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""" Spectral Algorithm for Nonlinear Equations """
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
""" 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
""" 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
return np.ascontiguousarray(x, dtype=float).view(np.complex128)
return np.ascontiguousarray(z, dtype=complex).view(np.float64) |