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