Fixed database typo and removed unnecessary class identifier.

venv/Lib/site-packages/scipy/optimize/_spectral.py

@@ -0,0 +1,257 @@
+"""
+Spectral Algorithm for Nonlinear Equations
+"""
+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 1-D 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)