From 2790a4079b0bd93e9f5e3a77cf30a9853078af05 Mon Sep 17 00:00:00 2001 From: Jean-Philippe ARGAUD Date: Wed, 6 Nov 2024 09:03:17 +0100 Subject: [PATCH] Documentation and code update for simple models --- ...le_MeasurementsOptimalPositioningTask3.png | Bin 233059 -> 232917 bytes ...le_MeasurementsOptimalPositioningTask3.rst | 23 +- doc/en/snippets/ModuleCompatibility.rst | 2 +- ...le_MeasurementsOptimalPositioningTask3.png | Bin 237609 -> 237436 bytes ...le_MeasurementsOptimalPositioningTask3.rst | 23 +- doc/fr/snippets/ModuleCompatibility.rst | 2 +- src/daComposant/daCore/BasicObjects.py | 283 +++++++++++------- .../daNumerics/Models/Lorenz1963.py | 64 ++-- .../daNumerics/Models/Lorenz1996.py | 162 ++++++++++ 9 files changed, 397 insertions(+), 162 deletions(-) create mode 100644 src/daComposant/daNumerics/Models/Lorenz1996.py diff --git a/doc/en/scripts/simple_MeasurementsOptimalPositioningTask3.png b/doc/en/scripts/simple_MeasurementsOptimalPositioningTask3.png index 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It therefore enables a pedagogical illustration of optimal measurement points. -This function depends on the position :math:`x=(x_1,x_2)\in\Omega=[0.1,0.9]^2` -in the 2D plane, and on the parameter +This function :math:`G` depends on the position +:math:`x=(x_1,x_2)\in\Omega=[0.1,0.9]^2` in the 2D plane, and on the parameter :math:`\mu=(\mu_1,\mu_2)\in\mathcal{D}=[-1,-0.01]^2` of dimension 2 : .. math:: G(x;\mu) = \frac{1}{\sqrt{(x_1 - \mu_1)^2 + (x_2 - \mu_2)^2 + 0.1^2}} @@ -20,15 +20,16 @@ The function is represented on a regular :math:`\Omega_G` spatial grid of size 20x20 points. It is available in ADAO built-in test models under the name ``TwoDimensionalInverseDistanceCS2010``, together with the spatial and parametric domain default definition. So here we first build a set of -simulations of :math:`G`, then we look for the best locations for measurements -to obtain an DEIM interpolation representation of the fields, by applying the -DEIM-type decomposition algorithm to it, and then derive some simple -illustrations. We choose to look for an arbitrary number ``nbmeasures`` of 15 -measurement locations. +simulations of :math:`G` for various :math:`\mu` parameter values, then we look +for the best locations for measurements to obtain an DEIM interpolation +representation of the fields, by applying the DEIM-type decomposition algorithm +to it, and then derive some simple illustrations. We choose to look for an +arbitrary number ``nbmeasures`` of 15 measurement locations. It can be seen that the singular values decrease steadily down to numerical noise, indicating that around a hundred basis elements are needed to fully -represent the information contained in the set of :math:`G` simulations. -Furthermore, the optimal measurement points in the :math:`\Omega_G` domain are -inhomogeneously distributed, favoring the spatial zone near the -:math:`(0.1,0.1)` corner in which the :math:`G` function varies more. +represent the information contained in the set of :math:`G` parametric +simulations. Furthermore, the optimal measurement points in the +:math:`\Omega_G` domain are inhomogeneously distributed, favoring the spatial +zone near the :math:`(0.1,0.1)` corner in which the :math:`G` function varies +more. diff --git a/doc/en/snippets/ModuleCompatibility.rst b/doc/en/snippets/ModuleCompatibility.rst index 44ddb74..eec5239 100644 --- a/doc/en/snippets/ModuleCompatibility.rst +++ b/doc/en/snippets/ModuleCompatibility.rst @@ -16,7 +16,7 @@ versions within the range described below. :align: center Python, 3.6.5, 3.12.6 - Numpy, 1.14.3, 2.1.2 + Numpy, 1.14.3, 2.1.3 Scipy, 0.19.1, 1.14.1 MatplotLib, 2.2.2, 3.9.2 GnuplotPy, 1.8, 1.8 diff --git a/doc/fr/scripts/simple_MeasurementsOptimalPositioningTask3.png b/doc/fr/scripts/simple_MeasurementsOptimalPositioningTask3.png index 7fa8e18f0d56430a2e09f99af8567bec4d1ad806..42e10355285140f8f58bf99e1b4dd99ebcd83177 100644 GIT binary patch delta 194123 zcmb5W2{cyi_XhkRLp+Ac@MMZaGCoS?S;&w~nP-wBnTHIAOvw~7l#q~=A*2kM38^TA 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zK37W%ok>y6NME|Y@moJ4*}22M=A(uWZ@_~Z3I`3yE0kpy|3^sueVY!hBKFdlXi8)s zg2LntlM)zBC+Mc|BhYAV8mO%JID}&yZ0q)!r9D~Ip;SS$(jJ1)h=?3FxPJ&645Du- z7Nhnq6&xg&;q7S>hM=Dd!cV2|?>#@6GWQZKBetrQs97+^e);gcZfI!eopM7+UbMv6 zzFi$R$H5np2JLptJKOYC<_I_{gGJ2J{ZPVWv4$ut{WlN#yCJ0scilihrVgHm{8b)ZMHlCLOnuwK^W>|`a|gtH)q}c z$}e_<>`KIS zt30oafJQsj{r~T;2+mQd_hQ7^e|$Nu`*XIr>U&!D>> can <<< be used: - - self.set_mu - - self.set_integrator - - self.set_dt - - self.set_t0 - - self.set_tf - - self.set_y0 - - self.set_autonomous + Setters/Getters that >>> can <<< be used: + - self.Parameters + - self.Integrator + - self.IntegrationStep + - self.ObservationStep + - self.InitialTime + - self.FinalTime + - self.InitialCondition + - self.Autonomous """ pass # -------------------------------------------------------------------------- - def set_description( - self, - mu=None, - integrator=None, - dt=None, - t0=None, - tf=None, - y0=None, - autonomous=False, - ): - "Explicit setting of EDO description" - self.set_mu(mu) - self.set_integrator(integrator) - self.set_dt(dt) - self.set_t0(t0) - self.set_tf(tf) - self.set_y0(y0) - self.set_autonomous(autonomous) - - def set_mu(self, mu=None): - "Set EDO intrinsic parameters µ" - if mu is not None: - self._mu = numpy.ravel(mu) + @property + def Parameters(self): return self._mu - def set_integrator(self, integrator=None): - "Set integration scheme name" - if integrator is None: + @Parameters.setter + def Parameters(self, value): + if value is None: + pass + else: + self._mu = numpy.ravel(value) + + # ------- + @property + def Integrator(self): + return self._integrator + + @Integrator.setter + def Integrator(self, value): + if value is None: pass - elif not (integrator in self.__integrator_list): + elif not (value in self.__integrator_list): raise ValueError( - "Wrong value %s for integrator in set_integrator call. \nAvailable integrators are: %s" - % (integrator, self.__integrator_list) + "wrong value %s set for the integrator scheme. \nAvailable integrator scheme are: %s" + % (value, self.__integrator_list) ) else: - self._integrator = str(integrator) - return self._integrator + self._integrator = str(value) - def set_dt(self, value=None): - "Set integration step size dt" - if value is not None: - self._dt = max(2.0e-16, abs(float(value))) + # ------- + @property + def IntegrationStep(self): return self._dt - def set_t0(self, value=None): - "Set initial integration time" - if value is not None: + @IntegrationStep.setter + def IntegrationStep(self, value): + if value is None: + pass + elif float(value) > 0: + self._dt = max(2.0e-16, float(value)) + else: + raise ValueError("integration step has to be strictly positive") + + # ------- + @property + def ObservationStep(self): + return self._do + + @ObservationStep.setter + def ObservationStep(self, value): + if value is None: + pass + elif float(value) > 0: + self._do = max(2.0e-16, float(value)) + if hasattr(self, "_dt") and self._do < self._dt: + raise ValueError( + "Observation step is inconsistent with integration one" + ) + else: + raise ValueError("observation step has to be strictly positive") + + # ------- + @property + def InitialTime(self): + return self._t0 + + @InitialTime.setter + def InitialTime(self, value): + if value is None: + pass + else: self._t0 = float(value) if hasattr(self, "_tf") and self._t0 > self._tf: - raise ValueError("Initial time has to remain less than final time") - return self._t0 + raise ValueError("initial time has to remain less than final time") + + # ------- + @property + def FinalTime(self): + return self._tf - def set_tf(self, value=None): - "Set final integration time" - if value is not None: + @FinalTime.setter + def FinalTime(self, value): + if value is None: + pass + else: self._tf = float(value) if hasattr(self, "_t0") and self._t0 > self._tf: - raise ValueError("Initial time has to remain less than final time") - return self._tf + raise ValueError("initial time has to remain less than final time") - def set_y0(self, value): - "Set integration initial condition" - if value is not None: - self._y0 = numpy.ravel(value) + # ------- + @property + def InitialCondition(self): return self._y0 - def set_autonomous(self, value=None): + @InitialCondition.setter + def InitialCondition(self, value): + if value is None: + pass + else: + self._y0 = numpy.ravel(value) + + # ------- + @property + def Autonomous(self): + return self._autonomous + + @Autonomous.setter + def Autonomous(self, value): "Declare the system to be autonomous" - if value is not None: + if value is None: + pass + else: self._autonomous = bool(value) - return self._autonomous - def set_do(self, value=None): - "Set observation step size do" - if value is not None: - self._do = max(2.0e-16, abs(float(value))) - return self._do + # -------------------------------------------------------------------------- + + def _description( + self, + mu=None, + integrator=None, + dt=None, + t0=None, + tf=None, + y0=None, + autonomous=False, + ): + "Explicit setting of EDO description at init time" + self.Parameters = mu + self.Integrator = integrator + self.IntegrationStep = dt + self.InitialTime = t0 + self.FinalTime = tf + self.InitialCondition = y0 + self.Autonomous = autonomous # -------------------------------------------------------------------------- @@ -3706,26 +3764,31 @@ class DynamicalSimulator(object): _euler_step = _rk1_step - def Integration(self, y0=None, t0=None, tf=None, mu=None): + def Integration(self, y1=None, t1=None, t2=None, mu=None): """ - Exécute l'intégration du modèle entre t0 et tf, en partant de y0, - via le schéma d'intégration choisi + Exécute l'intégration du modèle entre t1 et t2, en partant de y1, via + le schéma d'intégration choisi. """ - if y0 is not None: - self.set_y0(y0) - if t0 is not None: - self.set_t0(t0) - if tf is not None: - self.set_tf(tf) - if mu is not None: - self.set_mu(mu) + if y1 is None: + _ly0 = self._y0 + else: + _ly0 = numpy.ravel(y1) + if t1 is None: + _lt0 = self._t0 + else: + _lt0 = float(t1) + if t2 is None: + _ltf = self._tf + else: + _ltf = float(t2) + self.Parameters = mu if ( (self._mu is None) or (self._integrator is None) or (self._dt is None) - or (self._t0 is None) - or (self._tf is None) - or (self._y0 is None) + or (_lt0 is None) + or (_ltf is None) + or (_ly0 is None) ): raise ValueError( "Some integration input information are None and not defined\n(%s, %s, %s, %s, %s, %s)" @@ -3733,14 +3796,14 @@ class DynamicalSimulator(object): self._mu, self._integrator, self._dt, - self._t0, - self._tf, - self._y0, + _lt0, + _ltf, + _ly0, ) ) # ODE = self.ODEModel - times = numpy.arange(self._t0, self._tf + self._dt / 2, self._dt) + times = numpy.arange(_lt0, _ltf + self._dt / 2, self._dt) if self._integrator == "odeint": # intégration 'automatique' dans le cas d'un système pouvant être # problématique avec rk4 ou euler (comme Van Der Pol) @@ -3748,7 +3811,7 @@ class DynamicalSimulator(object): trajectory = odeint( ODE, - numpy.array(self._y0, dtype=float), + numpy.array(_ly0, dtype=float), times, tfirst=True, ) @@ -3759,8 +3822,8 @@ class DynamicalSimulator(object): sol = solve_ivp( ODE, - (self._t0, self._tf), - numpy.array(self._y0, dtype=float), + (_lt0, _ltf), + numpy.array(_ly0, dtype=float), t_eval=times, ) trajectory = sol.y.T @@ -3773,18 +3836,18 @@ class DynamicalSimulator(object): % self._integrator ) # - t = self._t0 - y = self._y0 - trajectory = numpy.array([self._y0]) + t = _lt0 + y = _ly0 + trajectory = numpy.array([_ly0]) # - while t < self._tf - self._dt / 2: + while t < _ltf - self._dt / 2: [t, y] = integration_step(t, y, self._dt, ODE) trajectory = numpy.concatenate((trajectory, numpy.array([y])), axis=0) # return [times, trajectory] def ForecastedPath(self, y1=None, t1=None, t2=None, mu=None): - "Trajectoire de t1 à t2, en partant de yn, pour un paramètre donné mu" + "Trajectoire de t1 à t2, en partant de y1, pour un paramètre donné mu" # _, trajectory_from_t1_to_t2 = self.Integration(y1, t1, t2, mu) # @@ -3798,42 +3861,52 @@ class DynamicalSimulator(object): return trajectory_from_t1_to_t2[-1, :] def StateTransition(self, y1=None): - "État y[n+1] intégré depuis y[n] sur pas constant ou non" - if self.set_autonomous(): + "État y[n+1] intégré depuis y[n] sur un pas self.ObservationStep" + if self.Autonomous: if not hasattr(self, "_do") or self._do is None: raise ValueError( " StateTransition requires an observation step size to be given" ) - return self.ForecastedState(y1, t1=0.0, t2=self.set_do()) + return self.ForecastedState(y1, 0.0, self.ObservationStep, self.Parameters) else: raise NotImplementedError( " StateTransition has to be provided by the user in case of non-autonomous ODE" ) - def HistoryBoard(self, t_s, i_s, y_s, filename="history_board_of_trajectory.pdf"): + def HistoryBoard( + self, t_s, y_s, i_s=None, filename="figure_of_trajectory.pdf", suptitle="", title="", xlabel="Time", ylabel="State variables", cmap="gist_gray_r", + ): """ t_s : série des instants t - i_s : série des indices i des variables y_s : série des valeurs 1D des variables du système dynamique, pour chaque pas de temps, sous forme d'un tableau 2D de type: SDyn(i,t) = SDyn[i][t] = [SDyn[i] pour chaque t] + i_s : série des indices i des variables """ import matplotlib import matplotlib.pyplot as plt from matplotlib.ticker import MaxNLocator + # + if i_s is None: + i_s = range(y_s.shape[0]) levels = MaxNLocator(nbins=25).tick_values( numpy.ravel(y_s).min(), numpy.ravel(y_s).max() ) fig, ax = plt.subplots(figsize=(15, 5)) fig.subplots_adjust(bottom=0.1, left=0.05, right=0.95, top=0.9) im = plt.contourf( - t_s, i_s, y_s, levels=levels, cmap=plt.get_cmap("gist_gray_r") + t_s, i_s, y_s, levels=levels, cmap=plt.get_cmap(cmap) ) fig.colorbar(im, ax=ax) - plt.title("Model trajectory with %i variables" % len(y_s[:, 0])) - plt.xlabel("Time") - plt.ylabel("State variables") + if len(suptitle) > 0: + plt.suptitle(suptitle) + if len(title) > 0: + plt.title(title) + else: + plt.title("Model trajectory with %i variables" % len(y_s[:, 0])) + plt.xlabel(xlabel) + plt.ylabel(ylabel) if filename is None: plt.show() else: diff --git a/src/daComposant/daNumerics/Models/Lorenz1963.py b/src/daComposant/daNumerics/Models/Lorenz1963.py index 8c8a774..e624320 100644 --- a/src/daComposant/daNumerics/Models/Lorenz1963.py +++ b/src/daComposant/daNumerics/Models/Lorenz1963.py @@ -22,52 +22,51 @@ import sys import unittest -import math import numpy from daCore.BasicObjects import DynamicalSimulator # ============================================================================== -class Lorenz1963(DynamicalSimulator): +class EDS(DynamicalSimulator): """ Three-dimensional parametrized nonlinear ODE system depending on µ=(σ,ρ,β): - ∂x/∂t = σ (y − x) - ∂y/∂t = ρ x − y − x z - ∂z/∂t = x y − β z + dx/dt = σ (y − x) + dy/dt = ρ x − y − x z + dz/dt = x y − β z with t ∈ [0, 40] the time interval, x(t), y(t), z(t) the dependent variables, and with σ=10, ρ=28, and β=8/3 the commonly used parameter values. The initial conditions for (x, y, z) at t=0 for the reference case are (0, 1, 0). - This is the well known parametrized coupled system of three nonlinear - ordinary differential equations: + This is a parametrized coupled system of three nonlinear ordinary + differential equations described in: Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20, 130–141. doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 """ - def set_canonical_description(self): - self.set_mu((10.0, 28.0, 8.0 / 3.0)) # µ = (σ, ρ, β) - self.set_integrator("rk4") - self.set_dt(0.01) - self.set_t0(0.0) - self.set_tf(40) - self.set_y0((0.0, 1.0, 0.0)) - self.set_autonomous(True) + def CanonicalDescription(self): + self.Parameters = (10.0, 28.0, 8.0 / 3.0) # µ = (σ, ρ, β) + self.Integrator = "rk4" + self.IntegrationStep = 0.01 + self.InitialTime = 0.0 + self.FinalTime = 40 + self.InitialCondition = (0.0, 1.0, 0.0) + self.Autonomous = True return True def ODEModel(self, t, Y): - "ODE dY/dt = F(Y,t)" - sigma, rho, beta = self.set_mu() + "ODE dy/dt = F_µ(t, y)" + sigma, rho, beta = self.Parameters x, y, z = map(float, Y) # - rx = sigma * (y - x) - ry = x * rho - y - x * z - rz = x * y - beta * z + dxdt = sigma * (y - x) + dydt = x * rho - y - x * z + dzdt = x * y - beta * z # - return numpy.array([rx, ry, rz]) + return numpy.array([dxdt, dydt, dzdt]) # ============================================================================== @@ -75,24 +74,23 @@ class LocalTest(unittest.TestCase): @classmethod def setUpClass(cls): print("\nAUTODIAGNOSTIC\n==============\n") - print(" " + Lorenz1963().__doc__.strip()) + print(" " + EDS().__doc__.strip()) def test001(self): - numpy.random.seed(123456789) - ODE = Lorenz1963() # Default parameters + ODE = EDS() # Default parameters + ODE.ObservationStep = 0.2 trajectory = ODE.ForecastedPath() + lastvalue = numpy.array([16.48799962, 14.01693428, 40.30448848]) # print() - self.assertTrue(trajectory.shape[0] == 1 + int(ODE.set_tf() / ODE.set_dt())) self.assertTrue( - abs( - max( - trajectory[-1] - - numpy.array([16.48799962, 14.01693428, 40.30448848]) - ) - ) - <= 1.0e-8, - msg=" Last value is not equal to the reference one", + trajectory.shape[0] == 1 + int(ODE.FinalTime / ODE.IntegrationStep) + ) + erreur = abs(max(trajectory[-1] - lastvalue)) + self.assertTrue( + erreur <= 1.0e-8, + msg=" Last value is not equal to the reference one. Error = %.2e" + % erreur, ) print(" Last value is equal to the reference one") diff --git a/src/daComposant/daNumerics/Models/Lorenz1996.py b/src/daComposant/daNumerics/Models/Lorenz1996.py new file mode 100644 index 0000000..0956f6a --- /dev/null +++ b/src/daComposant/daNumerics/Models/Lorenz1996.py @@ -0,0 +1,162 @@ +# -*- coding: utf-8 -*- +# +# Copyright (C) 2008-2024 EDF R&D +# +# This library is free software; you can redistribute it and/or +# modify it under the terms of the GNU Lesser General Public +# License as published by the Free Software Foundation; either +# version 2.1 of the License. +# +# This library is distributed in the hope that it will be useful, +# but WITHOUT ANY WARRANTY; without even the implied warranty of +# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU +# Lesser General Public License for more details. +# +# You should have received a copy of the GNU Lesser General Public +# License along with this library; if not, write to the Free Software +# Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA +# +# See http://www.salome-platform.org/ or email : webmaster.salome@opencascade.com +# +# Author: Jean-Philippe Argaud, jean-philippe.argaud@edf.fr, EDF R&D + +import sys +import unittest +import numpy as np +from daCore.BasicObjects import DynamicalSimulator + + +# ============================================================================== +class EDS(DynamicalSimulator): + """ + N-dimensional parametrized nonlinear ODE system depending on a F forcing + term: + + dy[i]/dt = (y[i+1] − y[i-2]) * y[i-1] − y[i] + F + + with i=1,...,N and periodic conditions coming from circle-like domain + for the N variables: y[0]=y[N], y[-1]=y[N-1], y[1]=y[N+1]. With a + forcing term F value of 8, the dynamics of this model is chaotic. + + This is a parametrized system of nonlinear ordinary differential equations, + first described during a seminar at the European Centre for Medium-Range + Weather Forecasts (ECMWF) in the Autumn of 1995 (Seminar on Predictability, + 4-8 September 1995), the proceedings of which were published as Lorenz + (1996): + Lorenz, E.N. Predictability: A problem partly solved. In Proceedings of + the Seminar on Predictability, ECMWF, Reading, UK, 9–11 September 1996, + Volume 1, pp. 1-18. + See: + https://www.ecmwf.int/en/elibrary/75462-predictability-problem-partly-solved + The system is chaotic and numerically very sensible, even on the order of + numerical calculations. + """ + + def CanonicalDescription(self): + N = 40 + self.Parameters = (N, 8.0) # N, F + self.Integrator = "odeint" + self.IntegrationStep = 0.05 + self.InitialTime = 0.0 + self.FinalTime = 1.0 + self.InitialCondition = np.arange(N) + self.Autonomous = True + return True + + def ODEModel(self, t, Y): + "ODE dy/dt = F_µ(t, y)" + N, F = self.Parameters + N = int(N) + F = float(F) + Y = np.ravel(Y) + dydt = np.zeros(np.shape(Y)) + assert len(Y) == N, "%i =/= %i" % (len(Y), N) + # + # Boundary case equations (rank 1, 2, N): + dydt[0] = (Y[1] - Y[N - 2]) * Y[N - 1] - Y[0] + dydt[1] = (Y[2] - Y[N - 1]) * Y[0] - Y[1] + dydt[-1] = (Y[0] - Y[-3]) * Y[-2] - Y[-1] + # General indices (rank 3 to N-1) + dydt[2:-1] = (Y[3:] - Y[:-3]) * Y[1:-2] - Y[2:-1] + # Adding forcing + dydt = dydt + F + # + return dydt + + +# ============================================================================== +class LocalTest(unittest.TestCase): + @classmethod + def setUpClass(cls): + print("\nAUTODIAGNOSTIC\n==============\n") + print(" " + EDS().__doc__.strip()) + + def test001(self): + ODE = EDS() # Default parameters + ODE.ObservationStep = 0.2 + trajectory = ODE.ForecastedPath() + lastvalue = np.array( + [ + 4.87526450e00, + 1.47956922e00, + -3.59900230e00, + 1.04256938e00, + 3.63094386e00, + 1.88075245e01, + -1.67896983e01, + 9.47857563e00, + -4.10761433e01, + -7.22094136e00, + 1.82999964e01, + -9.50258813e00, + 1.97970038e01, + -9.14157688e-01, + -3.23915931e00, + 9.08515096e00, + 1.90398329e01, + 6.63360465e00, + 3.62060706e00, + 1.40450966e00, + -4.23014939e00, + 3.12067035e00, + -7.45630836e00, + 5.29044964e00, + -4.95916975e00, + 1.49074059e00, + -5.34660284e00, + 7.88846593e00, + 9.82088697e00, + 9.01760941e-01, + 7.27891147e00, + 4.66438119e00, + -1.22290917e00, + 3.38177733e00, + 6.19234382e00, + 3.66288450e00, + 2.51247077e00, + 3.61265491e00, + 4.56523506e00, + 5.03793331e00, + ] + ) + # + print() + self.assertTrue( + trajectory.shape[0] == 1 + int(ODE.FinalTime / ODE.IntegrationStep) + ) + erreur = abs(max(trajectory[-1] - lastvalue)) + self.assertTrue( + erreur <= 1.0e-7, + msg=" Last value is not equal to the reference one. Error = %.2e" + % erreur, + ) + print(" Last value is equal to the reference one") + + def tearDown(cls): + print("\n Tests are finished\n") + + +# ============================================================================== +if __name__ == "__main__": + sys.stderr = sys.stdout + unittest.main(verbosity=0) -- 2.39.2

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