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Python

"""
Unit tests for optimization routines from minpack.py.
"""
from __future__ import division, print_function, absolute_import
from numpy.testing import (assert_, assert_almost_equal, assert_array_equal,
assert_array_almost_equal, assert_allclose)
from pytest import raises as assert_raises
import numpy as np
from numpy import array, float64, matrix
from multiprocessing.pool import ThreadPool
from scipy import optimize
from scipy.special import lambertw
from scipy.optimize.minpack import leastsq, curve_fit, fixed_point
from scipy._lib._numpy_compat import _assert_warns, suppress_warnings
from scipy.optimize import OptimizeWarning
class ReturnShape(object):
"""This class exists to create a callable that does not have a '__name__' attribute.
__init__ takes the argument 'shape', which should be a tuple of ints. When an instance
it called with a single argument 'x', it returns numpy.ones(shape).
"""
def __init__(self, shape):
self.shape = shape
def __call__(self, x):
return np.ones(self.shape)
def dummy_func(x, shape):
"""A function that returns an array of ones of the given shape.
`x` is ignored.
"""
return np.ones(shape)
def sequence_parallel(fs):
pool = ThreadPool(len(fs))
try:
return pool.map(lambda f: f(), fs)
finally:
pool.terminate()
# Function and jacobian for tests of solvers for systems of nonlinear
# equations
def pressure_network(flow_rates, Qtot, k):
"""Evaluate non-linear equation system representing
the pressures and flows in a system of n parallel pipes::
f_i = P_i - P_0, for i = 1..n
f_0 = sum(Q_i) - Qtot
Where Q_i is the flow rate in pipe i and P_i the pressure in that pipe.
Pressure is modeled as a P=kQ**2 where k is a valve coefficient and
Q is the flow rate.
Parameters
----------
flow_rates : float
A 1D array of n flow rates [kg/s].
k : float
A 1D array of n valve coefficients [1/kg m].
Qtot : float
A scalar, the total input flow rate [kg/s].
Returns
-------
F : float
A 1D array, F[i] == f_i.
"""
P = k * flow_rates**2
F = np.hstack((P[1:] - P[0], flow_rates.sum() - Qtot))
return F
def pressure_network_jacobian(flow_rates, Qtot, k):
"""Return the jacobian of the equation system F(flow_rates)
computed by `pressure_network` with respect to
*flow_rates*. See `pressure_network` for the detailed
description of parrameters.
Returns
-------
jac : float
*n* by *n* matrix ``df_i/dQ_i`` where ``n = len(flow_rates)``
and *f_i* and *Q_i* are described in the doc for `pressure_network`
"""
n = len(flow_rates)
pdiff = np.diag(flow_rates[1:] * 2 * k[1:] - 2 * flow_rates[0] * k[0])
jac = np.empty((n, n))
jac[:n-1, :n-1] = pdiff * 0
jac[:n-1, n-1] = 0
jac[n-1, :] = np.ones(n)
return jac
def pressure_network_fun_and_grad(flow_rates, Qtot, k):
return (pressure_network(flow_rates, Qtot, k),
pressure_network_jacobian(flow_rates, Qtot, k))
class TestFSolve(object):
def test_pressure_network_no_gradient(self):
# fsolve without gradient, equal pipes -> equal flows.
k = np.ones(4) * 0.5
Qtot = 4
initial_guess = array([2., 0., 2., 0.])
final_flows, info, ier, mesg = optimize.fsolve(
pressure_network, initial_guess, args=(Qtot, k),
full_output=True)
assert_array_almost_equal(final_flows, np.ones(4))
assert_(ier == 1, mesg)
def test_pressure_network_with_gradient(self):
# fsolve with gradient, equal pipes -> equal flows
k = np.ones(4) * 0.5
Qtot = 4
initial_guess = array([2., 0., 2., 0.])
final_flows = optimize.fsolve(
pressure_network, initial_guess, args=(Qtot, k),
fprime=pressure_network_jacobian)
assert_array_almost_equal(final_flows, np.ones(4))
def test_wrong_shape_func_callable(self):
func = ReturnShape(1)
# x0 is a list of two elements, but func will return an array with
# length 1, so this should result in a TypeError.
x0 = [1.5, 2.0]
assert_raises(TypeError, optimize.fsolve, func, x0)
def test_wrong_shape_func_function(self):
# x0 is a list of two elements, but func will return an array with
# length 1, so this should result in a TypeError.
x0 = [1.5, 2.0]
assert_raises(TypeError, optimize.fsolve, dummy_func, x0, args=((1,),))
def test_wrong_shape_fprime_callable(self):
func = ReturnShape(1)
deriv_func = ReturnShape((2,2))
assert_raises(TypeError, optimize.fsolve, func, x0=[0,1], fprime=deriv_func)
def test_wrong_shape_fprime_function(self):
func = lambda x: dummy_func(x, (2,))
deriv_func = lambda x: dummy_func(x, (3,3))
assert_raises(TypeError, optimize.fsolve, func, x0=[0,1], fprime=deriv_func)
def test_func_can_raise(self):
def func(*args):
raise ValueError('I raised')
with assert_raises(ValueError, match='I raised'):
optimize.fsolve(func, x0=[0])
def test_Dfun_can_raise(self):
func = lambda x: x - np.array([10])
def deriv_func(*args):
raise ValueError('I raised')
with assert_raises(ValueError, match='I raised'):
optimize.fsolve(func, x0=[0], fprime=deriv_func)
def test_float32(self):
func = lambda x: np.array([x[0] - 100, x[1] - 1000], dtype=np.float32)**2
p = optimize.fsolve(func, np.array([1, 1], np.float32))
assert_allclose(func(p), [0, 0], atol=1e-3)
def test_reentrant_func(self):
def func(*args):
self.test_pressure_network_no_gradient()
return pressure_network(*args)
# fsolve without gradient, equal pipes -> equal flows.
k = np.ones(4) * 0.5
Qtot = 4
initial_guess = array([2., 0., 2., 0.])
final_flows, info, ier, mesg = optimize.fsolve(
func, initial_guess, args=(Qtot, k),
full_output=True)
assert_array_almost_equal(final_flows, np.ones(4))
assert_(ier == 1, mesg)
def test_reentrant_Dfunc(self):
def deriv_func(*args):
self.test_pressure_network_with_gradient()
return pressure_network_jacobian(*args)
# fsolve with gradient, equal pipes -> equal flows
k = np.ones(4) * 0.5
Qtot = 4
initial_guess = array([2., 0., 2., 0.])
final_flows = optimize.fsolve(
pressure_network, initial_guess, args=(Qtot, k),
fprime=deriv_func)
assert_array_almost_equal(final_flows, np.ones(4))
def test_concurrent_no_gradient(self):
return sequence_parallel([self.test_pressure_network_no_gradient] * 10)
def test_concurrent_with_gradient(self):
return sequence_parallel([self.test_pressure_network_with_gradient] * 10)
class TestRootHybr(object):
def test_pressure_network_no_gradient(self):
# root/hybr without gradient, equal pipes -> equal flows
k = np.ones(4) * 0.5
Qtot = 4
initial_guess = array([2., 0., 2., 0.])
final_flows = optimize.root(pressure_network, initial_guess,
method='hybr', args=(Qtot, k)).x
assert_array_almost_equal(final_flows, np.ones(4))
def test_pressure_network_with_gradient(self):
# root/hybr with gradient, equal pipes -> equal flows
k = np.ones(4) * 0.5
Qtot = 4
initial_guess = matrix([2., 0., 2., 0.])
final_flows = optimize.root(pressure_network, initial_guess,
args=(Qtot, k), method='hybr',
jac=pressure_network_jacobian).x
assert_array_almost_equal(final_flows, np.ones(4))
def test_pressure_network_with_gradient_combined(self):
# root/hybr with gradient and function combined, equal pipes -> equal
# flows
k = np.ones(4) * 0.5
Qtot = 4
initial_guess = array([2., 0., 2., 0.])
final_flows = optimize.root(pressure_network_fun_and_grad,
initial_guess, args=(Qtot, k),
method='hybr', jac=True).x
assert_array_almost_equal(final_flows, np.ones(4))
class TestRootLM(object):
def test_pressure_network_no_gradient(self):
# root/lm without gradient, equal pipes -> equal flows
k = np.ones(4) * 0.5
Qtot = 4
initial_guess = array([2., 0., 2., 0.])
final_flows = optimize.root(pressure_network, initial_guess,
method='lm', args=(Qtot, k)).x
assert_array_almost_equal(final_flows, np.ones(4))
class TestLeastSq(object):
def setup_method(self):
x = np.linspace(0, 10, 40)
a,b,c = 3.1, 42, -304.2
self.x = x
self.abc = a,b,c
y_true = a*x**2 + b*x + c
np.random.seed(0)
self.y_meas = y_true + 0.01*np.random.standard_normal(y_true.shape)
def residuals(self, p, y, x):
a,b,c = p
err = y-(a*x**2 + b*x + c)
return err
def residuals_jacobian(self, _p, _y, x):
return -np.vstack([x**2, x, np.ones_like(x)]).T
def test_basic(self):
p0 = array([0,0,0])
params_fit, ier = leastsq(self.residuals, p0,
args=(self.y_meas, self.x))
assert_(ier in (1,2,3,4), 'solution not found (ier=%d)' % ier)
# low precision due to random
assert_array_almost_equal(params_fit, self.abc, decimal=2)
def test_basic_with_gradient(self):
p0 = array([0,0,0])
params_fit, ier = leastsq(self.residuals, p0,
args=(self.y_meas, self.x),
Dfun=self.residuals_jacobian)
assert_(ier in (1,2,3,4), 'solution not found (ier=%d)' % ier)
# low precision due to random
assert_array_almost_equal(params_fit, self.abc, decimal=2)
def test_full_output(self):
p0 = matrix([0,0,0])
full_output = leastsq(self.residuals, p0,
args=(self.y_meas, self.x),
full_output=True)
params_fit, cov_x, infodict, mesg, ier = full_output
assert_(ier in (1,2,3,4), 'solution not found: %s' % mesg)
def test_input_untouched(self):
p0 = array([0,0,0],dtype=float64)
p0_copy = array(p0, copy=True)
full_output = leastsq(self.residuals, p0,
args=(self.y_meas, self.x),
full_output=True)
params_fit, cov_x, infodict, mesg, ier = full_output
assert_(ier in (1,2,3,4), 'solution not found: %s' % mesg)
assert_array_equal(p0, p0_copy)
def test_wrong_shape_func_callable(self):
func = ReturnShape(1)
# x0 is a list of two elements, but func will return an array with
# length 1, so this should result in a TypeError.
x0 = [1.5, 2.0]
assert_raises(TypeError, optimize.leastsq, func, x0)
def test_wrong_shape_func_function(self):
# x0 is a list of two elements, but func will return an array with
# length 1, so this should result in a TypeError.
x0 = [1.5, 2.0]
assert_raises(TypeError, optimize.leastsq, dummy_func, x0, args=((1,),))
def test_wrong_shape_Dfun_callable(self):
func = ReturnShape(1)
deriv_func = ReturnShape((2,2))
assert_raises(TypeError, optimize.leastsq, func, x0=[0,1], Dfun=deriv_func)
def test_wrong_shape_Dfun_function(self):
func = lambda x: dummy_func(x, (2,))
deriv_func = lambda x: dummy_func(x, (3,3))
assert_raises(TypeError, optimize.leastsq, func, x0=[0,1], Dfun=deriv_func)
def test_float32(self):
# Regression test for gh-1447
def func(p,x,y):
q = p[0]*np.exp(-(x-p[1])**2/(2.0*p[2]**2))+p[3]
return q - y
x = np.array([1.475,1.429,1.409,1.419,1.455,1.519,1.472, 1.368,1.286,
1.231], dtype=np.float32)
y = np.array([0.0168,0.0193,0.0211,0.0202,0.0171,0.0151,0.0185,0.0258,
0.034,0.0396], dtype=np.float32)
p0 = np.array([1.0,1.0,1.0,1.0])
p1, success = optimize.leastsq(func, p0, args=(x,y))
assert_(success in [1,2,3,4])
assert_((func(p1,x,y)**2).sum() < 1e-4 * (func(p0,x,y)**2).sum())
def test_func_can_raise(self):
def func(*args):
raise ValueError('I raised')
with assert_raises(ValueError, match='I raised'):
optimize.leastsq(func, x0=[0])
def test_Dfun_can_raise(self):
func = lambda x: x - np.array([10])
def deriv_func(*args):
raise ValueError('I raised')
with assert_raises(ValueError, match='I raised'):
optimize.leastsq(func, x0=[0], Dfun=deriv_func)
def test_reentrant_func(self):
def func(*args):
self.test_basic()
return self.residuals(*args)
p0 = array([0,0,0])
params_fit, ier = leastsq(func, p0,
args=(self.y_meas, self.x))
assert_(ier in (1,2,3,4), 'solution not found (ier=%d)' % ier)
# low precision due to random
assert_array_almost_equal(params_fit, self.abc, decimal=2)
def test_reentrant_Dfun(self):
def deriv_func(*args):
self.test_basic()
return self.residuals_jacobian(*args)
p0 = array([0,0,0])
params_fit, ier = leastsq(self.residuals, p0,
args=(self.y_meas, self.x),
Dfun=deriv_func)
assert_(ier in (1,2,3,4), 'solution not found (ier=%d)' % ier)
# low precision due to random
assert_array_almost_equal(params_fit, self.abc, decimal=2)
def test_concurrent_no_gradient(self):
return sequence_parallel([self.test_basic] * 10)
def test_concurrent_with_gradient(self):
return sequence_parallel([self.test_basic_with_gradient] * 10)
class TestCurveFit(object):
def setup_method(self):
self.y = array([1.0, 3.2, 9.5, 13.7])
self.x = array([1.0, 2.0, 3.0, 4.0])
def test_one_argument(self):
def func(x,a):
return x**a
popt, pcov = curve_fit(func, self.x, self.y)
assert_(len(popt) == 1)
assert_(pcov.shape == (1,1))
assert_almost_equal(popt[0], 1.9149, decimal=4)
assert_almost_equal(pcov[0,0], 0.0016, decimal=4)
# Test if we get the same with full_output. Regression test for #1415.
res = curve_fit(func, self.x, self.y, full_output=1)
(popt2, pcov2, infodict, errmsg, ier) = res
assert_array_almost_equal(popt, popt2)
def test_two_argument(self):
def func(x, a, b):
return b*x**a
popt, pcov = curve_fit(func, self.x, self.y)
assert_(len(popt) == 2)
assert_(pcov.shape == (2,2))
assert_array_almost_equal(popt, [1.7989, 1.1642], decimal=4)
assert_array_almost_equal(pcov, [[0.0852, -0.1260], [-0.1260, 0.1912]],
decimal=4)
def test_func_is_classmethod(self):
class test_self(object):
"""This class tests if curve_fit passes the correct number of
arguments when the model function is a class instance method.
"""
def func(self, x, a, b):
return b * x**a
test_self_inst = test_self()
popt, pcov = curve_fit(test_self_inst.func, self.x, self.y)
assert_(pcov.shape == (2,2))
assert_array_almost_equal(popt, [1.7989, 1.1642], decimal=4)
assert_array_almost_equal(pcov, [[0.0852, -0.1260], [-0.1260, 0.1912]],
decimal=4)
def test_regression_2639(self):
# This test fails if epsfcn in leastsq is too large.
x = [574.14200000000005, 574.154, 574.16499999999996,
574.17700000000002, 574.18799999999999, 574.19899999999996,
574.21100000000001, 574.22199999999998, 574.23400000000004,
574.245]
y = [859.0, 997.0, 1699.0, 2604.0, 2013.0, 1964.0, 2435.0,
1550.0, 949.0, 841.0]
guess = [574.1861428571428, 574.2155714285715, 1302.0, 1302.0,
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)
def test_lambertw(self):
# python-list/2010-December/594592.html
xxroot = fixed_point(lambda xx: np.exp(-2.0*xx)/2.0, 1.0,
args=(), xtol=1e-12, maxiter=500)
assert_allclose(xxroot, np.exp(-2.0*xxroot)/2.0)
assert_allclose(xxroot, lambertw(1)/2)
def test_no_acceleration(self):
# github issue 5460
ks = 2
kl = 6
m = 1.3
n0 = 1.001
i0 = ((m-1)/m)*(kl/ks/m)**(1/(m-1))
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)