `scipy.sparse.linalg.lgmres` for details.
    """
    pass