def cardan_poly_test():
#*****************************************************************************80
#
## CARDAN_POLY_TEST tests CARDAN_POLY.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 28 February 2015
#
# Author:
#
# John Burkardt
#
from cardan_poly_coef import cardan_poly_coef
from r8poly_value_horner import r8poly_value_horner
n_max = 10
s = 0.5
x = 0.25
print ''
print 'CARDAN_POLY_TEST'
print ' CARDAN_POLY evaluates a Cardan polynomial directly.'
print ''
print ' Compare CARDAN_POLY_COEF + R8POLY_VAL_HORNER'
print ' versus CARDAN_POLY alone.'
print ''
print ' Evaluate polynomials at X = %f' % (x)
print ' We use the parameter S = %f' % (s)
print ''
print ' Order Horner Direct'
print ''
cx2 = cardan_poly(n_max, x, s)
for n in range(0, n_max + 1):
c = cardan_poly_coef(n, s)
cx1 = r8poly_value_horner(n, c, x)
print ' %2d %12g %12g' % (n, cx1, cx2[n])
print ''
print 'CARDAN_POLY_TEST'
print ' Normal end of execution.'
return
def normal_01_cdf_inverse ( p ):
#*****************************************************************************80
#
## NORMAL_01_CDF_INVERSE inverts the standard normal CDF.
#
# Discussion:
#
# The result is accurate to about 1 part in 10^16.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 15 February 2015
#
# Author:
#
# Original FORTRAN77 version by Michael Wichura.
# Python version by John Burkardt.
#
# Reference:
#
# Michael Wichura,
# The Percentage Points of the Normal Distribution,
# Algorithm AS 241,
# Applied Statistics,
# Volume 37, Number 3, pages 477-484, 1988.
#
# Parameters:
#
# Input, real P, the value of the cumulative probability
# densitity function. 0 < P < 1. If P is not in this range, an "infinite"
# result is returned.
#
# Output, real VALUE, the normal deviate value with the
# property that the probability of a standard normal deviate being
# less than or equal to the value is P.
#
import numpy as np
from r8_huge import r8_huge
from r8poly_value_horner import r8poly_value_horner
a = np.array ( (\
3.3871328727963666080, 1.3314166789178437745e+2, \
1.9715909503065514427e+3, 1.3731693765509461125e+4, \
4.5921953931549871457e+4, 6.7265770927008700853e+4, \
3.3430575583588128105e+4, 2.5090809287301226727e+3 ))
b = np.array ( (\
1.0, 4.2313330701600911252e+1, \
6.8718700749205790830e+2, 5.3941960214247511077e+3, \
2.1213794301586595867e+4, 3.9307895800092710610e+4, \
2.8729085735721942674e+4, 5.2264952788528545610e+3 ))
c = np.array ( (\
1.42343711074968357734, 4.63033784615654529590, \
5.76949722146069140550, 3.64784832476320460504, \
1.27045825245236838258, 2.41780725177450611770e-1, \
2.27238449892691845833e-2, 7.74545014278341407640e-4 ))
const1 = 0.180625
const2 = 1.6
d = np.array ( (\
1.0, 2.05319162663775882187, \
1.67638483018380384940, 6.89767334985100004550e-1, \
1.48103976427480074590e-1, 1.51986665636164571966e-2, \
5.47593808499534494600e-4, 1.05075007164441684324e-9 ))
e = np.array ( (\
6.65790464350110377720, 5.46378491116411436990, \
1.78482653991729133580, 2.96560571828504891230e-1, \
2.65321895265761230930e-2, 1.24266094738807843860e-3, \
2.71155556874348757815e-5, 2.01033439929228813265e-7 ))
f = np.array ( (\
1.0, 5.99832206555887937690e-1, \
1.36929880922735805310e-1, 1.48753612908506148525e-2, \
7.86869131145613259100e-4, 1.84631831751005468180e-5, \
1.42151175831644588870e-7, 2.04426310338993978564e-15 ))
split1 = 0.425
split2 = 5.0
if ( p <= 0.0 ):
value = - r8_huge ( )
return value
if ( 1.0 <= p ):
value = r8_huge ( )
return value
q = p - 0.5
if ( abs ( q ) <= split1 ):
r = const1 - q * q
value = q * r8poly_value_horner ( 7, a, r ) \
/ r8poly_value_horner ( 7, b, r )
else:
if ( q < 0.0 ):
r = p
else:
r = 1.0 - p
if ( r <= 0.0 ):
value = r8_huge ( )
else:
r = np.sqrt ( - np.log ( r ) )
if ( r <= split2 ):
r = r - const2
value = r8poly_value_horner ( 7, c, r ) \
/ r8poly_value_horner ( 7, d, r )
else:
r = r - split2
value = r8poly_value_horner ( 7, e, r ) \
/ r8poly_value_horner ( 7, f, r )
if ( q < 0.0 ):
value = - value
return value
def zernike_poly_test ( ):
#*****************************************************************************80
#
## ZERNIKE_POLY_TEST tests ZERNIKE_POLY.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 26 February 2015
#
# Author:
#
# John Burkardt
#
from r8poly_value_horner import r8poly_value_horner
from zernike_poly_coef import zernike_poly_coef
print ''
print 'ZERNIKE_POLY_TEST'
print ' ZERNIKE_POLY evaluates a Zernike polynomial directly.'
print ''
print ' Table of polynomial coefficients:'
print ''
print ' N M'
print ''
for n in range ( 0, 6 ):
print ''
for m in range ( 0, n + 1 ):
c = zernike_poly_coef ( m, n )
print ' %2d %2d' % ( n, m ),
for i in range ( 0, n + 1 ):
print ' %7f' % ( c[i] ),
print ''
rho = 0.987654321
print ''
print ' Z1: Compute polynomial coefficients,'
print ' then evaluate by Horner\'s method;'
print ' Z2: Evaluate directly by recursion.'
print ''
print ' N M Z1 Z2'
print ''
for n in range ( 0, 6 ):
print ''
for m in range ( 0, n + 1 ):
c = zernike_poly_coef ( m, n )
z1 = r8poly_value_horner ( n, c, rho )
z2 = zernike_poly ( m, n, rho )
print ' %2d %2d %16f %16f' % ( n, m, z1, z2 )
print ''
print 'ZERNIKE_POLY_TEST'
print ' Normal end of execution.'
return
def zernike_poly_test():
#*****************************************************************************80
#
## ZERNIKE_POLY_TEST tests ZERNIKE_POLY.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 26 February 2015
#
# Author:
#
# John Burkardt
#
from r8poly_value_horner import r8poly_value_horner
from zernike_poly_coef import zernike_poly_coef
print ''
print 'ZERNIKE_POLY_TEST'
print ' ZERNIKE_POLY evaluates a Zernike polynomial directly.'
print ''
print ' Table of polynomial coefficients:'
print ''
print ' N M'
print ''
for n in range(0, 6):
print ''
for m in range(0, n + 1):
c = zernike_poly_coef(m, n)
print ' %2d %2d' % (n, m),
for i in range(0, n + 1):
print ' %7f' % (c[i]),
print ''
rho = 0.987654321
print ''
print ' Z1: Compute polynomial coefficients,'
print ' then evaluate by Horner\'s method;'
print ' Z2: Evaluate directly by recursion.'
print ''
print ' N M Z1 Z2'
print ''
for n in range(0, 6):
print ''
for m in range(0, n + 1):
c = zernike_poly_coef(m, n)
z1 = r8poly_value_horner(n, c, rho)
z2 = zernike_poly(m, n, rho)
print ' %2d %2d %16f %16f' % (n, m, z1, z2)
print ''
print 'ZERNIKE_POLY_TEST'
print ' Normal end of execution.'
return
def normal_01_cdf_inverse(p):
#*****************************************************************************80
#
## NORMAL_01_CDF_INVERSE inverts the standard normal CDF.
#
# Discussion:
#
# The result is accurate to about 1 part in 10^16.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 15 February 2015
#
# Author:
#
# Original FORTRAN77 version by Michael Wichura.
# Python version by John Burkardt.
#
# Reference:
#
# Michael Wichura,
# The Percentage Points of the Normal Distribution,
# Algorithm AS 241,
# Applied Statistics,
# Volume 37, Number 3, pages 477-484, 1988.
#
# Parameters:
#
# Input, real P, the value of the cumulative probability
# densitity function. 0 < P < 1. If P is not in this range, an "infinite"
# result is returned.
#
# Output, real VALUE, the normal deviate value with the
# property that the probability of a standard normal deviate being
# less than or equal to the value is P.
#
import numpy as np
from r8_huge import r8_huge
from r8poly_value_horner import r8poly_value_horner
a = np.array ( (\
3.3871328727963666080, 1.3314166789178437745e+2, \
1.9715909503065514427e+3, 1.3731693765509461125e+4, \
4.5921953931549871457e+4, 6.7265770927008700853e+4, \
3.3430575583588128105e+4, 2.5090809287301226727e+3 ))
b = np.array ( (\
1.0, 4.2313330701600911252e+1, \
6.8718700749205790830e+2, 5.3941960214247511077e+3, \
2.1213794301586595867e+4, 3.9307895800092710610e+4, \
2.8729085735721942674e+4, 5.2264952788528545610e+3 ))
c = np.array ( (\
1.42343711074968357734, 4.63033784615654529590, \
5.76949722146069140550, 3.64784832476320460504, \
1.27045825245236838258, 2.41780725177450611770e-1, \
2.27238449892691845833e-2, 7.74545014278341407640e-4 ))
const1 = 0.180625
const2 = 1.6
d = np.array ( (\
1.0, 2.05319162663775882187, \
1.67638483018380384940, 6.89767334985100004550e-1, \
1.48103976427480074590e-1, 1.51986665636164571966e-2, \
5.47593808499534494600e-4, 1.05075007164441684324e-9 ))
e = np.array ( (\
6.65790464350110377720, 5.46378491116411436990, \
1.78482653991729133580, 2.96560571828504891230e-1, \
2.65321895265761230930e-2, 1.24266094738807843860e-3, \
2.71155556874348757815e-5, 2.01033439929228813265e-7 ))
f = np.array ( (\
1.0, 5.99832206555887937690e-1, \
1.36929880922735805310e-1, 1.48753612908506148525e-2, \
7.86869131145613259100e-4, 1.84631831751005468180e-5, \
1.42151175831644588870e-7, 2.04426310338993978564e-15 ))
split1 = 0.425
split2 = 5.0
if (p <= 0.0):
value = -r8_huge()
return value
if (1.0 <= p):
value = r8_huge()
return value
q = p - 0.5
if (abs(q) <= split1):
r = const1 - q * q
value = q * r8poly_value_horner ( 7, a, r ) \
/ r8poly_value_horner ( 7, b, r )
else:
if (q < 0.0):
r = p
else:
r = 1.0 - p
if (r <= 0.0):
value = r8_huge()
else:
r = np.sqrt(-np.log(r))
if (r <= split2):
r = r - const2
value = r8poly_value_horner ( 7, c, r ) \
/ r8poly_value_horner ( 7, d, r )
else:
r = r - split2
value = r8poly_value_horner ( 7, e, r ) \
/ r8poly_value_horner ( 7, f, r )
if (q < 0.0):
value = -value
return value
def normal_01_cdf_inv(cdf):
#*****************************************************************************80
#
## NORMAL_01_CDF evaluates the Normal 01 CDF.
#
# Discussion:
#
# The result is accurate to about 1 part in 10^16.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 28 February 2015
#
# Author:
#
# John Burkardt
#
# Reference:
#
# Michael Wichura,
# The Percentage Points of the Normal Distribution,
# Algorithm AS 241,
# Applied Statistics,
# Volume 37, Number 3, pages 477-484, 1988.
#
# Parameters:
#
# Input, real CDF, the value of the CDF.
#
# Output, real VALUE, the argument of the CDF.
#
import numpy as np
from r8poly_value_horner import r8poly_value_horner
a = np.array ( [ \
3.3871328727963666080, 1.3314166789178437745e+2, \
1.9715909503065514427e+3, 1.3731693765509461125e+4, \
4.5921953931549871457e+4, 6.7265770927008700853e+4, \
3.3430575583588128105e+4, 2.5090809287301226727e+3 ] )
b = np.array ( [ \
1.0, 4.2313330701600911252e+1, \
6.8718700749205790830e+2, 5.3941960214247511077e+3, \
2.1213794301586595867e+4, 3.9307895800092710610e+4, \
2.8729085735721942674e+4, 5.2264952788528545610e+3 ] )
c = np.array ( [ \
1.42343711074968357734, 4.63033784615654529590, \
5.76949722146069140550, 3.64784832476320460504, \
1.27045825245236838258, 2.41780725177450611770e-1, \
2.27238449892691845833e-2, 7.74545014278341407640e-4 ] )
const1 = 0.180625
const2 = 1.6
d = np.array ( [ \
1.0, 2.05319162663775882187, \
1.67638483018380384940, 6.89767334985100004550e-1, \
1.48103976427480074590e-1, 1.51986665636164571966e-2, \
5.47593808499534494600e-4, 1.05075007164441684324e-9 ] )
e = np.array ( [ \
6.65790464350110377720, 5.46378491116411436990, \
1.78482653991729133580, 2.96560571828504891230e-1, \
2.65321895265761230930e-2, 1.24266094738807843860e-3, \
2.71155556874348757815e-5, 2.01033439929228813265e-7 ] )
f = np.array ( [ \
1.0, 5.99832206555887937690e-1, \
1.36929880922735805310e-1, 1.48753612908506148525e-2, \
7.86869131145613259100e-4, 1.84631831751005468180e-5, \
1.42151175831644588870e-7, 2.04426310338993978564e-15 ] )
r8_huge = 1.79769313486231571E+308
split1 = 0.425
split2 = 5.0
if (cdf <= 0.0):
value = -r8_huge
return value
if (1.0 <= cdf):
value = r8_huge
return value
q = cdf - 0.5
if (abs(q) <= split1):
r = const1 - q * q
value = q * r8poly_value_horner ( 7, a, r ) \
/ r8poly_value_horner ( 7, b, r )
else:
if (q < 0.0):
r = cdf
else:
r = 1.0 - cdf
if (r <= 0.0):
value = r8_huge
else:
r = np.sqrt(-np.log(r))
if (r <= split2):
r = r - const2
value = r8poly_value_horner ( 7, c, r ) \
/ r8poly_value_horner ( 7, d, r )
else:
r = r - split2
value = r8poly_value_horner ( 7, e, r ) \
/ r8poly_value_horner ( 7, f, r )
if (q < 0.0):
value = -value
return value
def normal_01_cdf_inv ( cdf ):
#*****************************************************************************80
#
## NORMAL_01_CDF evaluates the Normal 01 CDF.
#
# Discussion:
#
# The result is accurate to about 1 part in 10^16.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 28 February 2015
#
# Author:
#
# John Burkardt
#
# Reference:
#
# Michael Wichura,
# The Percentage Points of the Normal Distribution,
# Algorithm AS 241,
# Applied Statistics,
# Volume 37, Number 3, pages 477-484, 1988.
#
# Parameters:
#
# Input, real CDF, the value of the CDF.
#
# Output, real VALUE, the argument of the CDF.
#
import numpy as np
from r8poly_value_horner import r8poly_value_horner
a = np.array ( [ \
3.3871328727963666080, 1.3314166789178437745e+2, \
1.9715909503065514427e+3, 1.3731693765509461125e+4, \
4.5921953931549871457e+4, 6.7265770927008700853e+4, \
3.3430575583588128105e+4, 2.5090809287301226727e+3 ] )
b = np.array ( [ \
1.0, 4.2313330701600911252e+1, \
6.8718700749205790830e+2, 5.3941960214247511077e+3, \
2.1213794301586595867e+4, 3.9307895800092710610e+4, \
2.8729085735721942674e+4, 5.2264952788528545610e+3 ] )
c = np.array ( [ \
1.42343711074968357734, 4.63033784615654529590, \
5.76949722146069140550, 3.64784832476320460504, \
1.27045825245236838258, 2.41780725177450611770e-1, \
2.27238449892691845833e-2, 7.74545014278341407640e-4 ] )
const1 = 0.180625
const2 = 1.6
d = np.array ( [ \
1.0, 2.05319162663775882187, \
1.67638483018380384940, 6.89767334985100004550e-1, \
1.48103976427480074590e-1, 1.51986665636164571966e-2, \
5.47593808499534494600e-4, 1.05075007164441684324e-9 ] )
e = np.array ( [ \
6.65790464350110377720, 5.46378491116411436990, \
1.78482653991729133580, 2.96560571828504891230e-1, \
2.65321895265761230930e-2, 1.24266094738807843860e-3, \
2.71155556874348757815e-5, 2.01033439929228813265e-7 ] )
f = np.array ( [ \
1.0, 5.99832206555887937690e-1, \
1.36929880922735805310e-1, 1.48753612908506148525e-2, \
7.86869131145613259100e-4, 1.84631831751005468180e-5, \
1.42151175831644588870e-7, 2.04426310338993978564e-15 ] )
r8_huge = 1.79769313486231571E+308
split1 = 0.425
split2 = 5.0
if ( cdf <= 0.0 ):
value = - r8_huge
return value
if ( 1.0 <= cdf ):
value = r8_huge
return value
q = cdf - 0.5
if ( abs ( q ) <= split1 ):
r = const1 - q * q
value = q * r8poly_value_horner ( 7, a, r ) \
/ r8poly_value_horner ( 7, b, r )
else:
if ( q < 0.0 ):
r = cdf
else:
r = 1.0 - cdf
if ( r <= 0.0 ):
value = r8_huge
else:
r = np.sqrt ( - np.log ( r ) )
if ( r <= split2 ):
r = r - const2
value = r8poly_value_horner ( 7, c, r ) \
/ r8poly_value_horner ( 7, d, r )
else:
r = r - split2
value = r8poly_value_horner ( 7, e, r ) \
/ r8poly_value_horner ( 7, f, r )
if ( q < 0.0 ):
value = - value
return value
