def fmin_tnc(func, x0, fprime=None, args=(), approx_grad=0,
bounds=None, epsilon=1e-8, scale=None, offset=None,
messages=MSG_ALL, maxCGit=-1, maxfun=None, eta=-1,
stepmx=0, accuracy=0, fmin=0, ftol=-1, xtol=-1, pgtol=-1,
rescale=-1, disp=None):
"""
Minimize a function with variables subject to bounds, using
gradient information in a truncated Newton algorithm. This
method wraps a C implementation of the algorithm.
Parameters
----------
func : callable ``func(x, *args)``
Function to minimize. Must do one of
1. Return f and g, where f is
the value of the function and g its gradient (a list of
floats).
2. Return the function value but supply gradient function
seperately as fprime
3. Return the function value and set approx_grad=True.
If the function returns None, the minimization
is aborted.
x0 : list of floats
Initial estimate of minimum.
fprime : callable ``fprime(x, *args)``
Gradient of func. If None, then either func must return the
function value and the gradient (``f,g = func(x, *args)``)
or approx_grad must be True.
args : tuple
Arguments to pass to function.
approx_grad : bool
If true, approximate the gradient numerically.
bounds : list
(min, max) pairs for each element in x0, defining the
bounds on that parameter. Use None or +/-inf for one of
min or max when there is no bound in that direction.
epsilon: float
Used if approx_grad is True. The stepsize in a finite
difference approximation for fprime.
scale : list of floats
Scaling factors to apply to each variable. If None, the
factors are up-low for interval bounded variables and
1+|x] fo the others. Defaults to None
offset : float
Value to substract from each variable. If None, the
offsets are (up+low)/2 for interval bounded variables
and x for the others.
messages :
Bit mask used to select messages display during
minimization values defined in the MSGS dict. Defaults to
MGS_ALL.
disp : int
Integer interface to messages. 0 = no message, 5 = all messages
maxCGit : int
Maximum number of hessian*vector evaluations per main
iteration. If maxCGit == 0, the direction chosen is
-gradient if maxCGit < 0, maxCGit is set to
max(1,min(50,n/2)). Defaults to -1.
maxfun : int
Maximum number of function evaluation. if None, maxfun is
set to max(100, 10*len(x0)). Defaults to None.
eta : float
Severity of the line search. if < 0 or > 1, set to 0.25.
Defaults to -1.
stepmx : float
Maximum step for the line search. May be increased during
call. If too small, it will be set to 10.0. Defaults to 0.
accuracy : float
Relative precision for finite difference calculations. If
<= machine_precision, set to sqrt(machine_precision).
Defaults to 0.
fmin : float
Minimum function value estimate. Defaults to 0.
ftol : float
Precision goal for the value of f in the stoping criterion.
If ftol < 0.0, ftol is set to 0.0 defaults to -1.
xtol : float
Precision goal for the value of x in the stopping
criterion (after applying x scaling factors). If xtol <
0.0, xtol is set to sqrt(machine_precision). Defaults to
-1.
pgtol : float
Precision goal for the value of the projected gradient in
the stopping criterion (after applying x scaling factors).
If pgtol < 0.0, pgtol is set to 1e-2 * sqrt(accuracy).
Setting it to 0.0 is not recommended. Defaults to -1.
rescale : float
Scaling factor (in log10) used to trigger f value
rescaling. If 0, rescale at each iteration. If a large
value, never rescale. If < 0, rescale is set to 1.3.
Returns
-------
x : list of floats
The solution.
nfeval : int
The number of function evaluations.
rc : int
Return code as defined in the RCSTRINGS dict.
See also
--------
minimize: Interface to minimization algorithms for multivariate
functions. See the 'TNC' `method` in particular.
Notes
-----
The underlying algorithm is truncated Newton, also called
Newton Conjugate-Gradient. This method differs from
scipy.optimize.fmin_ncg in that
1. It wraps a C implementation of the algorithm
2. It allows each variable to be given an upper and lower bound.
The algorithm incoporates the bound constraints by determining
the descent direction as in an unconstrained truncated Newton,
but never taking a step-size large enough to leave the space
of feasible x's. The algorithm keeps track of a set of
currently active constraints, and ignores them when computing
the minimum allowable step size. (The x's associated with the
active constraint are kept fixed.) If the maximum allowable
step size is zero then a new constraint is added. At the end
of each iteration one of the constraints may be deemed no
longer active and removed. A constraint is considered
no longer active is if it is currently active
but the gradient for that variable points inward from the
constraint. The specific constraint removed is the one
associated with the variable of largest index whose
constraint is no longer active.
References
----------
Wright S., Nocedal J. (2006), 'Numerical Optimization'
Nash S.G. (1984), "Newton-Type Minimization Via the Lanczos Method",
SIAM Journal of Numerical Analysis 21, pp. 770-778
"""
# handle fprime/approx_grad
if approx_grad:
fun = func
jac = None
elif fprime is None:
fun = MemoizeJac(func)
jac = fun.derivative
else:
fun = func
jac = fprime
if disp is not None: # disp takes precedence over messages
mesg_num = disp
else:
mesg_num = {0:MSG_NONE, 1:MSG_ITER, 2:MSG_INFO, 3:MSG_VERS,
4:MSG_EXIT, 5:MSG_ALL}.get(messages, MSG_ALL)
# build options
opts = {'eps' : epsilon,
'scale': scale,
'offset': offset,
'mesg_num': mesg_num,
'maxCGit': maxCGit,
'maxfev': maxfun,
'eta': eta,
'stepmx': stepmx,
'accuracy': accuracy,
'minfev': fmin,
'ftol': ftol,
'xtol': xtol,
'pgtol': pgtol,
'rescale': rescale,
'disp': False}
res = _minimize_tnc(fun, x0, args, jac, bounds, options=opts)
return res['x'], res['nfev'], res['status']
def minimize(fun, x0, args=(), method='BFGS', jac=None, hess=None,
hessp=None, bounds=None, constraints=(), tol=None,
callback=None, options=None):
"""
Minimization of scalar function of one or more variables.
.. versionadded:: 0.11.0
Parameters
----------
fun : callable
Objective function.
x0 : ndarray
Initial guess.
args : tuple, optional
Extra arguments passed to the objective function and its
derivatives (Jacobian, Hessian).
method : str, optional
Type of solver. Should be one of
- 'Nelder-Mead'
- 'Powell'
- 'CG'
- 'BFGS'
- 'Newton-CG'
- 'Anneal'
- 'L-BFGS-B'
- 'TNC'
- 'COBYLA'
- 'SLSQP'
jac : bool or callable, optional
Jacobian of objective function. Only for CG, BFGS, Newton-CG.
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 the
objective. In this case, it must accept the same arguments as `fun`.
hess, hessp : callable, optional
Hessian of objective function or Hessian of objective function
times an arbitrary vector p. Only for Newton-CG.
Only one of `hessp` or `hess` needs to be given. If `hess` is
provided, then `hessp` will be ignored. If neither `hess` nor
`hessp` is provided, then the hessian product will be approximated
using finite differences on `jac`. `hessp` must compute the Hessian
times an arbitrary vector.
bounds : sequence, optional
Bounds for variables (only for L-BFGS-B, TNC, COBYLA and SLSQP).
``(min, max)`` pairs for each element in ``x``, defining
the bounds on that parameter. Use None for one of ``min`` or
``max`` when there is no bound in that direction.
constraints : dict or sequence of dict, optional
Constraints definition (only for COBYLA and SLSQP).
Each constraint is defined in a dictionary with fields:
type: str
Constraint type: 'eq' for equality, 'ineq' for inequality.
fun: callable
The function defining the constraint.
jac: callable, optional
The Jacobian of `fun` (only for SLSQP).
args: sequence, optional
Extra arguments to be passed to the function and Jacobian.
Equality constraint means that the constraint function result is to
be zero whereas inequality means that it is to be non-negative.
Note that COBYLA only supports inequality constraints.
tol : float, optional
Tolerance for termination. For detailed control, use solver-specific
options.
options : dict, optional
A dictionary of solver options. All methods accept the following
generic options:
maxiter : int
Maximum number of iterations to perform.
disp : bool
Set to True to print convergence messages.
For method-specific options, see `show_options('minimize', method)`.
callback : callable, optional
Called after each iteration, as ``callback(xk)``, where ``xk`` is the
current parameter vector.
Returns
-------
res : Result
The optimization result represented as a ``Result`` object.
Important attributes are: ``x`` the solution array, ``success`` a
Boolean flag indicating if the optimizer exited successfully and
``message`` which describes the cause of the termination. See
`Result` for a description of other attributes.
See also
--------
minimize_scalar: Interface to minimization algorithms for scalar
univariate functions.
Notes
-----
This section describes the available solvers that can be selected by the
'method' parameter. The default method is *BFGS*.
**Unconstrained minimization**
Method *Nelder-Mead* uses the Simplex algorithm [1]_, [2]_. This
algorithm has been successful in many applications but other algorithms
using the first and/or second derivatives information might be preferred
for their better performances and robustness in general.
Method *Powell* is a modification of Powell's method [3]_, [4]_ which
is a conjugate direction method. It performs sequential one-dimensional
minimizations along each vector of the directions set (`direc` field in
`options` and `info`), which is updated at each iteration of the main
minimization loop. The function need not be differentiable, and no
derivatives are taken.
Method *CG* uses a nonlinear conjugate gradient algorithm by Polak and
Ribiere, a variant of the Fletcher-Reeves method described in [5]_ pp.
120-122. Only the first derivatives are used.
Method *BFGS* uses the quasi-Newton method of Broyden, Fletcher,
Goldfarb, and Shanno (BFGS) [5]_ pp. 136. It uses the first derivatives
only. BFGS has proven good performance even for non-smooth
optimizations
Method *Newton-CG* uses a Newton-CG algorithm [5]_ pp. 168 (also known
as the truncated Newton method). It uses a CG method to the compute the
search direction. See also *TNC* method for a box-constrained
minimization with a similar algorithm.
Method *Anneal* uses simulated annealing, which is a probabilistic
metaheuristic algorithm for global optimization. It uses no derivative
information from the function being optimized.
**Constrained minimization**
Method *L-BFGS-B* uses the L-BFGS-B algorithm [6]_, [7]_ for bound
constrained minimization.
Method *TNC* uses a truncated Newton algorithm [5]_, [8]_ to minimize a
function with variables subject to bounds. This algorithm is uses
gradient information; it is also called Newton Conjugate-Gradient. It
differs from the *Newton-CG* method described above as it wraps a C
implementation and allows each variable to be given upper and lower
bounds.
Method *COBYLA* uses the Constrained Optimization BY Linear
Approximation (COBYLA) method [9]_, [10]_, [11]_. The algorithm is
based on linear approximations to the objective function and each
constraint. The method wraps a FORTRAN implementation of the algorithm.
Method *SLSQP* uses Sequential Least SQuares Programming to minimize a
function of several variables with any combination of bounds, equality
and inequality constraints. The method wraps the SLSQP Optimization
subroutine originally implemented by Dieter Kraft [12]_.
References
----------
.. [1] Nelder, J A, and R Mead. 1965. A Simplex Method for Function
Minimization. The Computer Journal 7: 308-13.
.. [2] Wright M H. 1996. Direct search methods: Once scorned, now
respectable, in Numerical Analysis 1995: Proceedings of the 1995
Dundee Biennial Conference in Numerical Analysis (Eds. D F
Griffiths and G A Watson). Addison Wesley Longman, Harlow, UK.
191-208.
.. [3] Powell, M J D. 1964. An efficient method for finding the minimum of
a function of several variables without calculating derivatives. The
Computer Journal 7: 155-162.
.. [4] Press W, S A Teukolsky, W T Vetterling and B P Flannery.
Numerical Recipes (any edition), Cambridge University Press.
.. [5] Nocedal, J, and S J Wright. 2006. Numerical Optimization.
Springer New York.
.. [6] Byrd, R H and P Lu and J. Nocedal. 1995. A Limited Memory
Algorithm for Bound Constrained Optimization. SIAM Journal on
Scientific and Statistical Computing 16 (5): 1190-1208.
.. [7] Zhu, C and R H Byrd and J Nocedal. 1997. L-BFGS-B: Algorithm
778: L-BFGS-B, FORTRAN routines for large scale bound constrained
optimization. ACM Transactions on Mathematical Software 23 (4):
550-560.
.. [8] Nash, S G. Newton-Type Minimization Via the Lanczos Method.
1984. SIAM Journal of Numerical Analysis 21: 770-778.
.. [9] Powell, M J D. A direct search optimization method that models
the objective and constraint functions by linear interpolation.
1994. Advances in Optimization and Numerical Analysis, eds. S. Gomez
and J-P Hennart, Kluwer Academic (Dordrecht), 51-67.
.. [10] Powell M J D. Direct search algorithms for optimization
calculations. 1998. Acta Numerica 7: 287-336.
.. [11] Powell M J D. A view of algorithms for optimization without
derivatives. 2007.Cambridge University Technical Report DAMTP
2007/NA03
.. [12] Kraft, D. A software package for sequential quadratic
programming. 1988. Tech. Rep. DFVLR-FB 88-28, DLR German Aerospace
Center -- Institute for Flight Mechanics, Koln, Germany.
Examples
--------
Let us consider the problem of minimizing the Rosenbrock function. This
function (and its respective derivatives) is implemented in `rosen`
(resp. `rosen_der`, `rosen_hess`) in the `scipy.optimize`.
>>> from scipy.optimize import minimize, rosen, rosen_der
A simple application of the *Nelder-Mead* method is:
>>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
>>> res = minimize(rosen, x0, method='Nelder-Mead')
>>> res.x
[ 1. 1. 1. 1. 1.]
Now using the *BFGS* algorithm, using the first derivative and a few
options:
>>> res = minimize(rosen, x0, method='BFGS', jac=rosen_der,
... options={'gtol': 1e-6, 'disp': True})
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 52
Function evaluations: 64
Gradient evaluations: 64
>>> res.x
[ 1. 1. 1. 1. 1.]
>>> print res.message
Optimization terminated successfully.
>>> res.hess
[[ 0.00749589 0.01255155 0.02396251 0.04750988 0.09495377]
[ 0.01255155 0.02510441 0.04794055 0.09502834 0.18996269]
[ 0.02396251 0.04794055 0.09631614 0.19092151 0.38165151]
[ 0.04750988 0.09502834 0.19092151 0.38341252 0.7664427 ]
[ 0.09495377 0.18996269 0.38165151 0.7664427 1.53713523]]
Next, consider a minimization problem with several constraints (namely
Example 16.4 from [5]_). The objective function is:
>>> fun = lambda x: (x[0] - 1)**2 + (x[1] - 2.5)**2
There are three constraints defined as:
>>> cons = ({'type': 'ineq', 'fun': lambda x: x[0] - 2 * x[1] + 2},
... {'type': 'ineq', 'fun': lambda x: -x[0] - 2 * x[1] + 6},
... {'type': 'ineq', 'fun': lambda x: -x[0] + 2 * x[1] + 2})
And variables must be positive, hence the following bounds:
>>> bnds = ((0, None), (0, None))
The optimization problem is solved using the SLSQP method as:
>>> res = minimize(fun, (2, 0), method='SLSQP', bounds=bnds,
... constraints=cons)
It should converge to the theoretical solution (1.4 ,1.7).
"""
meth = method.lower()
if options is None:
options = {}
# check if optional parameters are supported by the selected method
# - jac
if meth in ['nelder-mead', 'powell', 'anneal', 'cobyla'] and bool(jac):
warn('Method %s does not use gradient information (jac).' % method,
RuntimeWarning)
# - hess
if meth != 'newton-cg' and hess is not None:
warn('Method %s does not use Hessian information (hess).' % method,
RuntimeWarning)
# - constraints or bounds
if (meth in ['nelder-mead', 'powell', 'cg', 'bfgs', 'newton-cg'] and
(bounds is not None or any(constraints))):
warn('Method %s cannot handle constraints nor bounds.' % method,
RuntimeWarning)
if meth in ['l-bfgs-b', 'tnc'] and any(constraints):
warn('Method %s cannot handle constraints.' % method,
RuntimeWarning)
if meth is 'cobyla' and bounds is not None:
warn('Method %s cannot handle bounds.' % method,
RuntimeWarning)
# - callback
if (meth in ['anneal', 'tnc', 'cobyla', 'slsqp'] and
callback is not None):
warn('Method %s does not support callback.' % method,
RuntimeWarning)
# - return_all
if (meth in ['anneal', 'l-bfgs-b', 'tnc', 'cobyla', 'slsqp'] and
options.get('return_all', False)):
warn('Method %s does not support the return_all option.' % method,
RuntimeWarning)
# fun also returns the jacobian
if not callable(jac):
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 ['nelder-mead', 'newton-cg', 'powell', 'tnc']:
options.setdefault('xtol', tol)
if meth in ['nelder-mead', 'powell', 'anneal', 'l-bfgs-b', 'tnc',
'slsqp']:
options.setdefault('ftol', tol)
if meth in ['bfgs', 'cg', 'l-bfgs-b', 'tnc']:
options.setdefault('gtol', tol)
if meth in ['cobyla']:
options.setdefault('tol', tol)
if meth == 'nelder-mead':
return _minimize_neldermead(fun, x0, args, callback, **options)
elif meth == 'powell':
return _minimize_powell(fun, x0, args, callback, **options)
elif meth == 'cg':
return _minimize_cg(fun, x0, args, jac, callback, **options)
elif meth == 'bfgs':
return _minimize_bfgs(fun, x0, args, jac, callback, **options)
elif meth == 'newton-cg':
return _minimize_newtoncg(fun, x0, args, jac, hess, hessp, callback,
**options)
elif meth == 'anneal':
return _minimize_anneal(fun, x0, args, **options)
elif meth == 'l-bfgs-b':
return _minimize_lbfgsb(fun, x0, args, jac, bounds, callback=callback,
**options)
elif meth == 'tnc':
return _minimize_tnc(fun, x0, args, jac, bounds, **options)
elif meth == 'cobyla':
return _minimize_cobyla(fun, x0, args, constraints, **options)
elif meth == 'slsqp':
return _minimize_slsqp(fun, x0, args, jac, bounds,
constraints, **options)
else:
raise ValueError('Unknown solver %s' % method)
def fmin_l_bfgs_b(func,
x0,
fprime=None,
args=(),
approx_grad=0,
bounds=None,
m=10,
factr=1e7,
pgtol=1e-5,
epsilon=1e-8,
iprint=-1,
maxfun=15000,
maxiter=15000,
disp=None,
callback=None):
"""
Minimize a function func using the L-BFGS-B algorithm.
Parameters
----------
func : callable f(x,*args)
Function to minimise.
x0 : ndarray
Initial guess.
fprime : callable fprime(x,*args)
The gradient of `func`. If None, then `func` returns the function
value and the gradient (``f, g = func(x, *args)``), unless
`approx_grad` is True in which case `func` returns only ``f``.
args : sequence
Arguments to pass to `func` and `fprime`.
approx_grad : bool
Whether to approximate the gradient numerically (in which case
`func` returns only the function value).
bounds : list
``(min, max)`` pairs for each element in ``x``, defining
the bounds on that parameter. Use None for one of ``min`` or
``max`` when there is no bound in that direction.
m : int
The maximum number of variable metric corrections
used to define the limited memory matrix. (The limited memory BFGS
method does not store the full hessian but uses this many terms in an
approximation to it.)
factr : float
The iteration stops when
``(f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr * eps``,
where ``eps`` is the machine precision, which is automatically
generated by the code. Typical values for `factr` are: 1e12 for
low accuracy; 1e7 for moderate accuracy; 10.0 for extremely
high accuracy.
pgtol : float
The iteration will stop when
``max{|proj g_i | i = 1, ..., n} <= pgtol``
where ``pg_i`` is the i-th component of the projected gradient.
epsilon : float
Step size used when `approx_grad` is True, for numerically
calculating the gradient
iprint : int
Controls the frequency of output. ``iprint < 0`` means no output;
``iprint == 0`` means write messages to stdout; ``iprint > 1`` in
addition means write logging information to a file named
``iterate.dat`` in the current working directory.
disp : int, optional
If zero, then no output. If a positive number, then this over-rides
`iprint` (i.e., `iprint` gets the value of `disp`).
maxfun : int
Maximum number of function evaluations.
maxiter : int
Maximum number of iterations.
callback : callable, optional
Called after each iteration, as ``callback(xk)``, where ``xk`` is the
current parameter vector.
Returns
-------
x : array_like
Estimated position of the minimum.
f : float
Value of `func` at the minimum.
d : dict
Information dictionary.
* d['warnflag'] is
- 0 if converged,
- 1 if too many function evaluations or too many iterations,
- 2 if stopped for another reason, given in d['task']
* d['grad'] is the gradient at the minimum (should be 0 ish)
* d['funcalls'] is the number of function calls made.
* d['nit'] is the number of iterations.
See also
--------
minimize: Interface to minimization algorithms for multivariate
functions. See the 'L-BFGS-B' `method` in particular.
Notes
-----
License of L-BFGS-B (Fortran code):
The version included here (in fortran code) is 3.0 (released April 25, 2011).
It was written by Ciyou Zhu, Richard Byrd, and Jorge Nocedal
<*****@*****.**>. It carries the following condition for use:
This software is freely available, but we expect that all publications
describing work using this software, or all commercial products using it,
quote at least one of the references given below. This software is released
under the BSD License.
References
----------
* R. H. Byrd, P. Lu and J. Nocedal. A Limited Memory Algorithm for Bound
Constrained Optimization, (1995), SIAM Journal on Scientific and
Statistical Computing, 16, 5, pp. 1190-1208.
* C. Zhu, R. H. Byrd and J. Nocedal. L-BFGS-B: Algorithm 778: L-BFGS-B,
FORTRAN routines for large scale bound constrained optimization (1997),
ACM Transactions on Mathematical Software, 23, 4, pp. 550 - 560.
* J.L. Morales and J. Nocedal. L-BFGS-B: Remark on Algorithm 778: L-BFGS-B,
FORTRAN routines for large scale bound constrained optimization (2011),
ACM Transactions on Mathematical Software, 38, 1.
"""
# handle fprime/approx_grad
if approx_grad:
fun = func
jac = None
elif fprime is None:
fun = MemoizeJac(func)
jac = fun.derivative
else:
fun = func
jac = fprime
# build options
if disp is None:
disp = iprint
opts = {
'disp': disp,
'iprint': iprint,
'maxcor': m,
'ftol': factr * np.finfo(float).eps,
'gtol': pgtol,
'eps': epsilon,
'maxfun': maxfun,
'maxiter': maxiter,
'callback': callback
}
res = _minimize_lbfgsb(fun, x0, args=args, jac=jac, bounds=bounds, **opts)
d = {
'grad': res['jac'],
'task': res['message'],
'funcalls': res['nfev'],
'nit': res['nit'],
'warnflag': res['status']
}
f = res['fun']
x = res['x']
return x, f, d
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(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