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655 lines
26 KiB
Python
655 lines
26 KiB
Python
"""
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Unified interfaces to root finding algorithms.
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Functions
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---------
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- root : find a root of a vector function.
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"""
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__all__ = ['root']
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import numpy as np
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ROOT_METHODS = ['hybr', 'lm', 'broyden1', 'broyden2', 'anderson',
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'linearmixing', 'diagbroyden', 'excitingmixing', 'krylov',
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'df-sane']
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from warnings import warn
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from .optimize import MemoizeJac, OptimizeResult, _check_unknown_options
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from .minpack import _root_hybr, leastsq
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from ._spectral import _root_df_sane
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from . import nonlin
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def root(fun, x0, args=(), method='hybr', jac=None, tol=None, callback=None,
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options=None):
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"""
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Find a root of a vector function.
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Parameters
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----------
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fun : callable
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A vector function to find a root of.
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x0 : ndarray
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Initial guess.
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args : tuple, optional
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Extra arguments passed to the objective function and its Jacobian.
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method : str, optional
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Type of solver. Should be one of
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- 'hybr' :ref:`(see here) <optimize.root-hybr>`
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- 'lm' :ref:`(see here) <optimize.root-lm>`
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- 'broyden1' :ref:`(see here) <optimize.root-broyden1>`
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- 'broyden2' :ref:`(see here) <optimize.root-broyden2>`
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- 'anderson' :ref:`(see here) <optimize.root-anderson>`
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- 'linearmixing' :ref:`(see here) <optimize.root-linearmixing>`
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- 'diagbroyden' :ref:`(see here) <optimize.root-diagbroyden>`
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- 'excitingmixing' :ref:`(see here) <optimize.root-excitingmixing>`
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- 'krylov' :ref:`(see here) <optimize.root-krylov>`
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- 'df-sane' :ref:`(see here) <optimize.root-dfsane>`
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jac : bool or callable, optional
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If `jac` is a Boolean and is True, `fun` is assumed to return the
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value of Jacobian along with the objective function. If False, the
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Jacobian will be estimated numerically.
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`jac` can also be a callable returning the Jacobian of `fun`. In
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this case, it must accept the same arguments as `fun`.
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tol : float, optional
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Tolerance for termination. For detailed control, use solver-specific
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options.
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callback : function, optional
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Optional callback function. It is called on every iteration as
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``callback(x, f)`` where `x` is the current solution and `f`
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the corresponding residual. For all methods but 'hybr' and 'lm'.
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options : dict, optional
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A dictionary of solver options. E.g., `xtol` or `maxiter`, see
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:obj:`show_options()` for details.
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Returns
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-------
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sol : OptimizeResult
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The solution represented as a ``OptimizeResult`` object.
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Important attributes are: ``x`` the solution array, ``success`` a
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Boolean flag indicating if the algorithm exited successfully and
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``message`` which describes the cause of the termination. See
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`OptimizeResult` for a description of other attributes.
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See also
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--------
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show_options : Additional options accepted by the solvers
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Notes
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-----
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This section describes the available solvers that can be selected by the
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'method' parameter. The default method is *hybr*.
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Method *hybr* uses a modification of the Powell hybrid method as
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implemented in MINPACK [1]_.
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Method *lm* solves the system of nonlinear equations in a least squares
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sense using a modification of the Levenberg-Marquardt algorithm as
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implemented in MINPACK [1]_.
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Method *df-sane* is a derivative-free spectral method. [3]_
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Methods *broyden1*, *broyden2*, *anderson*, *linearmixing*,
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*diagbroyden*, *excitingmixing*, *krylov* are inexact Newton methods,
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with backtracking or full line searches [2]_. Each method corresponds
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to a particular Jacobian approximations. See `nonlin` for details.
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- Method *broyden1* uses Broyden's first Jacobian approximation, it is
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known as Broyden's good method.
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- Method *broyden2* uses Broyden's second Jacobian approximation, it
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is known as Broyden's bad method.
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- Method *anderson* uses (extended) Anderson mixing.
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- Method *Krylov* uses Krylov approximation for inverse Jacobian. It
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is suitable for large-scale problem.
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- Method *diagbroyden* uses diagonal Broyden Jacobian approximation.
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- Method *linearmixing* uses a scalar Jacobian approximation.
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- Method *excitingmixing* uses a tuned diagonal Jacobian
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approximation.
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.. warning::
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The algorithms implemented for methods *diagbroyden*,
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*linearmixing* and *excitingmixing* may be useful for specific
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problems, but whether they will work may depend strongly on the
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problem.
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.. versionadded:: 0.11.0
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References
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----------
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.. [1] More, Jorge J., Burton S. Garbow, and Kenneth E. Hillstrom.
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1980. User Guide for MINPACK-1.
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.. [2] C. T. Kelley. 1995. Iterative Methods for Linear and Nonlinear
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Equations. Society for Industrial and Applied Mathematics.
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<https://archive.siam.org/books/kelley/fr16/>
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.. [3] W. La Cruz, J.M. Martinez, M. Raydan. Math. Comp. 75, 1429 (2006).
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Examples
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--------
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The following functions define a system of nonlinear equations and its
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jacobian.
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>>> def fun(x):
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... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0,
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... 0.5 * (x[1] - x[0])**3 + x[1]]
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>>> def jac(x):
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... return np.array([[1 + 1.5 * (x[0] - x[1])**2,
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... -1.5 * (x[0] - x[1])**2],
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... [-1.5 * (x[1] - x[0])**2,
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... 1 + 1.5 * (x[1] - x[0])**2]])
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A solution can be obtained as follows.
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>>> from scipy import optimize
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>>> sol = optimize.root(fun, [0, 0], jac=jac, method='hybr')
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>>> sol.x
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array([ 0.8411639, 0.1588361])
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"""
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if not isinstance(args, tuple):
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args = (args,)
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meth = method.lower()
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if options is None:
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options = {}
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if callback is not None and meth in ('hybr', 'lm'):
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warn('Method %s does not accept callback.' % method,
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RuntimeWarning)
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# fun also returns the Jacobian
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if not callable(jac) and meth in ('hybr', 'lm'):
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if bool(jac):
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fun = MemoizeJac(fun)
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jac = fun.derivative
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else:
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jac = None
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# set default tolerances
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if tol is not None:
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options = dict(options)
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if meth in ('hybr', 'lm'):
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options.setdefault('xtol', tol)
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elif meth in ('df-sane',):
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options.setdefault('ftol', tol)
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elif meth in ('broyden1', 'broyden2', 'anderson', 'linearmixing',
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'diagbroyden', 'excitingmixing', 'krylov'):
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options.setdefault('xtol', tol)
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options.setdefault('xatol', np.inf)
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options.setdefault('ftol', np.inf)
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options.setdefault('fatol', np.inf)
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if meth == 'hybr':
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sol = _root_hybr(fun, x0, args=args, jac=jac, **options)
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elif meth == 'lm':
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sol = _root_leastsq(fun, x0, args=args, jac=jac, **options)
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elif meth == 'df-sane':
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_warn_jac_unused(jac, method)
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sol = _root_df_sane(fun, x0, args=args, callback=callback,
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**options)
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elif meth in ('broyden1', 'broyden2', 'anderson', 'linearmixing',
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'diagbroyden', 'excitingmixing', 'krylov'):
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_warn_jac_unused(jac, method)
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sol = _root_nonlin_solve(fun, x0, args=args, jac=jac,
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_method=meth, _callback=callback,
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**options)
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else:
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raise ValueError('Unknown solver %s' % method)
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return sol
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def _warn_jac_unused(jac, method):
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if jac is not None:
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warn('Method %s does not use the jacobian (jac).' % (method,),
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RuntimeWarning)
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def _root_leastsq(fun, x0, args=(), jac=None,
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col_deriv=0, xtol=1.49012e-08, ftol=1.49012e-08,
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gtol=0.0, maxiter=0, eps=0.0, factor=100, diag=None,
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**unknown_options):
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"""
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Solve for least squares with Levenberg-Marquardt
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Options
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-------
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col_deriv : bool
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non-zero to specify that the Jacobian function computes derivatives
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down the columns (faster, because there is no transpose operation).
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ftol : float
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Relative error desired in the sum of squares.
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xtol : float
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Relative error desired in the approximate solution.
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gtol : float
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Orthogonality desired between the function vector and the columns
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of the Jacobian.
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maxiter : int
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The maximum number of calls to the function. If zero, then
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100*(N+1) is the maximum where N is the number of elements in x0.
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epsfcn : float
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A suitable step length for the forward-difference approximation of
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the Jacobian (for Dfun=None). If epsfcn is less than the machine
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precision, it is assumed that the relative errors in the functions
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are of the order of the machine precision.
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factor : float
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A parameter determining the initial step bound
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(``factor * || diag * x||``). Should be in interval ``(0.1, 100)``.
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diag : sequence
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N positive entries that serve as a scale factors for the variables.
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"""
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_check_unknown_options(unknown_options)
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x, cov_x, info, msg, ier = leastsq(fun, x0, args=args, Dfun=jac,
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full_output=True,
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col_deriv=col_deriv, xtol=xtol,
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ftol=ftol, gtol=gtol,
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maxfev=maxiter, epsfcn=eps,
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factor=factor, diag=diag)
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sol = OptimizeResult(x=x, message=msg, status=ier,
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success=ier in (1, 2, 3, 4), cov_x=cov_x,
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fun=info.pop('fvec'))
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sol.update(info)
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return sol
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def _root_nonlin_solve(fun, x0, args=(), jac=None,
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_callback=None, _method=None,
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nit=None, disp=False, maxiter=None,
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ftol=None, fatol=None, xtol=None, xatol=None,
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tol_norm=None, line_search='armijo', jac_options=None,
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**unknown_options):
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_check_unknown_options(unknown_options)
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f_tol = fatol
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f_rtol = ftol
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x_tol = xatol
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x_rtol = xtol
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verbose = disp
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if jac_options is None:
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jac_options = dict()
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jacobian = {'broyden1': nonlin.BroydenFirst,
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'broyden2': nonlin.BroydenSecond,
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'anderson': nonlin.Anderson,
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'linearmixing': nonlin.LinearMixing,
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'diagbroyden': nonlin.DiagBroyden,
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'excitingmixing': nonlin.ExcitingMixing,
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'krylov': nonlin.KrylovJacobian
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}[_method]
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if args:
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if jac:
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def f(x):
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return fun(x, *args)[0]
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else:
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def f(x):
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return fun(x, *args)
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else:
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f = fun
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x, info = nonlin.nonlin_solve(f, x0, jacobian=jacobian(**jac_options),
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iter=nit, verbose=verbose,
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maxiter=maxiter, f_tol=f_tol,
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f_rtol=f_rtol, x_tol=x_tol,
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x_rtol=x_rtol, tol_norm=tol_norm,
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line_search=line_search,
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callback=_callback, full_output=True,
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raise_exception=False)
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sol = OptimizeResult(x=x)
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sol.update(info)
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return sol
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def _root_broyden1_doc():
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"""
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Options
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-------
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nit : int, optional
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Number of iterations to make. If omitted (default), make as many
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as required to meet tolerances.
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disp : bool, optional
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Print status to stdout on every iteration.
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maxiter : int, optional
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Maximum number of iterations to make. If more are needed to
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meet convergence, `NoConvergence` is raised.
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ftol : float, optional
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Relative tolerance for the residual. If omitted, not used.
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fatol : float, optional
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Absolute tolerance (in max-norm) for the residual.
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If omitted, default is 6e-6.
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xtol : float, optional
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Relative minimum step size. If omitted, not used.
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xatol : float, optional
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Absolute minimum step size, as determined from the Jacobian
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approximation. If the step size is smaller than this, optimization
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is terminated as successful. If omitted, not used.
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tol_norm : function(vector) -> scalar, optional
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Norm to use in convergence check. Default is the maximum norm.
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line_search : {None, 'armijo' (default), 'wolfe'}, optional
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Which type of a line search to use to determine the step size in
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the direction given by the Jacobian approximation. Defaults to
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'armijo'.
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jac_options : dict, optional
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Options for the respective Jacobian approximation.
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alpha : float, optional
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Initial guess for the Jacobian is (-1/alpha).
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reduction_method : str or tuple, optional
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Method used in ensuring that the rank of the Broyden
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matrix stays low. Can either be a string giving the
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name of the method, or a tuple of the form ``(method,
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param1, param2, ...)`` that gives the name of the
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method and values for additional parameters.
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Methods available:
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- ``restart``
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Drop all matrix columns. Has no
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extra parameters.
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- ``simple``
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Drop oldest matrix column. Has no
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extra parameters.
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- ``svd``
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Keep only the most significant SVD
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components.
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Extra parameters:
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- ``to_retain``
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Number of SVD components to
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retain when rank reduction is done.
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Default is ``max_rank - 2``.
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max_rank : int, optional
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Maximum rank for the Broyden matrix.
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Default is infinity (i.e., no rank reduction).
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"""
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pass
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def _root_broyden2_doc():
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"""
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Options
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-------
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nit : int, optional
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Number of iterations to make. If omitted (default), make as many
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as required to meet tolerances.
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disp : bool, optional
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Print status to stdout on every iteration.
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maxiter : int, optional
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Maximum number of iterations to make. If more are needed to
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meet convergence, `NoConvergence` is raised.
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ftol : float, optional
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Relative tolerance for the residual. If omitted, not used.
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fatol : float, optional
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Absolute tolerance (in max-norm) for the residual.
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If omitted, default is 6e-6.
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xtol : float, optional
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Relative minimum step size. If omitted, not used.
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xatol : float, optional
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Absolute minimum step size, as determined from the Jacobian
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approximation. If the step size is smaller than this, optimization
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is terminated as successful. If omitted, not used.
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tol_norm : function(vector) -> scalar, optional
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Norm to use in convergence check. Default is the maximum norm.
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line_search : {None, 'armijo' (default), 'wolfe'}, optional
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Which type of a line search to use to determine the step size in
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the direction given by the Jacobian approximation. Defaults to
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'armijo'.
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jac_options : dict, optional
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Options for the respective Jacobian approximation.
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alpha : float, optional
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Initial guess for the Jacobian is (-1/alpha).
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reduction_method : str or tuple, optional
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Method used in ensuring that the rank of the Broyden
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matrix stays low. Can either be a string giving the
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name of the method, or a tuple of the form ``(method,
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param1, param2, ...)`` that gives the name of the
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|
method and values for additional parameters.
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|
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Methods available:
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- ``restart``
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Drop all matrix columns. Has no
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extra parameters.
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- ``simple``
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Drop oldest matrix column. Has no
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extra parameters.
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- ``svd``
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Keep only the most significant SVD
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components.
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Extra parameters:
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- ``to_retain``
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Number of SVD components to
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retain when rank reduction is done.
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Default is ``max_rank - 2``.
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max_rank : int, optional
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Maximum rank for the Broyden matrix.
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Default is infinity (i.e., no rank reduction).
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"""
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pass
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def _root_anderson_doc():
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"""
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Options
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-------
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nit : int, optional
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Number of iterations to make. If omitted (default), make as many
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as required to meet tolerances.
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disp : bool, optional
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Print status to stdout on every iteration.
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maxiter : int, optional
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Maximum number of iterations to make. If more are needed to
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meet convergence, `NoConvergence` is raised.
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ftol : float, optional
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Relative tolerance for the residual. If omitted, not used.
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fatol : float, optional
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Absolute tolerance (in max-norm) for the residual.
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If omitted, default is 6e-6.
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xtol : float, optional
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Relative minimum step size. If omitted, not used.
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xatol : float, optional
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Absolute minimum step size, as determined from the Jacobian
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approximation. If the step size is smaller than this, optimization
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is terminated as successful. If omitted, not used.
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tol_norm : function(vector) -> scalar, optional
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Norm to use in convergence check. Default is the maximum norm.
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line_search : {None, 'armijo' (default), 'wolfe'}, optional
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Which type of a line search to use to determine the step size in
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the direction given by the Jacobian approximation. Defaults to
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'armijo'.
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jac_options : dict, optional
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Options for the respective Jacobian approximation.
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|
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alpha : float, optional
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Initial guess for the Jacobian is (-1/alpha).
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M : float, optional
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Number of previous vectors to retain. Defaults to 5.
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w0 : float, optional
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Regularization parameter for numerical stability.
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Compared to unity, good values of the order of 0.01.
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"""
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pass
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def _root_linearmixing_doc():
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"""
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Options
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-------
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nit : int, optional
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Number of iterations to make. If omitted (default), make as many
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as required to meet tolerances.
|
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disp : bool, optional
|
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Print status to stdout on every iteration.
|
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maxiter : int, optional
|
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Maximum number of iterations to make. If more are needed to
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meet convergence, ``NoConvergence`` is raised.
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ftol : float, optional
|
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Relative tolerance for the residual. If omitted, not used.
|
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fatol : float, optional
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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
|