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_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
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
def _root_hybr(func, x0, args=(), jac=None,
col_deriv=0, xtol=1.49012e-08, maxfev=0, band=None, eps=0.0,
factor=100, diag=None, full_output=0, **unknown_options):
"""
Find the roots of a multivariate function using MINPACK's hybrd and
hybrj routines (modified Powell method).
Options for the hybrd algorithm are:
col_deriv : bool
Specify whether the Jacobian function computes derivatives down
the columns (faster, because there is no transpose operation).
xtol : float
The calculation will terminate if the relative error between two
consecutive iterates is at most `xtol`.
maxfev : 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`.
band : tuple
If set to a two-sequence containing the number of sub- and
super-diagonals within the band of the Jacobi matrix, the
Jacobi matrix is considered banded (only for ``fprime=None``).
eps : float
A suitable step length for the forward-difference
approximation of the Jacobian (for ``fprime=None``). If
`eps` 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 the interval
``(0.1, 100)``.
diag : sequence
N positive entries that serve as a scale factors for the
variables.
This function is called by the `root` function with `method=hybr`. It
is not supposed to be called directly.
"""
_check_unknown_options(unknown_options)
epsfcn = eps
x0 = array(x0, ndmin=1)
n = len(x0)
if type(args) != type(()):
args = (args,)
_check_func('fsolve', 'func', func, x0, args, n, (n,))
Dfun = jac
if Dfun is None:
if band is None:
ml, mu = -10,-10
else:
ml, mu = band[:2]
if (maxfev == 0):
maxfev = 200*(n + 1)
retval = _minpack._hybrd(func, x0, args, 1, xtol, maxfev,
ml, mu, epsfcn, factor, diag)
else:
_check_func('fsolve', 'fprime', Dfun, x0, args, n, (n,n))
if (maxfev == 0):
maxfev = 100*(n + 1)
retval = _minpack._hybrj(func, Dfun, x0, args, 1,
col_deriv, xtol, maxfev, factor,diag)
x, status = retval[0], retval[-1]
errors = {0:["Improper input parameters were entered.",TypeError],
1:["The solution converged.", None],
2:["The number of calls to function has "
"reached maxfev = %d." % maxfev, ValueError],
3:["xtol=%f is too small, no further improvement "
"in the approximate\n solution "
"is possible." % xtol, ValueError],
4:["The iteration is not making good progress, as measured "
"by the \n improvement from the last five "
"Jacobian evaluations.", ValueError],
5:["The iteration is not making good progress, "
"as measured by the \n improvement from the last "
"ten iterations.", ValueError],
'unknown': ["An error occurred.", TypeError]}
if status != 1 and not full_output:
if status in [2,3,4,5]:
msg = errors[status][0]
warnings.warn(msg, RuntimeWarning)
else:
try:
raise errors[status][1](errors[status][0])
except KeyError:
raise errors['unknown'][1](errors['unknown'][0])
info = retval[1]
info['fun'] = info.pop('fvec')
sol = Result(x=x, success=(status==1), status=status)
sol.update(info)
try:
sol['message'] = errors[status][0]
except KeyError:
info['message'] = errors['unknown'][0]
return sol
def _root_hybr(func,
x0,
args=(),
jac=None,
col_deriv=0,
xtol=1.49012e-08,
maxfev=0,
band=None,
eps=0.0,
factor=100,
diag=None,
full_output=0,
**unknown_options):
"""
Find the roots of a multivariate function using MINPACK's hybrd and
hybrj routines (modified Powell method).
Options for the hybrd algorithm are:
col_deriv : bool
Specify whether the Jacobian function computes derivatives down
the columns (faster, because there is no transpose operation).
xtol : float
The calculation will terminate if the relative error between two
consecutive iterates is at most `xtol`.
maxfev : 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`.
band : tuple
If set to a two-sequence containing the number of sub- and
super-diagonals within the band of the Jacobi matrix, the
Jacobi matrix is considered banded (only for ``fprime=None``).
eps : float
A suitable step length for the forward-difference
approximation of the Jacobian (for ``fprime=None``). If
`eps` 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 the interval
``(0.1, 100)``.
diag : sequence
N positive entries that serve as a scale factors for the
variables.
This function is called by the `root` function with `method=hybr`. It
is not supposed to be called directly.
"""
_check_unknown_options(unknown_options)
epsfcn = eps
x0 = array(x0, ndmin=1)
n = len(x0)
if type(args) != type(()):
args = (args, )
_check_func('fsolve', 'func', func, x0, args, n, (n, ))
Dfun = jac
if Dfun is None:
if band is None:
ml, mu = -10, -10
else:
ml, mu = band[:2]
if (maxfev == 0):
maxfev = 200 * (n + 1)
retval = _minpack._hybrd(func, x0, args, 1, xtol, maxfev, ml, mu,
epsfcn, factor, diag)
else:
_check_func('fsolve', 'fprime', Dfun, x0, args, n, (n, n))
if (maxfev == 0):
maxfev = 100 * (n + 1)
retval = _minpack._hybrj(func, Dfun, x0, args, 1, col_deriv, xtol,
maxfev, factor, diag)
x, status = retval[0], retval[-1]
errors = {
0: ["Improper input parameters were entered.", TypeError],
1: ["The solution converged.", None],
2: [
"The number of calls to function has "
"reached maxfev = %d." % maxfev, ValueError
],
3: [
"xtol=%f is too small, no further improvement "
"in the approximate\n solution "
"is possible." % xtol, ValueError
],
4: [
"The iteration is not making good progress, as measured "
"by the \n improvement from the last five "
"Jacobian evaluations.", ValueError
],
5: [
"The iteration is not making good progress, "
"as measured by the \n improvement from the last "
"ten iterations.", ValueError
],
'unknown': ["An error occurred.", TypeError]
}
if status != 1 and not full_output:
if status in [2, 3, 4, 5]:
msg = errors[status][0]
warnings.warn(msg, RuntimeWarning)
else:
try:
raise errors[status][1](errors[status][0])
except KeyError:
raise errors['unknown'][1](errors['unknown'][0])
info = retval[1]
info['fun'] = info.pop('fvec')
sol = Result(x=x, success=(status == 1), status=status)
sol.update(info)
try:
sol['message'] = errors[status][0]
except KeyError:
info['message'] = errors['unknown'][0]
return sol
def root(fun, x0, args=(), method="hybr", jac=None, options=None, callback=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`.
options : dict, optional
A dictionary of solver options. E.g. `xtol` or `maxiter`, see
``show_options('root', method)`` for details.
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'.
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
if meth == "hybr":
sol = _root_hybr(fun, x0, args=args, jac=jac, options=options)
elif meth == "lm":
col_deriv = options.get("col_deriv", 0)
xtol = options.get("xtol", 1.49012e-08)
ftol = options.get("ftol", 1.49012e-08)
gtol = options.get("gtol", 0.0)
maxfev = options.get("maxfev", 0)
epsfcn = options.get("epsfcn", 0.0)
factor = options.get("factor", 100)
diag = options.get("diag", None)
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=maxfev,
epsfcn=epsfcn,
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)
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)
jacobian = {
"broyden1": nonlin.BroydenFirst,
"broyden2": nonlin.BroydenSecond,
"anderson": nonlin.Anderson,
"linearmixing": nonlin.LinearMixing,
"diagbroyden": nonlin.DiagBroyden,
"excitingmixing": nonlin.ExcitingMixing,
"krylov": nonlin.KrylovJacobian,
}[meth]
nit = options.get("nit")
verbose = options.get("disp", False)
maxiter = options.get("maxiter")
f_tol = options.get("ftol")
f_rtol = options.get("frtol")
x_tol = options.get("xtol")
x_rtol = options.get("xrtol")
tol_norm = options.get("tol_norm")
line_search = options.get("line_search", "armijo")
jac_opts = options.get("jac_options", dict())
if args:
def f(x):
if jac == True:
r = fun(x, *args)[0]
else:
r = fun(x, *args)
return r
else:
f = fun
x, info = nonlin.nonlin_solve(
f,
x0,
jacobian=jacobian(**jac_opts),
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)
else:
raise ValueError("Unknown solver %s" % method)
return sol
def root(fun,
x0,
args=(),
method='hybr',
jac=None,
options=None,
callback=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`.
options : dict, optional
A dictionary of solver options. E.g. `xtol` or `maxiter`, see
``show_options('root', method)`` for details.
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'.
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
if meth == 'hybr':
sol = _root_hybr(fun, x0, args=args, jac=jac, options=options)
elif meth == 'lm':
col_deriv = options.get('col_deriv', 0)
xtol = options.get('xtol', 1.49012e-08)
ftol = options.get('ftol', 1.49012e-08)
gtol = options.get('gtol', 0.0)
maxfev = options.get('maxfev', 0)
epsfcn = options.get('epsfcn', 0.0)
factor = options.get('factor', 100)
diag = options.get('diag', None)
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=maxfev,
epsfcn=epsfcn,
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)
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)
jacobian = {
'broyden1': nonlin.BroydenFirst,
'broyden2': nonlin.BroydenSecond,
'anderson': nonlin.Anderson,
'linearmixing': nonlin.LinearMixing,
'diagbroyden': nonlin.DiagBroyden,
'excitingmixing': nonlin.ExcitingMixing,
'krylov': nonlin.KrylovJacobian
}[meth]
nit = options.get('nit')
verbose = options.get('disp', False)
maxiter = options.get('maxiter')
f_tol = options.get('ftol')
f_rtol = options.get('frtol')
x_tol = options.get('xtol')
x_rtol = options.get('xrtol')
tol_norm = options.get('tol_norm')
line_search = options.get('line_search', 'armijo')
jac_opts = options.get('jac_options', dict())
if args:
def f(x):
if jac == True:
r = fun(x, *args)[0]
else:
r = fun(x, *args)
return r
else:
f = fun
x, info = nonlin.nonlin_solve(f,
x0,
jacobian=jacobian(**jac_opts),
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)
else:
raise ValueError('Unknown solver %s' % method)
return sol