EVOLUTION-MANAGER

Edit File: _spectral.py

"""
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)