/
_root.py
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/
_root.py
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"""
Unified interfaces to root finding algorithms.
Functions
---------
- root : find a root of a vector function.
"""
__all__ = ['root']
import numpy as np
from warnings import warn
from optimize import MemoizeJac, Result, _check_unknown_options
from minpack import _root_hybr, leastsq
import nonlin
def root(fun, x0, args=(), method='hybr', jac=None, tol=None, callback=None,
options=None):
"""
Find a root of a vector function.
.. versionadded:: 0.11.0
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'
- 'lm'
- 'broyden1'
- 'broyden2'
- 'anderson'
- 'linearmixing'
- 'diagbroyden'
- 'excitingmixing'
- 'krylov'
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
``show_options('root', method)`` for details.
Returns
-------
sol : Result
The solution represented as a ``Result`` 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
`Result` for a description of other attributes.
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]_.
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. See `nonlin` for details.
- 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.
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.
<http://www.siam.org/books/kelley/>
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])
"""
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 ('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 in ('broyden1', 'broyden2', 'anderson', 'linearmixing',
'diagbroyden', 'excitingmixing', 'krylov'):
if jac is not None:
warn('Method %s does not use the jacobian (jac).' % method,
RuntimeWarning)
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 _root_leastsq(func, 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):
_check_unknown_options(unknown_options)
x, cov_x, info, msg, ier = leastsq(func, 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 = Result(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(func, 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 == True:
def f(x):
return func(x, *args)[0]
else:
def f(x):
return func(x, *args)
else:
f = func
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 = Result(x=x)
sol.update(info)
return sol