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test_models.py
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/
test_models.py
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import numpy as np
import psyneulink as pnl
import psyneulink.core.components.functions.distributionfunctions
import psyneulink.core.components.functions.statefulfunctions.integratorfunctions
import psyneulink.core.components.functions.transferfunctions
import psyneulink.core.globals.utilities
class TestModels:
def test_DDM(self):
myMechanism = pnl.DDM(
function=psyneulink.core.components.functions.distributionfunctions.DriftDiffusionAnalytical(
drift_rate=(1.0),
threshold=(10.0),
starting_point=0.0,
),
name='My_DDM',
)
myMechanism_2 = pnl.DDM(
function=psyneulink.core.components.functions.distributionfunctions.DriftDiffusionAnalytical(
drift_rate=2.0,
threshold=20.0),
name='My_DDM_2'
)
myMechanism_3 = pnl.DDM(
function=psyneulink.core.components.functions.distributionfunctions.DriftDiffusionAnalytical(
drift_rate=3.0,
threshold=30.0
),
name='My_DDM_3',
)
z = pnl.Composition()
z.add_linear_processing_pathway([myMechanism,
pnl.MappingProjection(matrix=pnl.IDENTITY_MATRIX),
myMechanism_2,
pnl.MappingProjection(matrix=pnl.FULL_CONNECTIVITY_MATRIX),
myMechanism_3])
result = z.run(inputs={myMechanism: [[40]]})[0][0]
expected_output = [
(myMechanism.input_ports[0].parameters.value.get(z), np.array([40.])),
(myMechanism.output_ports[0].parameters.value.get(z), np.array([10.])),
(myMechanism_2.input_ports[0].parameters.value.get(z), np.array([10.])),
(myMechanism_2.output_ports[0].parameters.value.get(z), np.array([20.])),
(myMechanism_3.input_ports[0].parameters.value.get(z), np.array([20.])),
(myMechanism_3.output_ports[0].parameters.value.get(z), np.array([30.])),
(result, np.array([30.])),
]
for i in range(len(expected_output)):
val, expected = expected_output[i]
# setting absolute tolerance to be in accordance with reference_output precision
# if you do not specify, assert_allcose will use a relative tolerance of 1e-07,
# which WILL FAIL unless you gather higher precision values to use as reference
np.testing.assert_allclose(val, expected, atol=1e-08, err_msg='Failed on expected_output[{0}]'.format(i))
# FIX: 5/8/20 [JDC] Needs assertions
def test_bustamante_Stroop_model(self):
# INPUT UNITS
# colors: ('red', 'green'), words: ('RED','GREEN')
colors_input_layer = pnl.TransferMechanism(size=2,
function=psyneulink.core.components.functions.transferfunctions.Linear,
name='COLORS_INPUT')
words_input_layer = pnl.TransferMechanism(size=2,
function=psyneulink.core.components.functions.transferfunctions.Linear,
name='WORDS_INPUT')
# Task layer, tasks: ('name the color', 'read the word')
task_layer = pnl.TransferMechanism(size=2,
function=psyneulink.core.components.functions.transferfunctions.Linear,
name='TASK')
# HIDDEN LAYER UNITS
# colors_hidden: ('red','green')
# Logistic activation function, Gain = 1.0, Bias = -4.0 (in PNL bias is subtracted so enter +4.0 to get negative bias)
# randomly distributed noise to the net input
# time averaging = integration_rate = 0.1
unit_noise = 0.005
colors_hidden_layer = pnl.TransferMechanism(size=2,
function=psyneulink.core.components.functions.transferfunctions
.Logistic(gain=1.0, x_0=4.0),
# should be able to get same result with offset = -4.0
integrator_mode=True,
noise=psyneulink.core.components.functions.distributionfunctions
.NormalDist(mean=0, standard_deviation=unit_noise).function,
integration_rate=0.1,
name='COLORS HIDDEN')
# words_hidden: ('RED','GREEN')
words_hidden_layer = pnl.TransferMechanism(size=2,
function=pnl.Logistic(gain=1.0, x_0=4.0),
integrator_mode=True,
noise=pnl.NormalDist(mean=0,
standard_deviation=unit_noise).function,
integration_rate=0.1,
name='WORDS HIDDEN')
# OUTPUT UNITS
# Response layer, provide input to accumulator, responses: ('red', 'green')
# time averaging = tau = 0.1
# randomly distributed noise to the net input
response_layer = pnl.TransferMechanism(size=2,
function=psyneulink.core.components.functions.transferfunctions.Logistic,
name='RESPONSE',
integrator_mode=True,
noise=psyneulink.core.components.functions.distributionfunctions.NormalDist(mean=0, standard_deviation=unit_noise).function,
integration_rate=0.1)
# Respond red accumulator
# alpha = rate of evidence accumlation = 0.1
# sigma = noise = 0.1
# noise will be: squareroot(time_step_size * noise) * a random sample from a normal distribution
accumulator_noise = 0.1
respond_red_accumulator = pnl.IntegratorMechanism(function=pnl.SimpleIntegrator(noise=pnl.NormalDist(mean=0,
standard_deviation=accumulator_noise).function,
rate=0.1),
name='respond_red_accumulator')
# Respond green accumulator
respond_green_accumulator = pnl.IntegratorMechanism(function=pnl.SimpleIntegrator(noise=pnl.NormalDist(mean=0,
standard_deviation=accumulator_noise).function,
rate=0.1),
name='respond_green_accumulator')
# LOGGING
colors_hidden_layer.set_log_conditions('value')
words_hidden_layer.set_log_conditions('value')
response_layer.set_log_conditions('value')
respond_red_accumulator.set_log_conditions('value')
respond_green_accumulator.set_log_conditions('value')
# SET UP CONNECTIONS
# rows correspond to sender
# columns correspond to: weighting of the contribution that a given sender makes to the receiver
# INPUT TO HIDDEN
# row 0: input_'red' to hidden_'red', hidden_'green'
# row 1: input_'green' to hidden_'red', hidden_'green'
color_weights = pnl.MappingProjection(matrix=np.atleast_2d([[2.2, -2.2],
[-2.2, 2.2]]),
name='COLOR_WEIGHTS')
# row 0: input_'RED' to hidden_'RED', hidden_'GREEN'
# row 1: input_'GREEN' to hidden_'RED', hidden_'GREEN'
word_weights = pnl.MappingProjection(matrix=np.atleast_2d([[2.6, -2.6],
[-2.6, 2.6]]),
name='WORD_WEIGHTS')
# HIDDEN TO RESPONSE
# row 0: hidden_'red' to response_'red', response_'green'
# row 1: hidden_'green' to response_'red', response_'green'
color_response_weights = pnl.MappingProjection(matrix=np.atleast_2d([[1.3, -1.3],
[-1.3, 1.3]]),
name='COLOR_RESPONSE_WEIGHTS')
# row 0: hidden_'RED' to response_'red', response_'green'
# row 1: hidden_'GREEN' to response_'red', response_'green'
word_response_weights = pnl.MappingProjection(matrix=np.atleast_2d([[2.5, -2.5],
[-2.5, 2.5]]),
name='WORD_RESPONSE_WEIGHTS')
# TASK TO HIDDEN LAYER
# row 0: task_CN to hidden_'red', hidden_'green'
# row 1: task_WR to hidden_'red', hidden_'green'
task_CN_weights = pnl.MappingProjection(matrix=np.atleast_2d([[4.0, 4.0],
[0, 0]]),
name='TASK_CN_WEIGHTS')
# row 0: task_CN to hidden_'RED', hidden_'GREEN'
# row 1: task_WR to hidden_'RED', hidden_'GREEN'
task_WR_weights = pnl.MappingProjection(matrix=np.atleast_2d([[0, 0],
[4.0, 4.0]]),
name='TASK_WR_WEIGHTS')
# RESPONSE UNITS TO ACCUMULATORS
# row 0: response_'red' to respond_red_accumulator
# row 1: response_'green' to respond_red_accumulator
respond_red_differencing_weights = pnl.MappingProjection(matrix=np.atleast_2d([[1.0], [-1.0]]),
name='RESPOND_RED_WEIGHTS')
# row 0: response_'red' to respond_green_accumulator
# row 1: response_'green' to respond_green_accumulator
respond_green_differencing_weights = pnl.MappingProjection(matrix=np.atleast_2d([[-1.0], [1.0]]),
name='RESPOND_GREEN_WEIGHTS')
# CREATE COMPOSITION FROM PATHWAYS
my_Stroop = pnl.Composition(pathways=[
{'WORD_PATHWAY':[words_input_layer,
word_weights,
words_hidden_layer,
word_response_weights,
response_layer]},
{'COLO_PATHWAY': [colors_input_layer,
color_weights,
colors_hidden_layer,
color_response_weights,
response_layer]},
{'TASK_CN_PATHWAY':[task_layer,
task_CN_weights,
colors_hidden_layer]},
{'TASK_WR_PATHWAY':[task_layer,
task_WR_weights,
words_hidden_layer]},
{'RESPOND_RED_PATHWAY':[response_layer,
respond_red_differencing_weights,
respond_red_accumulator]},
{'RESPOND_GREEN_PATHWAY': [response_layer,
respond_green_differencing_weights,
respond_green_accumulator]},
])
# my_Stroop.show()
# my_Stroop.show_graph(show_dimensions=pnl.ALL)
# Function to create test trials
# a RED word input is [1,0] to words_input_layer and GREEN word is [0,1]
# a red color input is [1,0] to colors_input_layer and green color is [0,1]
# a color-naming trial is [1,0] to task_layer and a word-reading trial is [0,1]
def trial_dict(red_color, green_color, red_word, green_word, CN, WR):
trialdict = {
colors_input_layer: [red_color, green_color],
words_input_layer: [red_word, green_word],
task_layer: [CN, WR]
}
return trialdict
# CREATE THRESHOLD FUNCTION
# first value of DDM's value is DECISION_VARIABLE
# context is always passed to Condition functions and is the context
# in which the function gets called - below, during system execution
def pass_threshold(mech1, mech2, thresh, context=None):
results1 = mech1.output_ports[0].parameters.value.get(context)
results2 = mech2.output_ports[0].parameters.value.get(context)
for val in results1:
if val >= thresh:
return True
for val in results2:
if val >= thresh:
return True
return False
accumulator_threshold = 1.0
mechanisms_to_update = [colors_hidden_layer, words_hidden_layer, response_layer]
def switch_integrator_mode(mechanisms, mode):
for mechanism in mechanisms:
mechanism.integrator_mode = mode
def switch_noise(mechanisms, noise):
for mechanism in mechanisms:
mechanism.noise.base = noise
def switch_to_initialization_trial(mechanisms):
# Turn off accumulation
switch_integrator_mode(mechanisms, False)
# Turn off noise
switch_noise(mechanisms, 0)
# Execute once per trial
my_Stroop.termination_processing = {pnl.TimeScale.TRIAL: pnl.AllHaveRun()}
def switch_to_processing_trial(mechanisms):
# Turn on accumulation
switch_integrator_mode(mechanisms, True)
# Turn on noise
switch_noise(mechanisms, psyneulink.core.components.functions.distributionfunctions.NormalDist(mean=0, standard_deviation=unit_noise).function)
# Execute until one of the accumulators crosses the threshold
my_Stroop.termination_processing = {
pnl.TimeScale.TRIAL: pnl.While(
pass_threshold,
respond_red_accumulator,
respond_green_accumulator,
accumulator_threshold
)
}
def switch_trial_type():
# Next trial will be a processing trial
if isinstance(my_Stroop.termination_processing[pnl.TimeScale.TRIAL], pnl.AllHaveRun):
switch_to_processing_trial(mechanisms_to_update)
# Next trial will be an initialization trial
else:
switch_to_initialization_trial(mechanisms_to_update)
CN_trial_initialize_input = trial_dict(0, 0, 0, 0, 1, 0)
WR_trial_initialize_input = trial_dict(0, 0, 0, 0, 0, 1)
# Start with an initialization trial
switch_to_initialization_trial(mechanisms_to_update)
my_Stroop.run(inputs=trial_dict(0, 1, 1, 0, 1, 0),
# termination_processing=change_termination_processing,
num_trials=4,
call_after_trial=switch_trial_type)
# {colors_input_layer: [[0, 0], [1, 0]],
# words_input_layer: [[0, 0], [1, 0]],
# task_layer: [[0, 1], [0, 1]]}
# This script implements Figure 1 of Botvinick, M. M., Braver, T. S., Barch, D. M., Carter, C. S., & Cohen, J. D. (2001).
# Conflict monitoring and cognitive control. Psychological Review, 108, 624–652.
# http://dx.doi.org/10.1037/0033-295X.108.3.624
# Figure 1 plots the ENERGY computed by a conflict mechanism. It is highest for incongruent trials,
# and similar for congruent and neutral trials.
# Noise is turned of and for each condition we ran one trial only. A response threshold was not defined. Responses were
# made at the marked * signs in the figure.
# Note that this script implements a slightly different Figure than in the original Figure in the paper.
# However, this implementation is identical with a plot we created with an old MATLAB code which was used for the
# conflict monitoring simulations.
# def test_botvinick_model(self):
#
# colors_input_layer = pnl.TransferMechanism(size=3,
# function=pnl.Linear,
# name='COLORS_INPUT')
#
# words_input_layer = pnl.TransferMechanism(size=3,
# function=pnl.Linear,
# name='WORDS_INPUT')
#
# task_input_layer = pnl.TransferMechanism(size=2,
# function=pnl.Linear,
# name='TASK_INPUT')
#
# task_layer = pnl.RecurrentTransferMechanism(size=2,
# function=pnl.Logistic(),
# hetero=-2,
# integrator_mode=True,
# integration_rate=0.01,
# name='TASK_LAYER')
#
# colors_hidden_layer = pnl.RecurrentTransferMechanism(size=3,
# function=pnl.Logistic(bias=4.0),
# # bias 4.0 is -4.0 in the paper see Docs for description
# integrator_mode=True,
# hetero=-2,
# integration_rate=0.01, # cohen-huston text says 0.01
# name='COLORS_HIDDEN')
#
# words_hidden_layer = pnl.RecurrentTransferMechanism(size=3,
# function=pnl.Logistic(bias=4.0),
# integrator_mode=True,
# hetero=-2,
# integration_rate=0.01,
# name='WORDS_HIDDEN')
#
# # Response layer, responses: ('red', 'green')
# response_layer = pnl.RecurrentTransferMechanism(size=2,
# function=pnl.Logistic(),
# hetero=-2.0,
# integrator_mode=True,
# integration_rate=0.01,
# output_ports=[pnl.RESULT,
# {pnl.NAME: 'DECISION_ENERGY',
# pnl.VARIABLE: (pnl.OWNER_VALUE, 0),
# pnl.FUNCTION: pnl.Stability(
# default_variable=np.array([0.0, 0.0]),
# metric=pnl.ENERGY,
# matrix=np.array([[0.0, -4.0],
# [-4.0, 0.0]]))}],
# name='RESPONSE', )
#
# response_layer.set_log_conditions('DECISION_ENERGY')
#
# color_input_weights = pnl.MappingProjection(matrix=np.array([[1.0, 0.0, 0.0],
# [0.0, 1.0, 0.0],
# [0.0, 0.0, 1.0]]))
#
# word_input_weights = pnl.MappingProjection(matrix=np.array([[1.0, 0.0, 0.0],
# [0.0, 1.0, 0.0],
# [0.0, 0.0, 1.0]]))
#
# task_input_weights = pnl.MappingProjection(matrix=np.array([[1.0, 0.0],
# [0.0, 1.0]]))
#
# color_task_weights = pnl.MappingProjection(matrix=np.array([[4.0, 0.0],
# [4.0, 0.0],
# [4.0, 0.0]]))
#
# task_color_weights = pnl.MappingProjection(matrix=np.array([[4.0, 4.0, 4.0],
# [0.0, 0.0, 0.0]]))
#
# response_color_weights = pnl.MappingProjection(matrix=np.array([[1.5, 0.0, 0.0],
# [0.0, 1.5, 0.0]]))
#
# response_word_weights = pnl.MappingProjection(matrix=np.array([[2.5, 0.0, 0.0],
# [0.0, 2.5, 0.0]]))
#
# color_response_weights = pnl.MappingProjection(matrix=np.array([[1.5, 0.0],
# [0.0, 1.5],
# [0.0, 0.0]]))
#
# word_response_weights = pnl.MappingProjection(matrix=np.array([[2.5, 0.0],
# [0.0, 2.5],
# [0.0, 0.0]]))
#
# word_task_weights = pnl.MappingProjection(matrix=np.array([[0.0, 4.0],
# [0.0, 4.0],
# [0.0, 4.0]]))
#
# task_word_weights = pnl.MappingProjection(matrix=np.array([[0.0, 0.0, 0.0],
# [4.0, 4.0, 4.0]]))
#
# # CREATE Composition
# comp = pnl.Composition()
#
# comp.add_linear_processing_pathway([colors_input_layer,
# color_input_weights,
# colors_hidden_layer,
# color_response_weights,
# response_layer])
# comp.add_projection(response_color_weights, response_layer, colors_hidden_layer)
#
# comp.add_linear_processing_pathway([words_input_layer,
# word_input_weights,
# words_hidden_layer,
# word_response_weights,
# response_layer])
# comp.add_projection(response_word_weights, response_layer, words_hidden_layer)
#
# comp.add_projection(task_input_weights, task_input_layer, task_layer)
#
# comp.add_projection(task_color_weights, task_layer, colors_hidden_layer)
# comp.add_projection(color_task_weights, colors_hidden_layer, task_layer)
#
# comp.add_projection(task_word_weights, task_layer, words_hidden_layer)
# comp.add_projection(word_task_weights, words_hidden_layer, task_layer)
#
# def trial_dict(red_color, green_color, neutral_color, red_word, green_word, neutral_word, CN, WR):
# trialdict = {
# colors_input_layer: [red_color, green_color, neutral_color],
# words_input_layer: [red_word, green_word, neutral_word],
# task_input_layer: [CN, WR]
# }
# return trialdict
#
# # Define initialization trials separately
# CN_trial_initialize_input = trial_dict(0, 0, 0, 0, 0, 0, 1,
# 0) # red_color, green color, red_word, green word, CN, WR
# CN_incongruent_trial_input = trial_dict(1, 0, 0, 0, 1, 0, 1,
# 0) # red_color, green color, red_word, green word, CN, WR
# CN_congruent_trial_input = trial_dict(1, 0, 0, 1, 0, 0, 1,
# 0) # red_color, green color, red_word, green word, CN, WR
# CN_control_trial_input = trial_dict(1, 0, 0, 0, 0, 1, 1,
# 0) # red_color, green color, red_word, green word, CN, WR
#
# Stimulus = [[CN_trial_initialize_input, CN_congruent_trial_input],
# [CN_trial_initialize_input, CN_incongruent_trial_input],
# [CN_trial_initialize_input, CN_control_trial_input]]
#
# # should be 500 and 1000
# ntrials0 = 50
# ntrials = 100
#
# def run():
# results = []
# for stim in Stimulus:
# # RUN the SYSTEM to initialize ----------------------------------------------------------------------------------------
# res = comp.run(inputs=stim[0], num_trials=ntrials0)
# results.append(res)
# res = comp.run(inputs=stim[1], num_trials=ntrials)
# results.append(res)
# # reset after condition was run
# colors_hidden_layer.reset([[0, 0, 0]])
# words_hidden_layer.reset([[0, 0, 0]])
# response_layer.reset([[0, 0]])
# task_layer.reset([[0, 0]])
# comp.reset()
#
# return results
#
# res = run()
# assert np.allclose(res[0], [0.04946301, 0.04946301, 0.03812533])
# assert np.allclose(res[1], [0.20351701, 0.11078586, 0.04995664])
# assert np.allclose(res[2], [0.04946301, 0.04946301, 0.03812533])
# assert np.allclose(res[3], [0.11168014, 0.20204928, 0.04996308])
# assert np.allclose(res[4], [0.05330691, 0.05330691, 0.03453411])
# assert np.allclose(res[5], [0.11327619, 0.11238362, 0.09399782])
#
# if mode == 'LLVM':
# return
# r2 = response_layer.log.nparray_dictionary('DECISION_ENERGY') # get logged DECISION_ENERGY dictionary
# energy = r2['DECISION_ENERGY'] # save logged DECISION_ENERGY
#
# assert np.allclose(energy[:450],
# [0.9907482, 0.98169891, 0.97284822, 0.96419228, 0.95572727, 0.94744946,
# 0.93935517, 0.93144078, 0.92370273, 0.91613752, 0.90874171, 0.90151191,
# 0.89444481, 0.88753715, 0.88078573, 0.8741874, 0.8677391, 0.86143779,
# 0.85528051, 0.84926435, 0.84338646, 0.83764405, 0.83203438, 0.82655476,
# 0.82120257, 0.81597521, 0.81087018, 0.805885, 0.80101724, 0.79626453,
# 0.79162455, 0.78709503, 0.78267373, 0.77835847, 0.77414712, 0.77003759,
# 0.76602783, 0.76211584, 0.75829965, 0.75457736, 0.75094707, 0.74740696,
# 0.74395521, 0.74059008, 0.73730983, 0.73411278, 0.73099728, 0.72796172,
# 0.7250045, 0.72212409, 0.71932195, 0.71660193, 0.7139626, 0.71140257,
# 0.70892047, 0.70651498, 0.70418478, 0.70192859, 0.69974515, 0.69763324,
# 0.69559166, 0.69361922, 0.69171476, 0.68987717, 0.68810533, 0.68639816,
# 0.68475458, 0.68317357, 0.6816541, 0.68019517, 0.67879579, 0.67745502,
# 0.6761719, 0.67494551, 0.67377494, 0.67265932, 0.67159776, 0.67058942,
# 0.66963346, 0.66872906, 0.66787541, 0.66707173, 0.66631723, 0.66561116,
# 0.66495278, 0.66434134, 0.66377614, 0.66325648, 0.66278164, 0.66235097,
# 0.66196379, 0.66161945, 0.66131731, 0.66105673, 0.66083709, 0.6606578,
# 0.66051824, 0.66041784, 0.66035601, 0.66033219, 0.66034583, 0.66039637,
# 0.66048328, 0.66060603, 0.6607641, 0.66095698, 0.66118416, 0.66144515,
# 0.66173948, 0.66206664, 0.66242619, 0.66281766, 0.66324058, 0.66369451,
# 0.66417902, 0.66469367, 0.66523802, 0.66581167, 0.66641419, 0.66704517,
# 0.66770422, 0.66839095, 0.66910495, 0.66984584, 0.67061326, 0.67140681,
# 0.67222614, 0.67307088, 0.67394067, 0.67483517, 0.67575402, 0.67669687,
# 0.67766339, 0.67865325, 0.67966612, 0.68070166, 0.68175955, 0.68283949,
# 0.68394114, 0.68506421, 0.68620839, 0.68737336, 0.68855885, 0.68976453,
# 0.69099013, 0.69223536, 0.69349992, 0.69478353, 0.69608592, 0.6974068,
# 0.9907482, 0.98169891, 0.97284822, 0.96419228, 0.95572727, 0.94744946,
# 0.93935517, 0.93144078, 0.92370273, 0.91613752, 0.90874171, 0.90151191,
# 0.89444481, 0.88753715, 0.88078573, 0.8741874, 0.8677391, 0.86143779,
# 0.85528051, 0.84926435, 0.84338646, 0.83764405, 0.83203438, 0.82655476,
# 0.82120257, 0.81597521, 0.81087018, 0.805885, 0.80101724, 0.79626453,
# 0.79162455, 0.78709503, 0.78267373, 0.77835847, 0.77414712, 0.77003759,
# 0.76602783, 0.76211584, 0.75829965, 0.75457736, 0.75094707, 0.74740696,
# 0.74395521, 0.74059008, 0.73730983, 0.73411278, 0.73099728, 0.72796172,
# 0.7250045, 0.72212409, 0.71932195, 0.71660193, 0.7139626, 0.71140257,
# 0.70892048, 0.70651499, 0.70418479, 0.70192861, 0.69974518, 0.69763329,
# 0.69559172, 0.69361931, 0.69171489, 0.68987734, 0.68810556, 0.68639845,
# 0.68475496, 0.68317405, 0.6816547, 0.68019591, 0.6787967, 0.67745612,
# 0.67617322, 0.67494709, 0.67377682, 0.67266154, 0.67160036, 0.67059245,
# 0.66963697, 0.6687331, 0.66788005, 0.66707703, 0.66632327, 0.66561801,
# 0.66496052, 0.66435007, 0.66378595, 0.66326745, 0.6627939, 0.66236463,
# 0.66197896, 0.66163626, 0.66133589, 0.66107724, 0.66085967, 0.66068261,
# 0.66054545, 0.66044762, 0.66038855, 0.66036769, 0.66038449, 0.66043841,
# 0.66052892, 0.66065551, 0.66081767, 0.6610149, 0.6612467, 0.6615126,
# 0.66181212, 0.66214479, 0.66251017, 0.6629078, 0.66333724, 0.66379805,
# 0.66428981, 0.66481211, 0.66536452, 0.66594665, 0.66655809, 0.66719846,
# 0.66786737, 0.66856444, 0.66928929, 0.67004157, 0.67082092, 0.67162697,
# 0.67245938, 0.67331781, 0.67420192, 0.67511137, 0.67604585, 0.67700502,
# 0.67798857, 0.67899619, 0.68002758, 0.68108242, 0.68216042, 0.6832613,
# 0.68438475, 0.68553049, 0.68669824, 0.68788774, 0.68909869, 0.69033084,
# 0.69158392, 0.69285767, 0.69415183, 0.69546615, 0.69680037, 0.69815425,
# 0.9907482, 0.98169891, 0.97284822, 0.96419228, 0.95572727, 0.94744946,
# 0.93935517, 0.93144078, 0.92370273, 0.91613752, 0.90874171, 0.90151191,
# 0.89444481, 0.88753715, 0.88078573, 0.8741874, 0.8677391, 0.86143779,
# 0.85528051, 0.84926435, 0.84338646, 0.83764405, 0.83203438, 0.82655476,
# 0.82120257, 0.81597521, 0.81087018, 0.805885, 0.80101724, 0.79626453,
# 0.79162455, 0.78709503, 0.78267373, 0.77835847, 0.77414712, 0.77003759,
# 0.76602783, 0.76211584, 0.75829965, 0.75457736, 0.75094707, 0.74740696,
# 0.74395521, 0.74059008, 0.73730983, 0.73411278, 0.73099728, 0.72796172,
# 0.7250045, 0.72212409, 0.71932195, 0.7165966, 0.71394661, 0.71137057,
# 0.70886708, 0.70643479, 0.70407238, 0.70177856, 0.69955206, 0.69739162,
# 0.69529605, 0.69326414, 0.69129474, 0.68938671, 0.68753892, 0.6857503,
# 0.68401976, 0.68234626, 0.68072878, 0.67916632, 0.67765789, 0.67620253,
# 0.67479929, 0.67344727, 0.67214554, 0.67089324, 0.66968949, 0.66853344,
# 0.66742427, 0.66636116, 0.66534331, 0.66436994, 0.66344028, 0.66255359,
# 0.66170913, 0.66090618, 0.66014404, 0.65942201, 0.65873941, 0.65809559,
# 0.65748989, 0.65692167, 0.6563903, 0.65589518, 0.6554357, 0.65501128,
# 0.65462132, 0.65426528, 0.65394258, 0.65365269, 0.65339506, 0.65316919,
# 0.65297454, 0.65281061, 0.65267692, 0.65257297, 0.65249828, 0.65245239,
# 0.65243484, 0.65244517, 0.65248294, 0.65254773, 0.65263909, 0.65275661,
# 0.65289989, 0.6530685, 0.65326207, 0.65348019, 0.65372249, 0.65398859,
# 0.65427811, 0.6545907, 0.65492599, 0.65528364, 0.6556633, 0.65606463,
# 0.65648729, 0.65693097, 0.65739533, 0.65788006, 0.65838485, 0.65890938,
# 0.65945336, 0.6600165, 0.66059849, 0.66119904, 0.66181789, 0.66245474,
# 0.66310932, 0.66378136, 0.6644706, 0.66517677, 0.66589961, 0.66663887,
# 0.66739429, 0.66816564, 0.66895265, 0.66975511, 0.67057276, 0.67140537],
# atol=1e-02)