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