Skip to content


Subversion checkout URL

You can clone with
Download ZIP
Fetching contributors…

Cannot retrieve contributors at this time

250 lines (213 sloc) 9.168 kB
Unified interfaces to root finding algorithms.
- root : find a root of a vector function.
from __future__ import division, print_function, absolute_import
__all__ = ['root']
import numpy as np
from scipy.lib.six import callable
from warnings import warn
from .optimize import MemoizeJac, Result, _check_unknown_options
from .minpack import _root_hybr, leastsq
from . import nonlin
def root(fun, x0, args=(), method='hybr', jac=None, tol=None, callback=None,
Find a root of a vector function.
.. versionadded:: 0.11.0
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
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.
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.
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
.. 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
.. [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.
The following functions define a system of nonlinear equations and its
>>> 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,
# fun also returns the jacobian
if not callable(jac) and meth in ('hybr', 'lm'):
if bool(jac):
fun = MemoizeJac(fun)
jac = fun.derivative
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,
sol = _root_nonlin_solve(fun, x0, args=args, jac=jac,
_method=meth, _callback=callback,
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,
x, cov_x, info, msg, ier = leastsq(func, x0, args=args, Dfun=jac,
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,
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,
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
if args:
if jac == True:
def f(x):
return func(x, *args)[0]
def f(x):
return func(x, *args)
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,
callback=_callback, full_output=True,
sol = Result(x=x)
return sol
Jump to Line
Something went wrong with that request. Please try again.