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optimize
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test__spectral.py
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import itertools import numpy as np from numpy import exp from numpy.testing import assert_, assert_equal from scipy.optimize import root def test_performance(): # Compare performance results to those listed in # [Cheng & Li, IMA J. Num. An. 29, 814 (2008)] # and # [W. La Cruz, J.M. Martinez, M. Raydan, Math. Comp. 75, 1429 (2006)]. # and those produced by dfsane.f from M. Raydan's website. # # Where the results disagree, the largest limits are taken. e_a = 1e-5 e_r = 1e-4 table_1 = [ dict(F=F_1, x0=x0_1, n=1000, nit=5, nfev=5), dict(F=F_1, x0=x0_1, n=10000, nit=2, nfev=2), dict(F=F_2, x0=x0_2, n=500, nit=11, nfev=11), dict(F=F_2, x0=x0_2, n=2000, nit=11, nfev=11), # dict(F=F_4, x0=x0_4, n=999, nit=243, nfev=1188), removed: too sensitive to rounding errors dict(F=F_6, x0=x0_6, n=100, nit=6, nfev=6), # Results from dfsane.f; papers list nit=3, nfev=3 dict(F=F_7, x0=x0_7, n=99, nit=23, nfev=29), # Must have n%3==0, typo in papers? dict(F=F_7, x0=x0_7, n=999, nit=23, nfev=29), # Must have n%3==0, typo in papers? dict(F=F_9, x0=x0_9, n=100, nit=12, nfev=18), # Results from dfsane.f; papers list nit=nfev=6? dict(F=F_9, x0=x0_9, n=1000, nit=12, nfev=18), dict(F=F_10, x0=x0_10, n=1000, nit=5, nfev=5), # Results from dfsane.f; papers list nit=2, nfev=12 ] # Check also scaling invariance for xscale, yscale, line_search in itertools.product([1.0, 1e-10, 1e10], [1.0, 1e-10, 1e10], ['cruz', 'cheng']): for problem in table_1: n = problem['n'] func = lambda x, n: yscale*problem['F'](x/xscale, n) args = (n,) x0 = problem['x0'](n) * xscale fatol = np.sqrt(n) * e_a * yscale + e_r * np.linalg.norm(func(x0, n)) sigma_eps = 1e-10 * min(yscale/xscale, xscale/yscale) sigma_0 = xscale/yscale with np.errstate(over='ignore'): sol = root(func, x0, args=args, options=dict(ftol=0, fatol=fatol, maxfev=problem['nfev'] + 1, sigma_0=sigma_0, sigma_eps=sigma_eps, line_search=line_search), method='DF-SANE') err_msg = repr([xscale, yscale, line_search, problem, np.linalg.norm(func(sol.x, n)), fatol, sol.success, sol.nit, sol.nfev]) assert_(sol.success, err_msg) assert_(sol.nfev <= problem['nfev'] + 1, err_msg) # nfev+1: dfsane.f doesn't count first eval assert_(sol.nit <= problem['nit'], err_msg) assert_(np.linalg.norm(func(sol.x, n)) <= fatol, err_msg) def test_complex(): def func(z): return z**2 - 1 + 2j x0 = 2.0j ftol = 1e-4 sol = root(func, x0, tol=ftol, method='DF-SANE') assert_(sol.success) f0 = np.linalg.norm(func(x0)) fx = np.linalg.norm(func(sol.x)) assert_(fx <= ftol*f0) def test_linear_definite(): # The DF-SANE paper proves convergence for "strongly isolated" # solutions. # # For linear systems F(x) = A x - b = 0, with A positive or # negative definite, the solution is strongly isolated. def check_solvability(A, b, line_search='cruz'): func = lambda x: A.dot(x) - b xp = np.linalg.solve(A, b) eps = np.linalg.norm(func(xp)) * 1e3 sol = root(func, b, options=dict(fatol=eps, ftol=0, maxfev=17523, line_search=line_search), method='DF-SANE') assert_(sol.success) assert_(np.linalg.norm(func(sol.x)) <= eps) n = 90 # Test linear pos.def. system np.random.seed(1234) A = np.arange(n*n).reshape(n, n) A = A + n*n * np.diag(1 + np.arange(n)) assert_(np.linalg.eigvals(A).min() > 0) b = np.arange(n) * 1.0 check_solvability(A, b, 'cruz') check_solvability(A, b, 'cheng') # Test linear neg.def. system check_solvability(-A, b, 'cruz') check_solvability(-A, b, 'cheng') def test_shape(): def f(x, arg): return x - arg for dt in [float, complex]: x = np.zeros([2,2]) arg = np.ones([2,2], dtype=dt) sol = root(f, x, args=(arg,), method='DF-SANE') assert_(sol.success) assert_equal(sol.x.shape, x.shape) # Some of the test functions and initial guesses listed in # [W. La Cruz, M. Raydan. Optimization Methods and Software, 18, 583 (2003)] def F_1(x, n): g = np.zeros([n]) i = np.arange(2, n+1) g[0] = exp(x[0] - 1) - 1 g[1:] = i*(exp(x[1:] - 1) - x[1:]) return g def x0_1(n): x0 = np.empty([n]) x0.fill(n/(n-1)) return x0 def F_2(x, n): g = np.zeros([n]) i = np.arange(2, n+1) g[0] = exp(x[0]) - 1 g[1:] = 0.1*i*(exp(x[1:]) + x[:-1] - 1) return g def x0_2(n): x0 = np.empty([n]) x0.fill(1/n**2) return x0 def F_4(x, n): assert_equal(n % 3, 0) g = np.zeros([n]) # Note: the first line is typoed in some of the references; # correct in original [Gasparo, Optimization Meth. 13, 79 (2000)] g[::3] = 0.6 * x[::3] + 1.6 * x[1::3]**3 - 7.2 * x[1::3]**2 + 9.6 * x[1::3] - 4.8 g[1::3] = 0.48 * x[::3] - 0.72 * x[1::3]**3 + 3.24 * x[1::3]**2 - 4.32 * x[1::3] - x[2::3] + 0.2 * x[2::3]**3 + 2.16 g[2::3] = 1.25 * x[2::3] - 0.25*x[2::3]**3 return g def x0_4(n): assert_equal(n % 3, 0) x0 = np.array([-1, 1/2, -1] * (n//3)) return x0 def F_6(x, n): c = 0.9 mu = (np.arange(1, n+1) - 0.5)/n return x - 1/(1 - c/(2*n) * (mu[:,None]*x / (mu[:,None] + mu)).sum(axis=1)) def x0_6(n): return np.ones([n]) def F_7(x, n): assert_equal(n % 3, 0) def phi(t): v = 0.5*t - 2 v[t > -1] = ((-592*t**3 + 888*t**2 + 4551*t - 1924)/1998)[t > -1] v[t >= 2] = (0.5*t + 2)[t >= 2] return v g = np.zeros([n]) g[::3] = 1e4 * x[1::3]**2 - 1 g[1::3] = exp(-x[::3]) + exp(-x[1::3]) - 1.0001 g[2::3] = phi(x[2::3]) return g def x0_7(n): assert_equal(n % 3, 0) return np.array([1e-3, 18, 1] * (n//3)) def F_9(x, n): g = np.zeros([n]) i = np.arange(2, n) g[0] = x[0]**3/3 + x[1]**2/2 g[1:-1] = -x[1:-1]**2/2 + i*x[1:-1]**3/3 + x[2:]**2/2 g[-1] = -x[-1]**2/2 + n*x[-1]**3/3 return g def x0_9(n): return np.ones([n]) def F_10(x, n): return np.log(1 + x) - x/n def x0_10(n): return np.ones([n])