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Python

"""
Spectral Algorithm for Nonlinear Equations
"""
from __future__ import division, absolute_import, print_function
import collections
import numpy as np
from scipy.optimize import OptimizeResult
from scipy.optimize.optimize import _check_unknown_options
from .linesearch import _nonmonotone_line_search_cruz, _nonmonotone_line_search_cheng
class _NoConvergence(Exception):
pass
def _root_df_sane(func, x0, args=(), ftol=1e-8, fatol=1e-300, maxfev=1000,
fnorm=None, callback=None, disp=False, M=10, eta_strategy=None,
sigma_eps=1e-10, sigma_0=1.0, line_search='cruz', **unknown_options):
r"""
Solve nonlinear equation with the DF-SANE method
Options
-------
ftol : float, optional
Relative norm tolerance.
fatol : float, optional
Absolute norm tolerance.
Algorithm terminates when ``||func(x)|| < fatol + ftol ||func(x_0)||``.
fnorm : callable, optional
Norm to use in the convergence check. If None, 2-norm is used.
maxfev : int, optional
Maximum number of function evaluations.
disp : bool, optional
Whether to print convergence process to stdout.
eta_strategy : callable, optional
Choice of the ``eta_k`` parameter, which gives slack for growth
of ``||F||**2``. Called as ``eta_k = eta_strategy(k, x, F)`` with
`k` the iteration number, `x` the current iterate and `F` the current
residual. Should satisfy ``eta_k > 0`` and ``sum(eta, k=0..inf) < inf``.
Default: ``||F||**2 / (1 + k)**2``.
sigma_eps : float, optional
The spectral coefficient is constrained to ``sigma_eps < sigma < 1/sigma_eps``.
Default: 1e-10
sigma_0 : float, optional
Initial spectral coefficient.
Default: 1.0
M : int, optional
Number of iterates to include in the nonmonotonic line search.
Default: 10
line_search : {'cruz', 'cheng'}
Type of line search to employ. 'cruz' is the original one defined in
[Martinez & Raydan. Math. Comp. 75, 1429 (2006)], 'cheng' is
a modified search defined in [Cheng & Li. IMA J. Numer. Anal. 29, 814 (2009)].
Default: 'cruz'
References
----------
.. [1] "Spectral residual method without gradient information for solving
large-scale nonlinear systems of equations." W. La Cruz,
J.M. Martinez, M. Raydan. Math. Comp. **75**, 1429 (2006).
.. [2] W. La Cruz, Opt. Meth. Software, 29, 24 (2014).
.. [3] W. Cheng, D.-H. Li. IMA J. Numer. Anal. **29**, 814 (2009).
"""
_check_unknown_options(unknown_options)
if line_search not in ('cheng', 'cruz'):
raise ValueError("Invalid value %r for 'line_search'" % (line_search,))
nexp = 2
if eta_strategy is None:
# Different choice from [1], as their eta is not invariant
# vs. scaling of F.
def eta_strategy(k, x, F):
# Obtain squared 2-norm of the initial residual from the outer scope
return f_0 / (1 + k)**2
if fnorm is None:
def fnorm(F):
# Obtain squared 2-norm of the current residual from the outer scope
return f_k**(1.0/nexp)
def fmerit(F):
return np.linalg.norm(F)**nexp
nfev = [0]
f, x_k, x_shape, f_k, F_k, is_complex = _wrap_func(func, x0, fmerit, nfev, maxfev, args)
k = 0
f_0 = f_k
sigma_k = sigma_0
F_0_norm = fnorm(F_k)
# For the 'cruz' line search
prev_fs = collections.deque([f_k], M)
# For the 'cheng' line search
Q = 1.0
C = f_0
converged = False
message = "too many function evaluations required"
while True:
F_k_norm = fnorm(F_k)
if disp:
print("iter %d: ||F|| = %g, sigma = %g" % (k, F_k_norm, sigma_k))
if callback is not None:
callback(x_k, F_k)
if F_k_norm < ftol * F_0_norm + fatol:
# Converged!
message = "successful convergence"
converged = True
break
# Control spectral parameter, from [2]
if abs(sigma_k) > 1/sigma_eps:
sigma_k = 1/sigma_eps * np.sign(sigma_k)
elif abs(sigma_k) < sigma_eps:
sigma_k = sigma_eps
# Line search direction
d = -sigma_k * F_k
# Nonmonotone line search
eta = eta_strategy(k, x_k, F_k)
try:
if line_search == 'cruz':
alpha, xp, fp, Fp = _nonmonotone_line_search_cruz(f, x_k, d, prev_fs, eta=eta)
elif line_search == 'cheng':
alpha, xp, fp, Fp, C, Q = _nonmonotone_line_search_cheng(f, x_k, d, f_k, C, Q, eta=eta)
except _NoConvergence:
break
# Update spectral parameter
s_k = xp - x_k
y_k = Fp - F_k
sigma_k = np.vdot(s_k, s_k) / np.vdot(s_k, y_k)
# Take step
x_k = xp
F_k = Fp
f_k = fp
# Store function value
if line_search == 'cruz':
prev_fs.append(fp)
k += 1
x = _wrap_result(x_k, is_complex, shape=x_shape)
F = _wrap_result(F_k, is_complex)
result = OptimizeResult(x=x, success=converged,
message=message,
fun=F, nfev=nfev[0], nit=k)
return result
def _wrap_func(func, x0, fmerit, nfev_list, maxfev, args=()):
"""
Wrap a function and an initial value so that (i) complex values
are wrapped to reals, and (ii) value for a merit function
fmerit(x, f) is computed at the same time, (iii) iteration count
is maintained and an exception is raised if it is exceeded.
Parameters
----------
func : callable
Function to wrap
x0 : ndarray
Initial value
fmerit : callable
Merit function fmerit(f) for computing merit value from residual.
nfev_list : list
List to store number of evaluations in. Should be [0] in the beginning.
maxfev : int
Maximum number of evaluations before _NoConvergence is raised.
args : tuple
Extra arguments to func
Returns
-------
wrap_func : callable
Wrapped function, to be called as
``F, fp = wrap_func(x0)``
x0_wrap : ndarray of float
Wrapped initial value; raveled to 1D and complex
values mapped to reals.
x0_shape : tuple
Shape of the initial value array
f : float
Merit function at F
F : ndarray of float
Residual at x0_wrap
is_complex : bool
Whether complex values were mapped to reals
"""
x0 = np.asarray(x0)
x0_shape = x0.shape
F = np.asarray(func(x0, *args)).ravel()
is_complex = np.iscomplexobj(x0) or np.iscomplexobj(F)
x0 = x0.ravel()
nfev_list[0] = 1
if is_complex:
def wrap_func(x):
if nfev_list[0] >= maxfev:
raise _NoConvergence()
nfev_list[0] += 1
z = _real2complex(x).reshape(x0_shape)
v = np.asarray(func(z, *args)).ravel()
F = _complex2real(v)
f = fmerit(F)
return f, F
x0 = _complex2real(x0)
F = _complex2real(F)
else:
def wrap_func(x):
if nfev_list[0] >= maxfev:
raise _NoConvergence()
nfev_list[0] += 1
x = x.reshape(x0_shape)
F = np.asarray(func(x, *args)).ravel()
f = fmerit(F)
return f, F
return wrap_func, x0, x0_shape, fmerit(F), F, is_complex
def _wrap_result(result, is_complex, shape=None):
"""
Convert from real to complex and reshape result arrays.
"""
if is_complex:
z = _real2complex(result)
else:
z = result
if shape is not None:
z = z.reshape(shape)
return z
def _real2complex(x):
return np.ascontiguousarray(x, dtype=float).view(np.complex128)
def _complex2real(z):
return np.ascontiguousarray(z, dtype=complex).view(np.float64)