1661 lines
50 KiB
Python
1661 lines
50 KiB
Python
r"""
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Nonlinear solvers
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-----------------
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.. currentmodule:: scipy.optimize
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This is a collection of general-purpose nonlinear multidimensional
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solvers. These solvers find *x* for which *F(x) = 0*. Both *x*
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and *F* can be multidimensional.
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Routines
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~~~~~~~~
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Large-scale nonlinear solvers:
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.. autosummary::
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newton_krylov
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anderson
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General nonlinear solvers:
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.. autosummary::
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broyden1
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broyden2
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Simple iterations:
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.. autosummary::
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excitingmixing
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linearmixing
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diagbroyden
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Examples
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~~~~~~~~
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**Small problem**
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>>> def F(x):
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... return np.cos(x) + x[::-1] - [1, 2, 3, 4]
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>>> import scipy.optimize
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>>> x = scipy.optimize.broyden1(F, [1,1,1,1], f_tol=1e-14)
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>>> x
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array([ 4.04674914, 3.91158389, 2.71791677, 1.61756251])
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>>> np.cos(x) + x[::-1]
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array([ 1., 2., 3., 4.])
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**Large problem**
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Suppose that we needed to solve the following integrodifferential
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equation on the square :math:`[0,1]\times[0,1]`:
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.. math::
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\nabla^2 P = 10 \left(\int_0^1\int_0^1\cosh(P)\,dx\,dy\right)^2
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with :math:`P(x,1) = 1` and :math:`P=0` elsewhere on the boundary of
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the square.
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The solution can be found using the `newton_krylov` solver:
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.. plot::
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import numpy as np
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from scipy.optimize import newton_krylov
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from numpy import cosh, zeros_like, mgrid, zeros
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# parameters
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nx, ny = 75, 75
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hx, hy = 1./(nx-1), 1./(ny-1)
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P_left, P_right = 0, 0
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P_top, P_bottom = 1, 0
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def residual(P):
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d2x = zeros_like(P)
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d2y = zeros_like(P)
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d2x[1:-1] = (P[2:] - 2*P[1:-1] + P[:-2]) / hx/hx
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d2x[0] = (P[1] - 2*P[0] + P_left)/hx/hx
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d2x[-1] = (P_right - 2*P[-1] + P[-2])/hx/hx
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d2y[:,1:-1] = (P[:,2:] - 2*P[:,1:-1] + P[:,:-2])/hy/hy
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d2y[:,0] = (P[:,1] - 2*P[:,0] + P_bottom)/hy/hy
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d2y[:,-1] = (P_top - 2*P[:,-1] + P[:,-2])/hy/hy
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return d2x + d2y - 10*cosh(P).mean()**2
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# solve
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guess = zeros((nx, ny), float)
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sol = newton_krylov(residual, guess, method='lgmres', verbose=1)
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print('Residual: %g' % abs(residual(sol)).max())
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# visualize
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import matplotlib.pyplot as plt
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x, y = mgrid[0:1:(nx*1j), 0:1:(ny*1j)]
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plt.pcolormesh(x, y, sol, shading='gouraud')
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plt.colorbar()
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plt.show()
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"""
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# Copyright (C) 2009, Pauli Virtanen <pav@iki.fi>
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# Distributed under the same license as SciPy.
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import sys
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import numpy as np
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from scipy.linalg import norm, solve, inv, qr, svd, LinAlgError
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from numpy import asarray, dot, vdot
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import scipy.sparse.linalg
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import scipy.sparse
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from scipy.linalg import get_blas_funcs
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import inspect
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from scipy._lib._util import getfullargspec_no_self as _getfullargspec
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from .linesearch import scalar_search_wolfe1, scalar_search_armijo
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__all__ = [
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'broyden1', 'broyden2', 'anderson', 'linearmixing',
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'diagbroyden', 'excitingmixing', 'newton_krylov']
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#------------------------------------------------------------------------------
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# Utility functions
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#------------------------------------------------------------------------------
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class NoConvergence(Exception):
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pass
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def maxnorm(x):
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return np.absolute(x).max()
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def _as_inexact(x):
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"""Return `x` as an array, of either floats or complex floats"""
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x = asarray(x)
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if not np.issubdtype(x.dtype, np.inexact):
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return asarray(x, dtype=np.float_)
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return x
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def _array_like(x, x0):
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"""Return ndarray `x` as same array subclass and shape as `x0`"""
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x = np.reshape(x, np.shape(x0))
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wrap = getattr(x0, '__array_wrap__', x.__array_wrap__)
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return wrap(x)
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def _safe_norm(v):
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if not np.isfinite(v).all():
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return np.array(np.inf)
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return norm(v)
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#------------------------------------------------------------------------------
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# Generic nonlinear solver machinery
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#------------------------------------------------------------------------------
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_doc_parts = dict(
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params_basic="""
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F : function(x) -> f
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Function whose root to find; should take and return an array-like
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object.
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xin : array_like
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Initial guess for the solution
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""".strip(),
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params_extra="""
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iter : int, optional
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Number of iterations to make. If omitted (default), make as many
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as required to meet tolerances.
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verbose : bool, optional
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Print status to stdout on every iteration.
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maxiter : int, optional
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Maximum number of iterations to make. If more are needed to
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meet convergence, `NoConvergence` is raised.
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f_tol : float, optional
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Absolute tolerance (in max-norm) for the residual.
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If omitted, default is 6e-6.
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f_rtol : float, optional
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Relative tolerance for the residual. If omitted, not used.
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x_tol : float, optional
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Absolute minimum step size, as determined from the Jacobian
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approximation. If the step size is smaller than this, optimization
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is terminated as successful. If omitted, not used.
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x_rtol : float, optional
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Relative minimum step size. If omitted, not used.
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tol_norm : function(vector) -> scalar, optional
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Norm to use in convergence check. Default is the maximum norm.
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line_search : {None, 'armijo' (default), 'wolfe'}, optional
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Which type of a line search to use to determine the step size in the
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direction given by the Jacobian approximation. Defaults to 'armijo'.
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callback : function, optional
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Optional callback function. It is called on every iteration as
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``callback(x, f)`` where `x` is the current solution and `f`
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the corresponding residual.
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Returns
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-------
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sol : ndarray
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An array (of similar array type as `x0`) containing the final solution.
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Raises
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------
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NoConvergence
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When a solution was not found.
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""".strip()
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)
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def _set_doc(obj):
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if obj.__doc__:
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obj.__doc__ = obj.__doc__ % _doc_parts
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def nonlin_solve(F, x0, jacobian='krylov', iter=None, verbose=False,
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maxiter=None, f_tol=None, f_rtol=None, x_tol=None, x_rtol=None,
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tol_norm=None, line_search='armijo', callback=None,
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full_output=False, raise_exception=True):
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"""
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Find a root of a function, in a way suitable for large-scale problems.
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Parameters
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----------
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%(params_basic)s
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jacobian : Jacobian
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A Jacobian approximation: `Jacobian` object or something that
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`asjacobian` can transform to one. Alternatively, a string specifying
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which of the builtin Jacobian approximations to use:
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krylov, broyden1, broyden2, anderson
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diagbroyden, linearmixing, excitingmixing
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%(params_extra)s
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full_output : bool
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If true, returns a dictionary `info` containing convergence
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information.
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raise_exception : bool
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If True, a `NoConvergence` exception is raise if no solution is found.
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See Also
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--------
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asjacobian, Jacobian
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Notes
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-----
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This algorithm implements the inexact Newton method, with
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backtracking or full line searches. Several Jacobian
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approximations are available, including Krylov and Quasi-Newton
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methods.
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References
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----------
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.. [KIM] C. T. Kelley, \"Iterative Methods for Linear and Nonlinear
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Equations\". Society for Industrial and Applied Mathematics. (1995)
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https://archive.siam.org/books/kelley/fr16/
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"""
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# Can't use default parameters because it's being explicitly passed as None
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# from the calling function, so we need to set it here.
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tol_norm = maxnorm if tol_norm is None else tol_norm
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condition = TerminationCondition(f_tol=f_tol, f_rtol=f_rtol,
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x_tol=x_tol, x_rtol=x_rtol,
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iter=iter, norm=tol_norm)
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x0 = _as_inexact(x0)
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func = lambda z: _as_inexact(F(_array_like(z, x0))).flatten()
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x = x0.flatten()
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dx = np.full_like(x, np.inf)
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Fx = func(x)
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Fx_norm = norm(Fx)
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jacobian = asjacobian(jacobian)
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jacobian.setup(x.copy(), Fx, func)
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if maxiter is None:
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if iter is not None:
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maxiter = iter + 1
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else:
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maxiter = 100*(x.size+1)
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if line_search is True:
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line_search = 'armijo'
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elif line_search is False:
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line_search = None
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if line_search not in (None, 'armijo', 'wolfe'):
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raise ValueError("Invalid line search")
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# Solver tolerance selection
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gamma = 0.9
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eta_max = 0.9999
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eta_treshold = 0.1
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eta = 1e-3
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for n in range(maxiter):
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status = condition.check(Fx, x, dx)
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if status:
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break
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# The tolerance, as computed for scipy.sparse.linalg.* routines
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tol = min(eta, eta*Fx_norm)
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dx = -jacobian.solve(Fx, tol=tol)
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if norm(dx) == 0:
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raise ValueError("Jacobian inversion yielded zero vector. "
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"This indicates a bug in the Jacobian "
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"approximation.")
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# Line search, or Newton step
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if line_search:
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s, x, Fx, Fx_norm_new = _nonlin_line_search(func, x, Fx, dx,
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line_search)
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else:
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s = 1.0
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x = x + dx
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Fx = func(x)
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Fx_norm_new = norm(Fx)
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jacobian.update(x.copy(), Fx)
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if callback:
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callback(x, Fx)
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# Adjust forcing parameters for inexact methods
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eta_A = gamma * Fx_norm_new**2 / Fx_norm**2
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if gamma * eta**2 < eta_treshold:
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eta = min(eta_max, eta_A)
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else:
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eta = min(eta_max, max(eta_A, gamma*eta**2))
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Fx_norm = Fx_norm_new
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# Print status
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if verbose:
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sys.stdout.write("%d: |F(x)| = %g; step %g\n" % (
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n, tol_norm(Fx), s))
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sys.stdout.flush()
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else:
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if raise_exception:
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raise NoConvergence(_array_like(x, x0))
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else:
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status = 2
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if full_output:
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info = {'nit': condition.iteration,
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'fun': Fx,
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'status': status,
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'success': status == 1,
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'message': {1: 'A solution was found at the specified '
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'tolerance.',
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2: 'The maximum number of iterations allowed '
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'has been reached.'
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}[status]
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}
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return _array_like(x, x0), info
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else:
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return _array_like(x, x0)
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_set_doc(nonlin_solve)
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def _nonlin_line_search(func, x, Fx, dx, search_type='armijo', rdiff=1e-8,
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smin=1e-2):
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tmp_s = [0]
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tmp_Fx = [Fx]
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tmp_phi = [norm(Fx)**2]
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s_norm = norm(x) / norm(dx)
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def phi(s, store=True):
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if s == tmp_s[0]:
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return tmp_phi[0]
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xt = x + s*dx
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v = func(xt)
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p = _safe_norm(v)**2
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if store:
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tmp_s[0] = s
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tmp_phi[0] = p
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tmp_Fx[0] = v
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return p
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def derphi(s):
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ds = (abs(s) + s_norm + 1) * rdiff
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return (phi(s+ds, store=False) - phi(s)) / ds
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if search_type == 'wolfe':
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s, phi1, phi0 = scalar_search_wolfe1(phi, derphi, tmp_phi[0],
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xtol=1e-2, amin=smin)
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elif search_type == 'armijo':
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s, phi1 = scalar_search_armijo(phi, tmp_phi[0], -tmp_phi[0],
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amin=smin)
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if s is None:
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# XXX: No suitable step length found. Take the full Newton step,
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# and hope for the best.
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s = 1.0
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x = x + s*dx
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if s == tmp_s[0]:
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Fx = tmp_Fx[0]
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else:
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Fx = func(x)
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Fx_norm = norm(Fx)
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return s, x, Fx, Fx_norm
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class TerminationCondition(object):
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"""
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Termination condition for an iteration. It is terminated if
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- |F| < f_rtol*|F_0|, AND
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- |F| < f_tol
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AND
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- |dx| < x_rtol*|x|, AND
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- |dx| < x_tol
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"""
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def __init__(self, f_tol=None, f_rtol=None, x_tol=None, x_rtol=None,
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iter=None, norm=maxnorm):
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if f_tol is None:
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f_tol = np.finfo(np.float_).eps ** (1./3)
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if f_rtol is None:
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f_rtol = np.inf
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if x_tol is None:
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x_tol = np.inf
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if x_rtol is None:
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x_rtol = np.inf
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self.x_tol = x_tol
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self.x_rtol = x_rtol
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self.f_tol = f_tol
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self.f_rtol = f_rtol
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self.norm = norm
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self.iter = iter
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self.f0_norm = None
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self.iteration = 0
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def check(self, f, x, dx):
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self.iteration += 1
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f_norm = self.norm(f)
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x_norm = self.norm(x)
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dx_norm = self.norm(dx)
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if self.f0_norm is None:
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self.f0_norm = f_norm
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if f_norm == 0:
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return 1
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if self.iter is not None:
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# backwards compatibility with SciPy 0.6.0
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return 2 * (self.iteration > self.iter)
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# NB: condition must succeed for rtol=inf even if norm == 0
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return int((f_norm <= self.f_tol
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and f_norm/self.f_rtol <= self.f0_norm)
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and (dx_norm <= self.x_tol
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and dx_norm/self.x_rtol <= x_norm))
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#------------------------------------------------------------------------------
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# Generic Jacobian approximation
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#------------------------------------------------------------------------------
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class Jacobian(object):
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"""
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Common interface for Jacobians or Jacobian approximations.
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The optional methods come useful when implementing trust region
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etc., algorithms that often require evaluating transposes of the
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Jacobian.
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Methods
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-------
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solve
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Returns J^-1 * v
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update
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Updates Jacobian to point `x` (where the function has residual `Fx`)
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matvec : optional
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Returns J * v
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rmatvec : optional
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Returns A^H * v
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rsolve : optional
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Returns A^-H * v
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matmat : optional
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Returns A * V, where V is a dense matrix with dimensions (N,K).
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todense : optional
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Form the dense Jacobian matrix. Necessary for dense trust region
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algorithms, and useful for testing.
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Attributes
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----------
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shape
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Matrix dimensions (M, N)
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dtype
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Data type of the matrix.
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func : callable, optional
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Function the Jacobian corresponds to
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"""
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def __init__(self, **kw):
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names = ["solve", "update", "matvec", "rmatvec", "rsolve",
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"matmat", "todense", "shape", "dtype"]
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for name, value in kw.items():
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if name not in names:
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raise ValueError("Unknown keyword argument %s" % name)
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if value is not None:
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setattr(self, name, kw[name])
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if hasattr(self, 'todense'):
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self.__array__ = lambda: self.todense()
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def aspreconditioner(self):
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return InverseJacobian(self)
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|
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def solve(self, v, tol=0):
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raise NotImplementedError
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def update(self, x, F):
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pass
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|
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def setup(self, x, F, func):
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self.func = func
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self.shape = (F.size, x.size)
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self.dtype = F.dtype
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if self.__class__.setup is Jacobian.setup:
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# Call on the first point unless overridden
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self.update(x, F)
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|
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)
|