671 lines
24 KiB
Python
671 lines
24 KiB
Python
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"""
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Unit tests for optimization routines from minpack.py.
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"""
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from __future__ import division, print_function, absolute_import
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from numpy.testing import (assert_, assert_almost_equal, assert_array_equal,
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assert_array_almost_equal, assert_allclose)
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from pytest import raises as assert_raises
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import numpy as np
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from numpy import array, float64, matrix
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from scipy import optimize
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from scipy.special import lambertw
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from scipy.optimize.minpack import leastsq, curve_fit, fixed_point
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from scipy._lib._numpy_compat import _assert_warns, suppress_warnings
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from scipy.optimize import OptimizeWarning
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class ReturnShape(object):
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"""This class exists to create a callable that does not have a '__name__' attribute.
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__init__ takes the argument 'shape', which should be a tuple of ints. When an instance
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it called with a single argument 'x', it returns numpy.ones(shape).
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"""
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def __init__(self, shape):
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self.shape = shape
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def __call__(self, x):
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return np.ones(self.shape)
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def dummy_func(x, shape):
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"""A function that returns an array of ones of the given shape.
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`x` is ignored.
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"""
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return np.ones(shape)
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# Function and jacobian for tests of solvers for systems of nonlinear
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# equations
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def pressure_network(flow_rates, Qtot, k):
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"""Evaluate non-linear equation system representing
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the pressures and flows in a system of n parallel pipes::
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f_i = P_i - P_0, for i = 1..n
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f_0 = sum(Q_i) - Qtot
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Where Q_i is the flow rate in pipe i and P_i the pressure in that pipe.
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Pressure is modeled as a P=kQ**2 where k is a valve coefficient and
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Q is the flow rate.
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Parameters
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----------
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flow_rates : float
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A 1D array of n flow rates [kg/s].
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k : float
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A 1D array of n valve coefficients [1/kg m].
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Qtot : float
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A scalar, the total input flow rate [kg/s].
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Returns
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-------
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F : float
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A 1D array, F[i] == f_i.
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"""
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P = k * flow_rates**2
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F = np.hstack((P[1:] - P[0], flow_rates.sum() - Qtot))
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return F
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def pressure_network_jacobian(flow_rates, Qtot, k):
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"""Return the jacobian of the equation system F(flow_rates)
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computed by `pressure_network` with respect to
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*flow_rates*. See `pressure_network` for the detailed
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description of parrameters.
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Returns
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-------
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jac : float
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*n* by *n* matrix ``df_i/dQ_i`` where ``n = len(flow_rates)``
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and *f_i* and *Q_i* are described in the doc for `pressure_network`
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"""
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n = len(flow_rates)
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pdiff = np.diag(flow_rates[1:] * 2 * k[1:] - 2 * flow_rates[0] * k[0])
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jac = np.empty((n, n))
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jac[:n-1, :n-1] = pdiff * 0
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jac[:n-1, n-1] = 0
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jac[n-1, :] = np.ones(n)
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return jac
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def pressure_network_fun_and_grad(flow_rates, Qtot, k):
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return (pressure_network(flow_rates, Qtot, k),
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pressure_network_jacobian(flow_rates, Qtot, k))
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class TestFSolve(object):
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def test_pressure_network_no_gradient(self):
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# fsolve without gradient, equal pipes -> equal flows.
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k = np.ones(4) * 0.5
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Qtot = 4
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initial_guess = array([2., 0., 2., 0.])
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final_flows, info, ier, mesg = optimize.fsolve(
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pressure_network, initial_guess, args=(Qtot, k),
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full_output=True)
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assert_array_almost_equal(final_flows, np.ones(4))
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assert_(ier == 1, mesg)
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def test_pressure_network_with_gradient(self):
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# fsolve with gradient, equal pipes -> equal flows
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k = np.ones(4) * 0.5
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Qtot = 4
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initial_guess = array([2., 0., 2., 0.])
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final_flows = optimize.fsolve(
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pressure_network, initial_guess, args=(Qtot, k),
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fprime=pressure_network_jacobian)
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assert_array_almost_equal(final_flows, np.ones(4))
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def test_wrong_shape_func_callable(self):
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func = ReturnShape(1)
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# x0 is a list of two elements, but func will return an array with
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# length 1, so this should result in a TypeError.
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x0 = [1.5, 2.0]
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assert_raises(TypeError, optimize.fsolve, func, x0)
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def test_wrong_shape_func_function(self):
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# x0 is a list of two elements, but func will return an array with
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# length 1, so this should result in a TypeError.
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x0 = [1.5, 2.0]
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assert_raises(TypeError, optimize.fsolve, dummy_func, x0, args=((1,),))
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def test_wrong_shape_fprime_callable(self):
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func = ReturnShape(1)
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deriv_func = ReturnShape((2,2))
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assert_raises(TypeError, optimize.fsolve, func, x0=[0,1], fprime=deriv_func)
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def test_wrong_shape_fprime_function(self):
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func = lambda x: dummy_func(x, (2,))
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deriv_func = lambda x: dummy_func(x, (3,3))
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assert_raises(TypeError, optimize.fsolve, func, x0=[0,1], fprime=deriv_func)
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def test_float32(self):
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func = lambda x: np.array([x[0] - 100, x[1] - 1000], dtype=np.float32)**2
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p = optimize.fsolve(func, np.array([1, 1], np.float32))
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assert_allclose(func(p), [0, 0], atol=1e-3)
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class TestRootHybr(object):
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def test_pressure_network_no_gradient(self):
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# root/hybr without gradient, equal pipes -> equal flows
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k = np.ones(4) * 0.5
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Qtot = 4
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initial_guess = array([2., 0., 2., 0.])
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final_flows = optimize.root(pressure_network, initial_guess,
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method='hybr', args=(Qtot, k)).x
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assert_array_almost_equal(final_flows, np.ones(4))
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def test_pressure_network_with_gradient(self):
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# root/hybr with gradient, equal pipes -> equal flows
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k = np.ones(4) * 0.5
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Qtot = 4
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initial_guess = matrix([2., 0., 2., 0.])
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final_flows = optimize.root(pressure_network, initial_guess,
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args=(Qtot, k), method='hybr',
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jac=pressure_network_jacobian).x
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assert_array_almost_equal(final_flows, np.ones(4))
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def test_pressure_network_with_gradient_combined(self):
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# root/hybr with gradient and function combined, equal pipes -> equal
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# flows
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k = np.ones(4) * 0.5
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Qtot = 4
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initial_guess = array([2., 0., 2., 0.])
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final_flows = optimize.root(pressure_network_fun_and_grad,
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initial_guess, args=(Qtot, k),
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method='hybr', jac=True).x
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assert_array_almost_equal(final_flows, np.ones(4))
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class TestRootLM(object):
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def test_pressure_network_no_gradient(self):
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# root/lm without gradient, equal pipes -> equal flows
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k = np.ones(4) * 0.5
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Qtot = 4
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initial_guess = array([2., 0., 2., 0.])
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final_flows = optimize.root(pressure_network, initial_guess,
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method='lm', args=(Qtot, k)).x
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assert_array_almost_equal(final_flows, np.ones(4))
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class TestLeastSq(object):
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def setup_method(self):
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x = np.linspace(0, 10, 40)
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a,b,c = 3.1, 42, -304.2
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self.x = x
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self.abc = a,b,c
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y_true = a*x**2 + b*x + c
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np.random.seed(0)
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self.y_meas = y_true + 0.01*np.random.standard_normal(y_true.shape)
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def residuals(self, p, y, x):
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a,b,c = p
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err = y-(a*x**2 + b*x + c)
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return err
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def test_basic(self):
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p0 = array([0,0,0])
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params_fit, ier = leastsq(self.residuals, p0,
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args=(self.y_meas, self.x))
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assert_(ier in (1,2,3,4), 'solution not found (ier=%d)' % ier)
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# low precision due to random
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assert_array_almost_equal(params_fit, self.abc, decimal=2)
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def test_full_output(self):
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p0 = matrix([0,0,0])
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full_output = leastsq(self.residuals, p0,
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args=(self.y_meas, self.x),
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full_output=True)
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params_fit, cov_x, infodict, mesg, ier = full_output
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assert_(ier in (1,2,3,4), 'solution not found: %s' % mesg)
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def test_input_untouched(self):
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p0 = array([0,0,0],dtype=float64)
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p0_copy = array(p0, copy=True)
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full_output = leastsq(self.residuals, p0,
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args=(self.y_meas, self.x),
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full_output=True)
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params_fit, cov_x, infodict, mesg, ier = full_output
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assert_(ier in (1,2,3,4), 'solution not found: %s' % mesg)
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assert_array_equal(p0, p0_copy)
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def test_wrong_shape_func_callable(self):
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func = ReturnShape(1)
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# x0 is a list of two elements, but func will return an array with
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# length 1, so this should result in a TypeError.
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x0 = [1.5, 2.0]
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assert_raises(TypeError, optimize.leastsq, func, x0)
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def test_wrong_shape_func_function(self):
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# x0 is a list of two elements, but func will return an array with
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# length 1, so this should result in a TypeError.
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x0 = [1.5, 2.0]
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assert_raises(TypeError, optimize.leastsq, dummy_func, x0, args=((1,),))
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def test_wrong_shape_Dfun_callable(self):
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func = ReturnShape(1)
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deriv_func = ReturnShape((2,2))
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assert_raises(TypeError, optimize.leastsq, func, x0=[0,1], Dfun=deriv_func)
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def test_wrong_shape_Dfun_function(self):
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func = lambda x: dummy_func(x, (2,))
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deriv_func = lambda x: dummy_func(x, (3,3))
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assert_raises(TypeError, optimize.leastsq, func, x0=[0,1], Dfun=deriv_func)
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def test_float32(self):
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# Regression test for gh-1447
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def func(p,x,y):
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q = p[0]*np.exp(-(x-p[1])**2/(2.0*p[2]**2))+p[3]
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return q - y
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x = np.array([1.475,1.429,1.409,1.419,1.455,1.519,1.472, 1.368,1.286,
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1.231], dtype=np.float32)
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y = np.array([0.0168,0.0193,0.0211,0.0202,0.0171,0.0151,0.0185,0.0258,
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0.034,0.0396], dtype=np.float32)
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p0 = np.array([1.0,1.0,1.0,1.0])
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p1, success = optimize.leastsq(func, p0, args=(x,y))
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assert_(success in [1,2,3,4])
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assert_((func(p1,x,y)**2).sum() < 1e-4 * (func(p0,x,y)**2).sum())
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class TestCurveFit(object):
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def setup_method(self):
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self.y = array([1.0, 3.2, 9.5, 13.7])
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self.x = array([1.0, 2.0, 3.0, 4.0])
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def test_one_argument(self):
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def func(x,a):
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return x**a
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popt, pcov = curve_fit(func, self.x, self.y)
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assert_(len(popt) == 1)
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assert_(pcov.shape == (1,1))
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assert_almost_equal(popt[0], 1.9149, decimal=4)
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assert_almost_equal(pcov[0,0], 0.0016, decimal=4)
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# Test if we get the same with full_output. Regression test for #1415.
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res = curve_fit(func, self.x, self.y, full_output=1)
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(popt2, pcov2, infodict, errmsg, ier) = res
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assert_array_almost_equal(popt, popt2)
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def test_two_argument(self):
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def func(x, a, b):
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return b*x**a
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popt, pcov = curve_fit(func, self.x, self.y)
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assert_(len(popt) == 2)
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assert_(pcov.shape == (2,2))
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assert_array_almost_equal(popt, [1.7989, 1.1642], decimal=4)
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assert_array_almost_equal(pcov, [[0.0852, -0.1260], [-0.1260, 0.1912]],
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decimal=4)
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def test_func_is_classmethod(self):
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class test_self(object):
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"""This class tests if curve_fit passes the correct number of
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arguments when the model function is a class instance method.
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"""
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def func(self, x, a, b):
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return b * x**a
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test_self_inst = test_self()
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popt, pcov = curve_fit(test_self_inst.func, self.x, self.y)
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assert_(pcov.shape == (2,2))
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assert_array_almost_equal(popt, [1.7989, 1.1642], decimal=4)
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assert_array_almost_equal(pcov, [[0.0852, -0.1260], [-0.1260, 0.1912]],
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decimal=4)
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def test_regression_2639(self):
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# This test fails if epsfcn in leastsq is too large.
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x = [574.14200000000005, 574.154, 574.16499999999996,
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574.17700000000002, 574.18799999999999, 574.19899999999996,
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574.21100000000001, 574.22199999999998, 574.23400000000004,
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574.245]
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y = [859.0, 997.0, 1699.0, 2604.0, 2013.0, 1964.0, 2435.0,
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1550.0, 949.0, 841.0]
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guess = [574.1861428571428, 574.2155714285715, 1302.0, 1302.0,
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0.0035019999999983615, 859.0]
|
||
|
|
good = [5.74177150e+02, 5.74209188e+02, 1.74187044e+03, 1.58646166e+03,
|
||
|
|
1.0068462e-02, 8.57450661e+02]
|
||
|
|
|
||
|
|
def f_double_gauss(x, x0, x1, A0, A1, sigma, c):
|
||
|
|
return (A0*np.exp(-(x-x0)**2/(2.*sigma**2))
|
||
|
|
+ A1*np.exp(-(x-x1)**2/(2.*sigma**2)) + c)
|
||
|
|
popt, pcov = curve_fit(f_double_gauss, x, y, guess, maxfev=10000)
|
||
|
|
assert_allclose(popt, good, rtol=1e-5)
|
||
|
|
|
||
|
|
def test_pcov(self):
|
||
|
|
xdata = np.array([0, 1, 2, 3, 4, 5])
|
||
|
|
ydata = np.array([1, 1, 5, 7, 8, 12])
|
||
|
|
sigma = np.array([1, 2, 1, 2, 1, 2])
|
||
|
|
|
||
|
|
def f(x, a, b):
|
||
|
|
return a*x + b
|
||
|
|
|
||
|
|
for method in ['lm', 'trf', 'dogbox']:
|
||
|
|
popt, pcov = curve_fit(f, xdata, ydata, p0=[2, 0], sigma=sigma,
|
||
|
|
method=method)
|
||
|
|
perr_scaled = np.sqrt(np.diag(pcov))
|
||
|
|
assert_allclose(perr_scaled, [0.20659803, 0.57204404], rtol=1e-3)
|
||
|
|
|
||
|
|
popt, pcov = curve_fit(f, xdata, ydata, p0=[2, 0], sigma=3*sigma,
|
||
|
|
method=method)
|
||
|
|
perr_scaled = np.sqrt(np.diag(pcov))
|
||
|
|
assert_allclose(perr_scaled, [0.20659803, 0.57204404], rtol=1e-3)
|
||
|
|
|
||
|
|
popt, pcov = curve_fit(f, xdata, ydata, p0=[2, 0], sigma=sigma,
|
||
|
|
absolute_sigma=True, method=method)
|
||
|
|
perr = np.sqrt(np.diag(pcov))
|
||
|
|
assert_allclose(perr, [0.30714756, 0.85045308], rtol=1e-3)
|
||
|
|
|
||
|
|
popt, pcov = curve_fit(f, xdata, ydata, p0=[2, 0], sigma=3*sigma,
|
||
|
|
absolute_sigma=True, method=method)
|
||
|
|
perr = np.sqrt(np.diag(pcov))
|
||
|
|
assert_allclose(perr, [3*0.30714756, 3*0.85045308], rtol=1e-3)
|
||
|
|
|
||
|
|
# infinite variances
|
||
|
|
|
||
|
|
def f_flat(x, a, b):
|
||
|
|
return a*x
|
||
|
|
|
||
|
|
pcov_expected = np.array([np.inf]*4).reshape(2, 2)
|
||
|
|
|
||
|
|
with suppress_warnings() as sup:
|
||
|
|
sup.filter(OptimizeWarning,
|
||
|
|
"Covariance of the parameters could not be estimated")
|
||
|
|
popt, pcov = curve_fit(f_flat, xdata, ydata, p0=[2, 0], sigma=sigma)
|
||
|
|
popt1, pcov1 = curve_fit(f, xdata[:2], ydata[:2], p0=[2, 0])
|
||
|
|
|
||
|
|
assert_(pcov.shape == (2, 2))
|
||
|
|
assert_array_equal(pcov, pcov_expected)
|
||
|
|
|
||
|
|
assert_(pcov1.shape == (2, 2))
|
||
|
|
assert_array_equal(pcov1, pcov_expected)
|
||
|
|
|
||
|
|
def test_array_like(self):
|
||
|
|
# Test sequence input. Regression test for gh-3037.
|
||
|
|
def f_linear(x, a, b):
|
||
|
|
return a*x + b
|
||
|
|
|
||
|
|
x = [1, 2, 3, 4]
|
||
|
|
y = [3, 5, 7, 9]
|
||
|
|
assert_allclose(curve_fit(f_linear, x, y)[0], [2, 1], atol=1e-10)
|
||
|
|
|
||
|
|
def test_indeterminate_covariance(self):
|
||
|
|
# Test that a warning is returned when pcov is indeterminate
|
||
|
|
xdata = np.array([1, 2, 3, 4, 5, 6])
|
||
|
|
ydata = np.array([1, 2, 3, 4, 5.5, 6])
|
||
|
|
_assert_warns(OptimizeWarning, curve_fit,
|
||
|
|
lambda x, a, b: a*x, xdata, ydata)
|
||
|
|
|
||
|
|
def test_NaN_handling(self):
|
||
|
|
# Test for correct handling of NaNs in input data: gh-3422
|
||
|
|
|
||
|
|
# create input with NaNs
|
||
|
|
xdata = np.array([1, np.nan, 3])
|
||
|
|
ydata = np.array([1, 2, 3])
|
||
|
|
|
||
|
|
assert_raises(ValueError, curve_fit,
|
||
|
|
lambda x, a, b: a*x + b, xdata, ydata)
|
||
|
|
assert_raises(ValueError, curve_fit,
|
||
|
|
lambda x, a, b: a*x + b, ydata, xdata)
|
||
|
|
|
||
|
|
assert_raises(ValueError, curve_fit, lambda x, a, b: a*x + b,
|
||
|
|
xdata, ydata, **{"check_finite": True})
|
||
|
|
|
||
|
|
def test_method_argument(self):
|
||
|
|
def f(x, a, b):
|
||
|
|
return a * np.exp(-b*x)
|
||
|
|
|
||
|
|
xdata = np.linspace(0, 1, 11)
|
||
|
|
ydata = f(xdata, 2., 2.)
|
||
|
|
|
||
|
|
for method in ['trf', 'dogbox', 'lm', None]:
|
||
|
|
popt, pcov = curve_fit(f, xdata, ydata, method=method)
|
||
|
|
assert_allclose(popt, [2., 2.])
|
||
|
|
|
||
|
|
assert_raises(ValueError, curve_fit, f, xdata, ydata, method='unknown')
|
||
|
|
|
||
|
|
def test_bounds(self):
|
||
|
|
def f(x, a, b):
|
||
|
|
return a * np.exp(-b*x)
|
||
|
|
|
||
|
|
xdata = np.linspace(0, 1, 11)
|
||
|
|
ydata = f(xdata, 2., 2.)
|
||
|
|
|
||
|
|
# The minimum w/out bounds is at [2., 2.],
|
||
|
|
# and with bounds it's at [1.5, smth].
|
||
|
|
bounds = ([1., 0], [1.5, 3.])
|
||
|
|
for method in [None, 'trf', 'dogbox']:
|
||
|
|
popt, pcov = curve_fit(f, xdata, ydata, bounds=bounds,
|
||
|
|
method=method)
|
||
|
|
assert_allclose(popt[0], 1.5)
|
||
|
|
|
||
|
|
# With bounds, the starting estimate is feasible.
|
||
|
|
popt, pcov = curve_fit(f, xdata, ydata, method='trf',
|
||
|
|
bounds=([0., 0], [0.6, np.inf]))
|
||
|
|
assert_allclose(popt[0], 0.6)
|
||
|
|
|
||
|
|
# method='lm' doesn't support bounds.
|
||
|
|
assert_raises(ValueError, curve_fit, f, xdata, ydata, bounds=bounds,
|
||
|
|
method='lm')
|
||
|
|
|
||
|
|
def test_bounds_p0(self):
|
||
|
|
# This test is for issue #5719. The problem was that an initial guess
|
||
|
|
# was ignored when 'trf' or 'dogbox' methods were invoked.
|
||
|
|
def f(x, a):
|
||
|
|
return np.sin(x + a)
|
||
|
|
|
||
|
|
xdata = np.linspace(-2*np.pi, 2*np.pi, 40)
|
||
|
|
ydata = np.sin(xdata)
|
||
|
|
bounds = (-3 * np.pi, 3 * np.pi)
|
||
|
|
for method in ['trf', 'dogbox']:
|
||
|
|
popt_1, _ = curve_fit(f, xdata, ydata, p0=2.1*np.pi)
|
||
|
|
popt_2, _ = curve_fit(f, xdata, ydata, p0=2.1*np.pi,
|
||
|
|
bounds=bounds, method=method)
|
||
|
|
|
||
|
|
# If the initial guess is ignored, then popt_2 would be close 0.
|
||
|
|
assert_allclose(popt_1, popt_2)
|
||
|
|
|
||
|
|
def test_jac(self):
|
||
|
|
# Test that Jacobian callable is handled correctly and
|
||
|
|
# weighted if sigma is provided.
|
||
|
|
def f(x, a, b):
|
||
|
|
return a * np.exp(-b*x)
|
||
|
|
|
||
|
|
def jac(x, a, b):
|
||
|
|
e = np.exp(-b*x)
|
||
|
|
return np.vstack((e, -a * x * e)).T
|
||
|
|
|
||
|
|
xdata = np.linspace(0, 1, 11)
|
||
|
|
ydata = f(xdata, 2., 2.)
|
||
|
|
|
||
|
|
# Test numerical options for least_squares backend.
|
||
|
|
for method in ['trf', 'dogbox']:
|
||
|
|
for scheme in ['2-point', '3-point', 'cs']:
|
||
|
|
popt, pcov = curve_fit(f, xdata, ydata, jac=scheme,
|
||
|
|
method=method)
|
||
|
|
assert_allclose(popt, [2, 2])
|
||
|
|
|
||
|
|
# Test the analytic option.
|
||
|
|
for method in ['lm', 'trf', 'dogbox']:
|
||
|
|
popt, pcov = curve_fit(f, xdata, ydata, method=method, jac=jac)
|
||
|
|
assert_allclose(popt, [2, 2])
|
||
|
|
|
||
|
|
# Now add an outlier and provide sigma.
|
||
|
|
ydata[5] = 100
|
||
|
|
sigma = np.ones(xdata.shape[0])
|
||
|
|
sigma[5] = 200
|
||
|
|
for method in ['lm', 'trf', 'dogbox']:
|
||
|
|
popt, pcov = curve_fit(f, xdata, ydata, sigma=sigma, method=method,
|
||
|
|
jac=jac)
|
||
|
|
# Still the optimization process is influenced somehow,
|
||
|
|
# have to set rtol=1e-3.
|
||
|
|
assert_allclose(popt, [2, 2], rtol=1e-3)
|
||
|
|
|
||
|
|
def test_maxfev_and_bounds(self):
|
||
|
|
# gh-6340: with no bounds, curve_fit accepts parameter maxfev (via leastsq)
|
||
|
|
# but with bounds, the parameter is `max_nfev` (via least_squares)
|
||
|
|
x = np.arange(0, 10)
|
||
|
|
y = 2*x
|
||
|
|
popt1, _ = curve_fit(lambda x,p: p*x, x, y, bounds=(0, 3), maxfev=100)
|
||
|
|
popt2, _ = curve_fit(lambda x,p: p*x, x, y, bounds=(0, 3), max_nfev=100)
|
||
|
|
|
||
|
|
assert_allclose(popt1, 2, atol=1e-14)
|
||
|
|
assert_allclose(popt2, 2, atol=1e-14)
|
||
|
|
|
||
|
|
def test_curvefit_simplecovariance(self):
|
||
|
|
|
||
|
|
def func(x, a, b):
|
||
|
|
return a * np.exp(-b*x)
|
||
|
|
|
||
|
|
def jac(x, a, b):
|
||
|
|
e = np.exp(-b*x)
|
||
|
|
return np.vstack((e, -a * x * e)).T
|
||
|
|
|
||
|
|
np.random.seed(0)
|
||
|
|
xdata = np.linspace(0, 4, 50)
|
||
|
|
y = func(xdata, 2.5, 1.3)
|
||
|
|
ydata = y + 0.2 * np.random.normal(size=len(xdata))
|
||
|
|
|
||
|
|
sigma = np.zeros(len(xdata)) + 0.2
|
||
|
|
covar = np.diag(sigma**2)
|
||
|
|
|
||
|
|
for jac1, jac2 in [(jac, jac), (None, None)]:
|
||
|
|
for absolute_sigma in [False, True]:
|
||
|
|
popt1, pcov1 = curve_fit(func, xdata, ydata, sigma=sigma,
|
||
|
|
jac=jac1, absolute_sigma=absolute_sigma)
|
||
|
|
popt2, pcov2 = curve_fit(func, xdata, ydata, sigma=covar,
|
||
|
|
jac=jac2, absolute_sigma=absolute_sigma)
|
||
|
|
|
||
|
|
assert_allclose(popt1, popt2, atol=1e-14)
|
||
|
|
assert_allclose(pcov1, pcov2, atol=1e-14)
|
||
|
|
|
||
|
|
def test_curvefit_covariance(self):
|
||
|
|
|
||
|
|
def funcp(x, a, b):
|
||
|
|
rotn = np.array([[1./np.sqrt(2), -1./np.sqrt(2), 0], [1./np.sqrt(2), 1./np.sqrt(2), 0], [0, 0, 1.0]])
|
||
|
|
return rotn.dot(a * np.exp(-b*x))
|
||
|
|
|
||
|
|
def jacp(x, a, b):
|
||
|
|
rotn = np.array([[1./np.sqrt(2), -1./np.sqrt(2), 0], [1./np.sqrt(2), 1./np.sqrt(2), 0], [0, 0, 1.0]])
|
||
|
|
e = np.exp(-b*x)
|
||
|
|
return rotn.dot(np.vstack((e, -a * x * e)).T)
|
||
|
|
|
||
|
|
def func(x, a, b):
|
||
|
|
return a * np.exp(-b*x)
|
||
|
|
|
||
|
|
def jac(x, a, b):
|
||
|
|
e = np.exp(-b*x)
|
||
|
|
return np.vstack((e, -a * x * e)).T
|
||
|
|
|
||
|
|
np.random.seed(0)
|
||
|
|
xdata = np.arange(1, 4)
|
||
|
|
y = func(xdata, 2.5, 1.0)
|
||
|
|
ydata = y + 0.2 * np.random.normal(size=len(xdata))
|
||
|
|
sigma = np.zeros(len(xdata)) + 0.2
|
||
|
|
covar = np.diag(sigma**2)
|
||
|
|
# Get a rotation matrix, and obtain ydatap = R ydata
|
||
|
|
# Chisq = ydata^T C^{-1} ydata
|
||
|
|
# = ydata^T R^T R C^{-1} R^T R ydata
|
||
|
|
# = ydatap^T Cp^{-1} ydatap
|
||
|
|
# Cp^{-1} = R C^{-1} R^T
|
||
|
|
# Cp = R C R^T, since R^-1 = R^T
|
||
|
|
rotn = np.array([[1./np.sqrt(2), -1./np.sqrt(2), 0], [1./np.sqrt(2), 1./np.sqrt(2), 0], [0, 0, 1.0]])
|
||
|
|
ydatap = rotn.dot(ydata)
|
||
|
|
covarp = rotn.dot(covar).dot(rotn.T)
|
||
|
|
|
||
|
|
for jac1, jac2 in [(jac, jacp), (None, None)]:
|
||
|
|
for absolute_sigma in [False, True]:
|
||
|
|
popt1, pcov1 = curve_fit(func, xdata, ydata, sigma=sigma,
|
||
|
|
jac=jac1, absolute_sigma=absolute_sigma)
|
||
|
|
popt2, pcov2 = curve_fit(funcp, xdata, ydatap, sigma=covarp,
|
||
|
|
jac=jac2, absolute_sigma=absolute_sigma)
|
||
|
|
|
||
|
|
assert_allclose(popt1, popt2, atol=1e-14)
|
||
|
|
assert_allclose(pcov1, pcov2, atol=1e-14)
|
||
|
|
|
||
|
|
|
||
|
|
class TestFixedPoint(object):
|
||
|
|
|
||
|
|
def test_scalar_trivial(self):
|
||
|
|
# f(x) = 2x; fixed point should be x=0
|
||
|
|
def func(x):
|
||
|
|
return 2.0*x
|
||
|
|
x0 = 1.0
|
||
|
|
x = fixed_point(func, x0)
|
||
|
|
assert_almost_equal(x, 0.0)
|
||
|
|
|
||
|
|
def test_scalar_basic1(self):
|
||
|
|
# f(x) = x**2; x0=1.05; fixed point should be x=1
|
||
|
|
def func(x):
|
||
|
|
return x**2
|
||
|
|
x0 = 1.05
|
||
|
|
x = fixed_point(func, x0)
|
||
|
|
assert_almost_equal(x, 1.0)
|
||
|
|
|
||
|
|
def test_scalar_basic2(self):
|
||
|
|
# f(x) = x**0.5; x0=1.05; fixed point should be x=1
|
||
|
|
def func(x):
|
||
|
|
return x**0.5
|
||
|
|
x0 = 1.05
|
||
|
|
x = fixed_point(func, x0)
|
||
|
|
assert_almost_equal(x, 1.0)
|
||
|
|
|
||
|
|
def test_array_trivial(self):
|
||
|
|
def func(x):
|
||
|
|
return 2.0*x
|
||
|
|
x0 = [0.3, 0.15]
|
||
|
|
olderr = np.seterr(all='ignore')
|
||
|
|
try:
|
||
|
|
x = fixed_point(func, x0)
|
||
|
|
finally:
|
||
|
|
np.seterr(**olderr)
|
||
|
|
assert_almost_equal(x, [0.0, 0.0])
|
||
|
|
|
||
|
|
def test_array_basic1(self):
|
||
|
|
# f(x) = c * x**2; fixed point should be x=1/c
|
||
|
|
def func(x, c):
|
||
|
|
return c * x**2
|
||
|
|
c = array([0.75, 1.0, 1.25])
|
||
|
|
x0 = [1.1, 1.15, 0.9]
|
||
|
|
olderr = np.seterr(all='ignore')
|
||
|
|
try:
|
||
|
|
x = fixed_point(func, x0, args=(c,))
|
||
|
|
finally:
|
||
|
|
np.seterr(**olderr)
|
||
|
|
assert_almost_equal(x, 1.0/c)
|
||
|
|
|
||
|
|
def test_array_basic2(self):
|
||
|
|
# f(x) = c * x**0.5; fixed point should be x=c**2
|
||
|
|
def func(x, c):
|
||
|
|
return c * x**0.5
|
||
|
|
c = array([0.75, 1.0, 1.25])
|
||
|
|
x0 = [0.8, 1.1, 1.1]
|
||
|
|
x = fixed_point(func, x0, args=(c,))
|
||
|
|
assert_almost_equal(x, c**2)
|
||
|
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def test_lambertw(self):
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# python-list/2010-December/594592.html
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xxroot = fixed_point(lambda xx: np.exp(-2.0*xx)/2.0, 1.0,
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args=(), xtol=1e-12, maxiter=500)
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assert_allclose(xxroot, np.exp(-2.0*xxroot)/2.0)
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assert_allclose(xxroot, lambertw(1)/2)
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def test_no_acceleration(self):
|
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|
|
# github issue 5460
|
||
|
|
ks = 2
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||
|
|
kl = 6
|
||
|
|
m = 1.3
|
||
|
|
n0 = 1.001
|
||
|
|
i0 = ((m-1)/m)*(kl/ks/m)**(1/(m-1))
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||
|
|
|
||
|
|
def func(n):
|
||
|
|
return np.log(kl/ks/n) / np.log((i0*n/(n - 1))) + 1
|
||
|
|
|
||
|
|
n = fixed_point(func, n0, method='iteration')
|
||
|
|
assert_allclose(n, m)
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||
|
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