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r"""
Nonlinear solvers
-----------------
.. currentmodule:: scipy.optimize
This is a collection of general-purpose nonlinear multidimensional
solvers. These solvers find *x* for which *F(x) = 0*. Both *x*
and *F* can be multidimensional.
Routines
~~~~~~~~
Large-scale nonlinear solvers:
.. autosummary::
newton_krylov
anderson
General nonlinear solvers:
.. autosummary::
broyden1
broyden2
Simple iterations:
.. autosummary::
excitingmixing
linearmixing
diagbroyden
Examples
~~~~~~~~
**Small problem**
>>> def F(x):
... return np.cos(x) + x[::-1] - [1, 2, 3, 4]
>>> import scipy.optimize
>>> x = scipy.optimize.broyden1(F, [1,1,1,1], f_tol=1e-14)
>>> x
array([ 4.04674914, 3.91158389, 2.71791677, 1.61756251])
>>> np.cos(x) + x[::-1]
array([ 1., 2., 3., 4.])
**Large problem**
Suppose that we needed to solve the following integrodifferential
equation on the square :math:`[0,1]\times[0,1]`:
.. math::
\nabla^2 P = 10 \left(\int_0^1\int_0^1\cosh(P)\,dx\,dy\right)^2
with :math:`P(x,1) = 1` and :math:`P=0` elsewhere on the boundary of
the square.
The solution can be found using the `newton_krylov` solver:
.. plot::
import numpy as np
from scipy.optimize import newton_krylov
from numpy import cosh, zeros_like, mgrid, zeros
# parameters
nx, ny = 75, 75
hx, hy = 1./(nx-1), 1./(ny-1)
P_left, P_right = 0, 0
P_top, P_bottom = 1, 0
def residual(P):
d2x = zeros_like(P)
d2y = zeros_like(P)
d2x[1:-1] = (P[2:] - 2*P[1:-1] + P[:-2]) / hx/hx
d2x[0] = (P[1] - 2*P[0] + P_left)/hx/hx
d2x[-1] = (P_right - 2*P[-1] + P[-2])/hx/hx
d2y[:,1:-1] = (P[:,2:] - 2*P[:,1:-1] + P[:,:-2])/hy/hy
d2y[:,0] = (P[:,1] - 2*P[:,0] + P_bottom)/hy/hy
d2y[:,-1] = (P_top - 2*P[:,-1] + P[:,-2])/hy/hy
return d2x + d2y - 10*cosh(P).mean()**2
# solve
guess = zeros((nx, ny), float)
sol = newton_krylov(residual, guess, method='lgmres', verbose=1)
print('Residual: %g' % abs(residual(sol)).max())
# visualize
import matplotlib.pyplot as plt
x, y = mgrid[0:1:(nx*1j), 0:1:(ny*1j)]
plt.pcolormesh(x, y, sol, shading='gouraud')
plt.colorbar()
plt.show()
"""
# Copyright (C) 2009, Pauli Virtanen <pav@iki.fi>
# Distributed under the same license as SciPy.
import sys
import numpy as np
from scipy.linalg import norm, solve, inv, qr, svd, LinAlgError
from numpy import asarray, dot, vdot
import scipy.sparse.linalg
import scipy.sparse
from scipy.linalg import get_blas_funcs
import inspect
from scipy._lib._util import getfullargspec_no_self as _getfullargspec
from .linesearch import scalar_search_wolfe1, scalar_search_armijo
__all__ = [
'broyden1', 'broyden2', 'anderson', 'linearmixing',
'diagbroyden', 'excitingmixing', 'newton_krylov']
#------------------------------------------------------------------------------
# Utility functions
#------------------------------------------------------------------------------
class NoConvergence(Exception):
pass
def maxnorm(x):
return np.absolute(x).max()
def _as_inexact(x):
"""Return `x` as an array, of either floats or complex floats"""
x = asarray(x)
if not np.issubdtype(x.dtype, np.inexact):
return asarray(x, dtype=np.float_)
return x
def _array_like(x, x0):
"""Return ndarray `x` as same array subclass and shape as `x0`"""
x = np.reshape(x, np.shape(x0))
wrap = getattr(x0, '__array_wrap__', x.__array_wrap__)
return wrap(x)
def _safe_norm(v):
if not np.isfinite(v).all():
return np.array(np.inf)
return norm(v)
#------------------------------------------------------------------------------
# Generic nonlinear solver machinery
#------------------------------------------------------------------------------
_doc_parts = dict(
params_basic="""
F : function(x) -> f
Function whose root to find; should take and return an array-like
object.
xin : array_like
Initial guess for the solution
""".strip(),
params_extra="""
iter : int, optional
Number of iterations to make. If omitted (default), make as many
as required to meet tolerances.
verbose : 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.
f_tol : float, optional
Absolute tolerance (in max-norm) for the residual.
If omitted, default is 6e-6.
f_rtol : float, optional
Relative tolerance for the residual. If omitted, not used.
x_tol : 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.
x_rtol : float, optional
Relative minimum step size. 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'.
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.
Returns
-------
sol : ndarray
An array (of similar array type as `x0`) containing the final solution.
Raises
------
NoConvergence
When a solution was not found.
""".strip()
)
def _set_doc(obj):
if obj.__doc__:
obj.__doc__ = obj.__doc__ % _doc_parts
def nonlin_solve(F, x0, jacobian='krylov', iter=None, verbose=False,
maxiter=None, f_tol=None, f_rtol=None, x_tol=None, x_rtol=None,
tol_norm=None, line_search='armijo', callback=None,
full_output=False, raise_exception=True):
"""
Find a root of a function, in a way suitable for large-scale problems.
Parameters
----------
%(params_basic)s
jacobian : Jacobian
A Jacobian approximation: `Jacobian` object or something that
`asjacobian` can transform to one. Alternatively, a string specifying
which of the builtin Jacobian approximations to use:
krylov, broyden1, broyden2, anderson
diagbroyden, linearmixing, excitingmixing
%(params_extra)s
full_output : bool
If true, returns a dictionary `info` containing convergence
information.
raise_exception : bool
If True, a `NoConvergence` exception is raise if no solution is found.
See Also
--------
asjacobian, Jacobian
Notes
-----
This algorithm implements the inexact Newton method, with
backtracking or full line searches. Several Jacobian
approximations are available, including Krylov and Quasi-Newton
methods.
References
----------
.. [KIM] C. T. Kelley, \"Iterative Methods for Linear and Nonlinear
Equations\". Society for Industrial and Applied Mathematics. (1995)
https://archive.siam.org/books/kelley/fr16/
"""
# Can't use default parameters because it's being explicitly passed as None
# from the calling function, so we need to set it here.
tol_norm = maxnorm if tol_norm is None else tol_norm
condition = TerminationCondition(f_tol=f_tol, f_rtol=f_rtol,
x_tol=x_tol, x_rtol=x_rtol,
iter=iter, norm=tol_norm)
x0 = _as_inexact(x0)
func = lambda z: _as_inexact(F(_array_like(z, x0))).flatten()
x = x0.flatten()
dx = np.full_like(x, np.inf)
Fx = func(x)
Fx_norm = norm(Fx)
jacobian = asjacobian(jacobian)
jacobian.setup(x.copy(), Fx, func)
if maxiter is None:
if iter is not None:
maxiter = iter + 1
else:
maxiter = 100*(x.size+1)
if line_search is True:
line_search = 'armijo'
elif line_search is False:
line_search = None
if line_search not in (None, 'armijo', 'wolfe'):
raise ValueError("Invalid line search")
# Solver tolerance selection
gamma = 0.9
eta_max = 0.9999
eta_treshold = 0.1
eta = 1e-3
for n in range(maxiter):
status = condition.check(Fx, x, dx)
if status:
break
# The tolerance, as computed for scipy.sparse.linalg.* routines
tol = min(eta, eta*Fx_norm)
dx = -jacobian.solve(Fx, tol=tol)
if norm(dx) == 0:
raise ValueError("Jacobian inversion yielded zero vector. "
"This indicates a bug in the Jacobian "
"approximation.")
# Line search, or Newton step
if line_search:
s, x, Fx, Fx_norm_new = _nonlin_line_search(func, x, Fx, dx,
line_search)
else:
s = 1.0
x = x + dx
Fx = func(x)
Fx_norm_new = norm(Fx)
jacobian.update(x.copy(), Fx)
if callback:
callback(x, Fx)
# Adjust forcing parameters for inexact methods
eta_A = gamma * Fx_norm_new**2 / Fx_norm**2
if gamma * eta**2 < eta_treshold:
eta = min(eta_max, eta_A)
else:
eta = min(eta_max, max(eta_A, gamma*eta**2))
Fx_norm = Fx_norm_new
# Print status
if verbose:
sys.stdout.write("%d: |F(x)| = %g; step %g\n" % (
n, tol_norm(Fx), s))
sys.stdout.flush()
else:
if raise_exception:
raise NoConvergence(_array_like(x, x0))
else:
status = 2
if full_output:
info = {'nit': condition.iteration,
'fun': Fx,
'status': status,
'success': status == 1,
'message': {1: 'A solution was found at the specified '
'tolerance.',
2: 'The maximum number of iterations allowed '
'has been reached.'
}[status]
}
return _array_like(x, x0), info
else:
return _array_like(x, x0)
_set_doc(nonlin_solve)
def _nonlin_line_search(func, x, Fx, dx, search_type='armijo', rdiff=1e-8,
smin=1e-2):
tmp_s = [0]
tmp_Fx = [Fx]
tmp_phi = [norm(Fx)**2]
s_norm = norm(x) / norm(dx)
def phi(s, store=True):
if s == tmp_s[0]:
return tmp_phi[0]
xt = x + s*dx
v = func(xt)
p = _safe_norm(v)**2
if store:
tmp_s[0] = s
tmp_phi[0] = p
tmp_Fx[0] = v
return p
def derphi(s):
ds = (abs(s) + s_norm + 1) * rdiff
return (phi(s+ds, store=False) - phi(s)) / ds
if search_type == 'wolfe':
s, phi1, phi0 = scalar_search_wolfe1(phi, derphi, tmp_phi[0],
xtol=1e-2, amin=smin)
elif search_type == 'armijo':
s, phi1 = scalar_search_armijo(phi, tmp_phi[0], -tmp_phi[0],
amin=smin)
if s is None:
# XXX: No suitable step length found. Take the full Newton step,
# and hope for the best.
s = 1.0
x = x + s*dx
if s == tmp_s[0]:
Fx = tmp_Fx[0]
else:
Fx = func(x)
Fx_norm = norm(Fx)
return s, x, Fx, Fx_norm
class TerminationCondition(object):
"""
Termination condition for an iteration. It is terminated if
- |F| < f_rtol*|F_0|, AND
- |F| < f_tol
AND
- |dx| < x_rtol*|x|, AND
- |dx| < x_tol
"""
def __init__(self, f_tol=None, f_rtol=None, x_tol=None, x_rtol=None,
iter=None, norm=maxnorm):
if f_tol is None:
f_tol = np.finfo(np.float_).eps ** (1./3)
if f_rtol is None:
f_rtol = np.inf
if x_tol is None:
x_tol = np.inf
if x_rtol is None:
x_rtol = np.inf
self.x_tol = x_tol
self.x_rtol = x_rtol
self.f_tol = f_tol
self.f_rtol = f_rtol
self.norm = norm
self.iter = iter
self.f0_norm = None
self.iteration = 0
def check(self, f, x, dx):
self.iteration += 1
f_norm = self.norm(f)
x_norm = self.norm(x)
dx_norm = self.norm(dx)
if self.f0_norm is None:
self.f0_norm = f_norm
if f_norm == 0:
return 1
if self.iter is not None:
# backwards compatibility with SciPy 0.6.0
return 2 * (self.iteration > self.iter)
# NB: condition must succeed for rtol=inf even if norm == 0
return int((f_norm <= self.f_tol
and f_norm/self.f_rtol <= self.f0_norm)
and (dx_norm <= self.x_tol
and dx_norm/self.x_rtol <= x_norm))
#------------------------------------------------------------------------------
# Generic Jacobian approximation
#------------------------------------------------------------------------------
class Jacobian(object):
"""
Common interface for Jacobians or Jacobian approximations.
The optional methods come useful when implementing trust region
etc., algorithms that often require evaluating transposes of the
Jacobian.
Methods
-------
solve
Returns J^-1 * v
update
Updates Jacobian to point `x` (where the function has residual `Fx`)
matvec : optional
Returns J * v
rmatvec : optional
Returns A^H * v
rsolve : optional
Returns A^-H * v
matmat : optional
Returns A * V, where V is a dense matrix with dimensions (N,K).
todense : optional
Form the dense Jacobian matrix. Necessary for dense trust region
algorithms, and useful for testing.
Attributes
----------
shape
Matrix dimensions (M, N)
dtype
Data type of the matrix.
func : callable, optional
Function the Jacobian corresponds to
"""
def __init__(self, **kw):
names = ["solve", "update", "matvec", "rmatvec", "rsolve",
"matmat", "todense", "shape", "dtype"]
for name, value in kw.items():
if name not in names:
raise ValueError("Unknown keyword argument %s" % name)
if value is not None:
setattr(self, name, kw[name])
if hasattr(self, 'todense'):
self.__array__ = lambda: self.todense()
def aspreconditioner(self):
return InverseJacobian(self)
def solve(self, v, tol=0):
raise NotImplementedError
def update(self, x, F):
pass
def setup(self, x, F, func):
self.func = func
self.shape = (F.size, x.size)
self.dtype = F.dtype
if self.__class__.setup is Jacobian.setup:
# Call on the first point unless overridden
self.update(x, F)
class InverseJacobian(object):
def __init__(self, jacobian):
self.jacobian = jacobian
self.matvec = jacobian.solve
self.update = jacobian.update
if hasattr(jacobian, 'setup'):
self.setup = jacobian.setup
if hasattr(jacobian, 'rsolve'):
self.rmatvec = jacobian.rsolve
@property
def shape(self):
return self.jacobian.shape
@property
def dtype(self):
return self.jacobian.dtype
def asjacobian(J):
"""
Convert given object to one suitable for use as a Jacobian.
"""
spsolve = scipy.sparse.linalg.spsolve
if isinstance(J, Jacobian):
return J
elif inspect.isclass(J) and issubclass(J, Jacobian):
return J()
elif isinstance(J, np.ndarray):
if J.ndim > 2:
raise ValueError('array must have rank <= 2')
J = np.atleast_2d(np.asarray(J))
if J.shape[0] != J.shape[1]:
raise ValueError('array must be square')
return Jacobian(matvec=lambda v: dot(J, v),
rmatvec=lambda v: dot(J.conj().T, v),
solve=lambda v: solve(J, v),
rsolve=lambda v: solve(J.conj().T, v),
dtype=J.dtype, shape=J.shape)
elif scipy.sparse.isspmatrix(J):
if J.shape[0] != J.shape[1]:
raise ValueError('matrix must be square')
return Jacobian(matvec=lambda v: J*v,
rmatvec=lambda v: J.conj().T * v,
solve=lambda v: spsolve(J, v),
rsolve=lambda v: spsolve(J.conj().T, v),
dtype=J.dtype, shape=J.shape)
elif hasattr(J, 'shape') and hasattr(J, 'dtype') and hasattr(J, 'solve'):
return Jacobian(matvec=getattr(J, 'matvec'),
rmatvec=getattr(J, 'rmatvec'),
solve=J.solve,
rsolve=getattr(J, 'rsolve'),
update=getattr(J, 'update'),
setup=getattr(J, 'setup'),
dtype=J.dtype,
shape=J.shape)
elif callable(J):
# Assume it's a function J(x) that returns the Jacobian
class Jac(Jacobian):
def update(self, x, F):
self.x = x
def solve(self, v, tol=0):
m = J(self.x)
if isinstance(m, np.ndarray):
return solve(m, v)
elif scipy.sparse.isspmatrix(m):
return spsolve(m, v)
else:
raise ValueError("Unknown matrix type")
def matvec(self, v):
m = J(self.x)
if isinstance(m, np.ndarray):
return dot(m, v)
elif scipy.sparse.isspmatrix(m):
return m*v
else:
raise ValueError("Unknown matrix type")
def rsolve(self, v, tol=0):
m = J(self.x)
if isinstance(m, np.ndarray):
return solve(m.conj().T, v)
elif scipy.sparse.isspmatrix(m):
return spsolve(m.conj().T, v)
else:
raise ValueError("Unknown matrix type")
def rmatvec(self, v):
m = J(self.x)
if isinstance(m, np.ndarray):
return dot(m.conj().T, v)
elif scipy.sparse.isspmatrix(m):
return m.conj().T * v
else:
raise ValueError("Unknown matrix type")
return Jac()
elif isinstance(J, str):
return dict(broyden1=BroydenFirst,
broyden2=BroydenSecond,
anderson=Anderson,
diagbroyden=DiagBroyden,
linearmixing=LinearMixing,
excitingmixing=ExcitingMixing,
krylov=KrylovJacobian)[J]()
else:
raise TypeError('Cannot convert object to a Jacobian')
#------------------------------------------------------------------------------
# Broyden
#------------------------------------------------------------------------------
class GenericBroyden(Jacobian):
def setup(self, x0, f0, func):
Jacobian.setup(self, x0, f0, func)
self.last_f = f0
self.last_x = x0
if hasattr(self, 'alpha') and self.alpha is None:
# Autoscale the initial Jacobian parameter
# unless we have already guessed the solution.
normf0 = norm(f0)
if normf0:
self.alpha = 0.5*max(norm(x0), 1) / normf0
else:
self.alpha = 1.0
def _update(self, x, f, dx, df, dx_norm, df_norm):
raise NotImplementedError
def update(self, x, f):
df = f - self.last_f
dx = x - self.last_x
self._update(x, f, dx, df, norm(dx), norm(df))
self.last_f = f
self.last_x = x
class LowRankMatrix(object):
r"""
A matrix represented as
.. math:: \alpha I + \sum_{n=0}^{n=M} c_n d_n^\dagger
However, if the rank of the matrix reaches the dimension of the vectors,
full matrix representation will be used thereon.
"""
def __init__(self, alpha, n, dtype):
self.alpha = alpha
self.cs = []
self.ds = []
self.n = n
self.dtype = dtype
self.collapsed = None
@staticmethod
def _matvec(v, alpha, cs, ds):
axpy, scal, dotc = get_blas_funcs(['axpy', 'scal', 'dotc'],
cs[:1] + [v])
w = alpha * v
for c, d in zip(cs, ds):
a = dotc(d, v)
w = axpy(c, w, w.size, a)
return w
@staticmethod
def _solve(v, alpha, cs, ds):
"""Evaluate w = M^-1 v"""
if len(cs) == 0:
return v/alpha
# (B + C D^H)^-1 = B^-1 - B^-1 C (I + D^H B^-1 C)^-1 D^H B^-1
axpy, dotc = get_blas_funcs(['axpy', 'dotc'], cs[:1] + [v])
c0 = cs[0]
A = alpha * np.identity(len(cs), dtype=c0.dtype)
for i, d in enumerate(ds):
for j, c in enumerate(cs):
A[i,j] += dotc(d, c)
q = np.zeros(len(cs), dtype=c0.dtype)
for j, d in enumerate(ds):
q[j] = dotc(d, v)
q /= alpha
q = solve(A, q)
w = v/alpha
for c, qc in zip(cs, q):
w = axpy(c, w, w.size, -qc)
return w
def matvec(self, v):
"""Evaluate w = M v"""
if self.collapsed is not None:
return np.dot(self.collapsed, v)
return LowRankMatrix._matvec(v, self.alpha, self.cs, self.ds)
def rmatvec(self, v):
"""Evaluate w = M^H v"""
if self.collapsed is not None:
return np.dot(self.collapsed.T.conj(), v)
return LowRankMatrix._matvec(v, np.conj(self.alpha), self.ds, self.cs)
def solve(self, v, tol=0):
"""Evaluate w = M^-1 v"""
if self.collapsed is not None:
return solve(self.collapsed, v)
return LowRankMatrix._solve(v, self.alpha, self.cs, self.ds)
def rsolve(self, v, tol=0):
"""Evaluate w = M^-H v"""
if self.collapsed is not None:
return solve(self.collapsed.T.conj(), v)
return LowRankMatrix._solve(v, np.conj(self.alpha), self.ds, self.cs)
def append(self, c, d):
if self.collapsed is not None:
self.collapsed += c[:,None] * d[None,:].conj()
return
self.cs.append(c)
self.ds.append(d)
if len(self.cs) > c.size:
self.collapse()
def __array__(self):
if self.collapsed is not None:
return self.collapsed
Gm = self.alpha*np.identity(self.n, dtype=self.dtype)
for c, d in zip(self.cs, self.ds):
Gm += c[:,None]*d[None,:].conj()
return Gm
def collapse(self):
"""Collapse the low-rank matrix to a full-rank one."""
self.collapsed = np.array(self)
self.cs = None
self.ds = None
self.alpha = None
def restart_reduce(self, rank):
"""
Reduce the rank of the matrix by dropping all vectors.
"""
if self.collapsed is not None:
return
assert rank > 0
if len(self.cs) > rank:
del self.cs[:]
del self.ds[:]
def simple_reduce(self, rank):
"""
Reduce the rank of the matrix by dropping oldest vectors.
"""
if self.collapsed is not None:
return
assert rank > 0
while len(self.cs) > rank:
del self.cs[0]
del self.ds[0]
def svd_reduce(self, max_rank, to_retain=None):
"""
Reduce the rank of the matrix by retaining some SVD components.
This corresponds to the \"Broyden Rank Reduction Inverse\"
algorithm described in [1]_.
Note that the SVD decomposition can be done by solving only a
problem whose size is the effective rank of this matrix, which
is viable even for large problems.
Parameters
----------
max_rank : int
Maximum rank of this matrix after reduction.
to_retain : int, optional
Number of SVD components to retain when reduction is done
(ie. rank > max_rank). Default is ``max_rank - 2``.
References
----------
.. [1] B.A. van der Rotten, PhD thesis,
\"A limited memory Broyden method to solve high-dimensional
systems of nonlinear equations\". Mathematisch Instituut,
Universiteit Leiden, The Netherlands (2003).
https://web.archive.org/web/20161022015821/http://www.math.leidenuniv.nl/scripties/Rotten.pdf
"""
if self.collapsed is not None:
return
p = max_rank
if to_retain is not None:
q = to_retain
else:
q = p - 2
if self.cs:
p = min(p, len(self.cs[0]))
q = max(0, min(q, p-1))
m = len(self.cs)
if m < p:
# nothing to do
return
C = np.array(self.cs).T
D = np.array(self.ds).T
D, R = qr(D, mode='economic')
C = dot(C, R.T.conj())
U, S, WH = svd(C, full_matrices=False, compute_uv=True)
C = dot(C, inv(WH))
D = dot(D, WH.T.conj())
for k in range(q):
self.cs[k] = C[:,k].copy()
self.ds[k] = D[:,k].copy()
del self.cs[q:]
del self.ds[q:]
_doc_parts['broyden_params'] = """
alpha : float, optional
Initial guess for the Jacobian is ``(-1/alpha)``.
reduction_method : str or tuple, optional
Method used in ensuring that the rank of the Broyden matrix
stays low. Can either be a string giving the name of the method,
or a tuple of the form ``(method, param1, param2, ...)``
that gives the name of the method and values for additional parameters.
Methods available:
- ``restart``: drop all matrix columns. Has no extra parameters.
- ``simple``: drop oldest matrix column. Has no extra parameters.
- ``svd``: keep only the most significant SVD components.
Takes an extra parameter, ``to_retain``, which determines the
number of SVD components to retain when rank reduction is done.
Default is ``max_rank - 2``.
max_rank : int, optional
Maximum rank for the Broyden matrix.
Default is infinity (i.e., no rank reduction).
""".strip()
class BroydenFirst(GenericBroyden):
r"""
Find a root of a function, using Broyden's first Jacobian approximation.
This method is also known as \"Broyden's good method\".
Parameters
----------
%(params_basic)s
%(broyden_params)s
%(params_extra)s
See Also
--------
root : Interface to root finding algorithms for multivariate
functions. See ``method=='broyden1'`` in particular.
Notes
-----
This algorithm implements the inverse Jacobian Quasi-Newton update
.. math:: H_+ = H + (dx - H df) dx^\dagger H / ( dx^\dagger H df)
which corresponds to Broyden's first Jacobian update
.. math:: J_+ = J + (df - J dx) dx^\dagger / dx^\dagger dx
References
----------
.. [1] B.A. van der Rotten, PhD thesis,
\"A limited memory Broyden method to solve high-dimensional
systems of nonlinear equations\". Mathematisch Instituut,
Universiteit Leiden, The Netherlands (2003).
https://web.archive.org/web/20161022015821/http://www.math.leidenuniv.nl/scripties/Rotten.pdf
Examples
--------
The following functions define a system of nonlinear equations
>>> def fun(x):
... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0,
... 0.5 * (x[1] - x[0])**3 + x[1]]
A solution can be obtained as follows.
>>> from scipy import optimize
>>> sol = optimize.broyden1(fun, [0, 0])
>>> sol
array([0.84116396, 0.15883641])
"""
def __init__(self, alpha=None, reduction_method='restart', max_rank=None):
GenericBroyden.__init__(self)
self.alpha = alpha
self.Gm = None
if max_rank is None:
max_rank = np.inf
self.max_rank = max_rank
if isinstance(reduction_method, str):
reduce_params = ()
else:
reduce_params = reduction_method[1:]
reduction_method = reduction_method[0]
reduce_params = (max_rank - 1,) + reduce_params
if reduction_method == 'svd':
self._reduce = lambda: self.Gm.svd_reduce(*reduce_params)
elif reduction_method == 'simple':
self._reduce = lambda: self.Gm.simple_reduce(*reduce_params)
elif reduction_method == 'restart':
self._reduce = lambda: self.Gm.restart_reduce(*reduce_params)
else:
raise ValueError("Unknown rank reduction method '%s'" %
reduction_method)
def setup(self, x, F, func):
GenericBroyden.setup(self, x, F, func)
self.Gm = LowRankMatrix(-self.alpha, self.shape[0], self.dtype)
def todense(self):
return inv(self.Gm)
def solve(self, f, tol=0):
r = self.Gm.matvec(f)
if not np.isfinite(r).all():
# singular; reset the Jacobian approximation
self.setup(self.last_x, self.last_f, self.func)
return self.Gm.matvec(f)
def matvec(self, f):
return self.Gm.solve(f)
def rsolve(self, f, tol=0):
return self.Gm.rmatvec(f)
def rmatvec(self, f):
return self.Gm.rsolve(f)
def _update(self, x, f, dx, df, dx_norm, df_norm):
self._reduce() # reduce first to preserve secant condition
v = self.Gm.rmatvec(dx)
c = dx - self.Gm.matvec(df)
d = v / vdot(df, v)
self.Gm.append(c, d)
class BroydenSecond(BroydenFirst):
"""
Find a root of a function, using Broyden\'s second Jacobian approximation.
This method is also known as \"Broyden's bad method\".
Parameters
----------
%(params_basic)s
%(broyden_params)s
%(params_extra)s
See Also
--------
root : Interface to root finding algorithms for multivariate
functions. See ``method=='broyden2'`` in particular.
Notes
-----
This algorithm implements the inverse Jacobian Quasi-Newton update
.. math:: H_+ = H + (dx - H df) df^\\dagger / ( df^\\dagger df)
corresponding to Broyden's second method.
References
----------
.. [1] B.A. van der Rotten, PhD thesis,
\"A limited memory Broyden method to solve high-dimensional
systems of nonlinear equations\". Mathematisch Instituut,
Universiteit Leiden, The Netherlands (2003).
https://web.archive.org/web/20161022015821/http://www.math.leidenuniv.nl/scripties/Rotten.pdf
Examples
--------
The following functions define a system of nonlinear equations
>>> def fun(x):
... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0,
... 0.5 * (x[1] - x[0])**3 + x[1]]
A solution can be obtained as follows.
>>> from scipy import optimize
>>> sol = optimize.broyden2(fun, [0, 0])
>>> sol
array([0.84116365, 0.15883529])
"""
def _update(self, x, f, dx, df, dx_norm, df_norm):
self._reduce() # reduce first to preserve secant condition
v = df
c = dx - self.Gm.matvec(df)
d = v / df_norm**2
self.Gm.append(c, d)
#------------------------------------------------------------------------------
# Broyden-like (restricted memory)
#------------------------------------------------------------------------------
class Anderson(GenericBroyden):
"""
Find a root of a function, using (extended) Anderson mixing.
The Jacobian is formed by for a 'best' solution in the space
spanned by last `M` vectors. As a result, only a MxM matrix
inversions and MxN multiplications are required. [Ey]_
Parameters
----------
%(params_basic)s
alpha : float, optional
Initial guess for the Jacobian is (-1/alpha).
M : float, optional
Number of previous vectors to retain. Defaults to 5.
w0 : float, optional
Regularization parameter for numerical stability.
Compared to unity, good values of the order of 0.01.
%(params_extra)s
See Also
--------
root : Interface to root finding algorithms for multivariate
functions. See ``method=='anderson'`` in particular.
References
----------
.. [Ey] V. Eyert, J. Comp. Phys., 124, 271 (1996).
Examples
--------
The following functions define a system of nonlinear equations
>>> def fun(x):
... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0,
... 0.5 * (x[1] - x[0])**3 + x[1]]
A solution can be obtained as follows.
>>> from scipy import optimize
>>> sol = optimize.anderson(fun, [0, 0])
>>> sol
array([0.84116588, 0.15883789])
"""
# Note:
#
# Anderson method maintains a rank M approximation of the inverse Jacobian,
#
# J^-1 v ~ -v*alpha + (dX + alpha dF) A^-1 dF^H v
# A = W + dF^H dF
# W = w0^2 diag(dF^H dF)
#
# so that for w0 = 0 the secant condition applies for last M iterates, i.e.,
#
# J^-1 df_j = dx_j
#
# for all j = 0 ... M-1.
#
# Moreover, (from Sherman-Morrison-Woodbury formula)
#
# J v ~ [ b I - b^2 C (I + b dF^H A^-1 C)^-1 dF^H ] v
# C = (dX + alpha dF) A^-1
# b = -1/alpha
#
# and after simplification
#
# J v ~ -v/alpha + (dX/alpha + dF) (dF^H dX - alpha W)^-1 dF^H v
#
def __init__(self, alpha=None, w0=0.01, M=5):
GenericBroyden.__init__(self)
self.alpha = alpha
self.M = M
self.dx = []
self.df = []
self.gamma = None
self.w0 = w0
def solve(self, f, tol=0):
dx = -self.alpha*f
n = len(self.dx)
if n == 0:
return dx
df_f = np.empty(n, dtype=f.dtype)
for k in range(n):
df_f[k] = vdot(self.df[k], f)
try:
gamma = solve(self.a, df_f)
except LinAlgError:
# singular; reset the Jacobian approximation
del self.dx[:]
del self.df[:]
return dx
for m in range(n):
dx += gamma[m]*(self.dx[m] + self.alpha*self.df[m])
return dx
def matvec(self, f):
dx = -f/self.alpha
n = len(self.dx)
if n == 0:
return dx
df_f = np.empty(n, dtype=f.dtype)
for k in range(n):
df_f[k] = vdot(self.df[k], f)
b = np.empty((n, n), dtype=f.dtype)
for i in range(n):
for j in range(n):
b[i,j] = vdot(self.df[i], self.dx[j])
if i == j and self.w0 != 0:
b[i,j] -= vdot(self.df[i], self.df[i])*self.w0**2*self.alpha
gamma = solve(b, df_f)
for m in range(n):
dx += gamma[m]*(self.df[m] + self.dx[m]/self.alpha)
return dx
def _update(self, x, f, dx, df, dx_norm, df_norm):
if self.M == 0:
return
self.dx.append(dx)
self.df.append(df)
while len(self.dx) > self.M:
self.dx.pop(0)
self.df.pop(0)
n = len(self.dx)
a = np.zeros((n, n), dtype=f.dtype)
for i in range(n):
for j in range(i, n):
if i == j:
wd = self.w0**2
else:
wd = 0
a[i,j] = (1+wd)*vdot(self.df[i], self.df[j])
a += np.triu(a, 1).T.conj()
self.a = a
#------------------------------------------------------------------------------
# Simple iterations
#------------------------------------------------------------------------------
class DiagBroyden(GenericBroyden):
"""
Find a root of a function, using diagonal Broyden Jacobian approximation.
The Jacobian approximation is derived from previous iterations, by
retaining only the diagonal of Broyden matrices.
.. warning::
This algorithm may be useful for specific problems, but whether
it will work may depend strongly on the problem.
Parameters
----------
%(params_basic)s
alpha : float, optional
Initial guess for the Jacobian is (-1/alpha).
%(params_extra)s
See Also
--------
root : Interface to root finding algorithms for multivariate
functions. See ``method=='diagbroyden'`` in particular.
Examples
--------
The following functions define a system of nonlinear equations
>>> def fun(x):
... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0,
... 0.5 * (x[1] - x[0])**3 + x[1]]
A solution can be obtained as follows.
>>> from scipy import optimize
>>> sol = optimize.diagbroyden(fun, [0, 0])
>>> sol
array([0.84116403, 0.15883384])
"""
def __init__(self, alpha=None):
GenericBroyden.__init__(self)
self.alpha = alpha
def setup(self, x, F, func):
GenericBroyden.setup(self, x, F, func)
self.d = np.full((self.shape[0],), 1 / self.alpha, dtype=self.dtype)
def solve(self, f, tol=0):
return -f / self.d
def matvec(self, f):
return -f * self.d
def rsolve(self, f, tol=0):
return -f / self.d.conj()
def rmatvec(self, f):
return -f * self.d.conj()
def todense(self):
return np.diag(-self.d)
def _update(self, x, f, dx, df, dx_norm, df_norm):
self.d -= (df + self.d*dx)*dx/dx_norm**2
class LinearMixing(GenericBroyden):
"""
Find a root of a function, using a scalar Jacobian approximation.
.. warning::
This algorithm may be useful for specific problems, but whether
it will work may depend strongly on the problem.
Parameters
----------
%(params_basic)s
alpha : float, optional
The Jacobian approximation is (-1/alpha).
%(params_extra)s
See Also
--------
root : Interface to root finding algorithms for multivariate
functions. See ``method=='linearmixing'`` in particular.
"""
def __init__(self, alpha=None):
GenericBroyden.__init__(self)
self.alpha = alpha
def solve(self, f, tol=0):
return -f*self.alpha
def matvec(self, f):
return -f/self.alpha
def rsolve(self, f, tol=0):
return -f*np.conj(self.alpha)
def rmatvec(self, f):
return -f/np.conj(self.alpha)
def todense(self):
return np.diag(np.full(self.shape[0], -1/self.alpha))
def _update(self, x, f, dx, df, dx_norm, df_norm):
pass
class ExcitingMixing(GenericBroyden):
"""
Find a root of a function, using a tuned diagonal Jacobian approximation.
The Jacobian matrix is diagonal and is tuned on each iteration.
.. warning::
This algorithm may be useful for specific problems, but whether
it will work may depend strongly on the problem.
See Also
--------
root : Interface to root finding algorithms for multivariate
functions. See ``method=='excitingmixing'`` in particular.
Parameters
----------
%(params_basic)s
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]``.
%(params_extra)s
"""
def __init__(self, alpha=None, alphamax=1.0):
GenericBroyden.__init__(self)
self.alpha = alpha
self.alphamax = alphamax
self.beta = None
def setup(self, x, F, func):
GenericBroyden.setup(self, x, F, func)
self.beta = np.full((self.shape[0],), self.alpha, dtype=self.dtype)
def solve(self, f, tol=0):
return -f*self.beta
def matvec(self, f):
return -f/self.beta
def rsolve(self, f, tol=0):
return -f*self.beta.conj()
def rmatvec(self, f):
return -f/self.beta.conj()
def todense(self):
return np.diag(-1/self.beta)
def _update(self, x, f, dx, df, dx_norm, df_norm):
incr = f*self.last_f > 0
self.beta[incr] += self.alpha
self.beta[~incr] = self.alpha
np.clip(self.beta, 0, self.alphamax, out=self.beta)
#------------------------------------------------------------------------------
# Iterative/Krylov approximated Jacobians
#------------------------------------------------------------------------------
class KrylovJacobian(Jacobian):
r"""
Find a root of a function, using Krylov approximation for inverse Jacobian.
This method is suitable for solving large-scale problems.
Parameters
----------
%(params_basic)s
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_maxiter : int, optional
Parameter to pass to the "inner" Krylov solver: maximum number of
iterations. Iteration will stop after maxiter steps even if the
specified tolerance has not been achieved.
inner_M : LinearOperator or InverseJacobian
Preconditioner for the inner Krylov iteration.
Note that you can use also inverse Jacobians as (adaptive)
preconditioners. For example,
>>> from scipy.optimize.nonlin import BroydenFirst, KrylovJacobian
>>> from scipy.optimize.nonlin import InverseJacobian
>>> jac = BroydenFirst()
>>> kjac = KrylovJacobian(inner_M=InverseJacobian(jac))
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.
outer_k : int, optional
Size of the subspace kept across LGMRES nonlinear iterations.
See `scipy.sparse.linalg.lgmres` for details.
inner_kwargs : kwargs
Keyword parameters for the "inner" Krylov solver
(defined with `method`). Parameter names must start with
the `inner_` prefix which will be stripped before passing on
the inner method. See, e.g., `scipy.sparse.linalg.gmres` for details.
%(params_extra)s
See Also
--------
root : Interface to root finding algorithms for multivariate
functions. See ``method=='krylov'`` in particular.
scipy.sparse.linalg.gmres
scipy.sparse.linalg.lgmres
Notes
-----
This function implements a Newton-Krylov solver. The basic idea is
to compute the inverse of the Jacobian with an iterative Krylov
method. These methods require only evaluating the Jacobian-vector
products, which are conveniently approximated by a finite difference:
.. math:: J v \approx (f(x + \omega*v/|v|) - f(x)) / \omega
Due to the use of iterative matrix inverses, these methods can
deal with large nonlinear problems.
SciPy's `scipy.sparse.linalg` module offers a selection of Krylov
solvers to choose from. The default here is `lgmres`, which is a
variant of restarted GMRES iteration that reuses some of the
information obtained in the previous Newton steps to invert
Jacobians in subsequent steps.
For a review on Newton-Krylov methods, see for example [1]_,
and for the LGMRES sparse inverse method, see [2]_.
References
----------
.. [1] D.A. Knoll and D.E. Keyes, J. Comp. Phys. 193, 357 (2004).
:doi:`10.1016/j.jcp.2003.08.010`
.. [2] A.H. Baker and E.R. Jessup and T. Manteuffel,
SIAM J. Matrix Anal. Appl. 26, 962 (2005).
:doi:`10.1137/S0895479803422014`
Examples
--------
The following functions define a system of nonlinear equations
>>> def fun(x):
... return [x[0] + 0.5 * x[1] - 1.0,
... 0.5 * (x[1] - x[0]) ** 2]
A solution can be obtained as follows.
>>> from scipy import optimize
>>> sol = optimize.newton_krylov(fun, [0, 0])
>>> sol
array([0.66731771, 0.66536458])
"""
def __init__(self, rdiff=None, method='lgmres', inner_maxiter=20,
inner_M=None, outer_k=10, **kw):
self.preconditioner = inner_M
self.rdiff = rdiff
self.method = dict(
bicgstab=scipy.sparse.linalg.bicgstab,
gmres=scipy.sparse.linalg.gmres,
lgmres=scipy.sparse.linalg.lgmres,
cgs=scipy.sparse.linalg.cgs,
minres=scipy.sparse.linalg.minres,
).get(method, method)
self.method_kw = dict(maxiter=inner_maxiter, M=self.preconditioner)
if self.method is scipy.sparse.linalg.gmres:
# Replace GMRES's outer iteration with Newton steps
self.method_kw['restrt'] = inner_maxiter
self.method_kw['maxiter'] = 1
self.method_kw.setdefault('atol', 0)
elif self.method is scipy.sparse.linalg.gcrotmk:
self.method_kw.setdefault('atol', 0)
elif self.method is scipy.sparse.linalg.lgmres:
self.method_kw['outer_k'] = outer_k
# Replace LGMRES's outer iteration with Newton steps
self.method_kw['maxiter'] = 1
# Carry LGMRES's `outer_v` vectors across nonlinear iterations
self.method_kw.setdefault('outer_v', [])
self.method_kw.setdefault('prepend_outer_v', True)
# But don't carry the corresponding Jacobian*v products, in case
# the Jacobian changes a lot in the nonlinear step
#
# XXX: some trust-region inspired ideas might be more efficient...
# See e.g., Brown & Saad. But needs to be implemented separately
# since it's not an inexact Newton method.
self.method_kw.setdefault('store_outer_Av', False)
self.method_kw.setdefault('atol', 0)
for key, value in kw.items():
if not key.startswith('inner_'):
raise ValueError("Unknown parameter %s" % key)
self.method_kw[key[6:]] = value
def _update_diff_step(self):
mx = abs(self.x0).max()
mf = abs(self.f0).max()
self.omega = self.rdiff * max(1, mx) / max(1, mf)
def matvec(self, v):
nv = norm(v)
if nv == 0:
return 0*v
sc = self.omega / nv
r = (self.func(self.x0 + sc*v) - self.f0) / sc
if not np.all(np.isfinite(r)) and np.all(np.isfinite(v)):
raise ValueError('Function returned non-finite results')
return r
def solve(self, rhs, tol=0):
if 'tol' in self.method_kw:
sol, info = self.method(self.op, rhs, **self.method_kw)
else:
sol, info = self.method(self.op, rhs, tol=tol, **self.method_kw)
return sol
def update(self, x, f):
self.x0 = x
self.f0 = f
self._update_diff_step()
# Update also the preconditioner, if possible
if self.preconditioner is not None:
if hasattr(self.preconditioner, 'update'):
self.preconditioner.update(x, f)
def setup(self, x, f, func):
Jacobian.setup(self, x, f, func)
self.x0 = x
self.f0 = f
self.op = scipy.sparse.linalg.aslinearoperator(self)
if self.rdiff is None:
self.rdiff = np.finfo(x.dtype).eps ** (1./2)
self._update_diff_step()
# Setup also the preconditioner, if possible
if self.preconditioner is not None:
if hasattr(self.preconditioner, 'setup'):
self.preconditioner.setup(x, f, func)
#------------------------------------------------------------------------------
# Wrapper functions
#------------------------------------------------------------------------------
def _nonlin_wrapper(name, jac):
"""
Construct a solver wrapper with given name and Jacobian approx.
It inspects the keyword arguments of ``jac.__init__``, and allows to
use the same arguments in the wrapper function, in addition to the
keyword arguments of `nonlin_solve`
"""
signature = _getfullargspec(jac.__init__)
args, varargs, varkw, defaults, kwonlyargs, kwdefaults, _ = signature
kwargs = list(zip(args[-len(defaults):], defaults))
kw_str = ", ".join(["%s=%r" % (k, v) for k, v in kwargs])
if kw_str:
kw_str = ", " + kw_str
kwkw_str = ", ".join(["%s=%s" % (k, k) for k, v in kwargs])
if kwkw_str:
kwkw_str = kwkw_str + ", "
if kwonlyargs:
raise ValueError('Unexpected signature %s' % signature)
# Construct the wrapper function so that its keyword arguments
# are visible in pydoc.help etc.
wrapper = """
def %(name)s(F, xin, iter=None %(kw)s, verbose=False, maxiter=None,
f_tol=None, f_rtol=None, x_tol=None, x_rtol=None,
tol_norm=None, line_search='armijo', callback=None, **kw):
jac = %(jac)s(%(kwkw)s **kw)
return nonlin_solve(F, xin, jac, iter, verbose, maxiter,
f_tol, f_rtol, x_tol, x_rtol, tol_norm, line_search,
callback)
"""
wrapper = wrapper % dict(name=name, kw=kw_str, jac=jac.__name__,
kwkw=kwkw_str)
ns = {}
ns.update(globals())
exec(wrapper, ns)
func = ns[name]
func.__doc__ = jac.__doc__
_set_doc(func)
return func
broyden1 = _nonlin_wrapper('broyden1', BroydenFirst)
broyden2 = _nonlin_wrapper('broyden2', BroydenSecond)
anderson = _nonlin_wrapper('anderson', Anderson)
linearmixing = _nonlin_wrapper('linearmixing', LinearMixing)
diagbroyden = _nonlin_wrapper('diagbroyden', DiagBroyden)
excitingmixing = _nonlin_wrapper('excitingmixing', ExcitingMixing)
newton_krylov = _nonlin_wrapper('newton_krylov', KrylovJacobian)