def evaluateTotalError(f, g, h, a, b, n):
'''
args: f - function being interpolated
g - equally spaced polynomial
h - Chebyshev polynomial
a - left endpoint of integral
b - right endpoint of integral
n - number of
returns: total error
Numerically integrates |f(x)-g(x)|
from a to b
This is exactly the "difference"
between the original function and
the interpolation polynomial that
is currently being analyzed.
'''
g_error = definite_integral(f - g, x, a, b)
h_error = definite_integral(f - h, x, a, b)
print("|f(x)-g(x)| from " + a + " to " + b + ": " + g_error)
print("|f(x)-h(x)| from " + a + " to " + b + ": " + h_error)
return total_error
def I4_GST(C_in, b):
"""
This function calculates the 4th invariant (a.k.a., GST invariant)
# Input:
# C_in -- Right Cauchy-Green deformation tensor
# b -- controlling parameter for fiber dispersion
# The mean fiber direciton is assumed to be aligned along the x direction
# The fiber distribution rho=e^(b*(cos(2theta)+1))
"""
from sage.symbolic.integration.integral import definite_integral
var('theta, phi')
wlambda=(lambda_f(C_in, theta=theta, phi=phi)
*rhom(theta, phi, b=b)).simplify_full()
e_int_phi=definite_integral(wlambda, phi, 0, 2*pi)
# KKK here actually equals to 4*pi*K
kkk=numerical_integral(rhom(theta, phi=0, b=b)
*sin(theta), 0, pi)[0]*2*pi
ops=e_int_phi.expand().operands()
E_i=0
for iop in ops:
# separate each term into a part that can be numerically integrated and a part that can't be numerically integrated
(fun_wvar, fun_wovar)=separate(iop, theta)
tobeitgd=sin(theta)*fun_wvar
itgd=numerical_integral(tobeitgd, 0, pi)[0]/kkk * fun_wovar
E_i+=itgd
return (E_i-1.)
def I4_CH(C_in, b):
r"""
This function calculates the characteristic invariant
Lazy bone version -- using numerical integral :)
"""
from sage.symbolic.integration.integral import definite_integral
var('theta, phi')
wlambda=((lambda_f(C_in, theta=theta, phi=phi)-1)**2
*rhom(theta, phi, b=b))
e_int_phi=definite_integral(wlambda, phi, 0, 2*pi)
# KKK here actually equals to 4*pi*K
kkk=numerical_integral(rhom(theta, phi=0, b=b)
*sin(theta), 0, pi)[0]*2*pi
ops=e_int_phi.expand().operands()
Lambda_i=0
for iop in ops:
# separate each term into a part that can be numerically integrated and a part that can't be numerically integrated
(fun_wvar, fun_wovar)=separate(iop, theta)
tobeitgd=sin(theta)*fun_wvar
itgd=numerical_integral(tobeitgd, 0, pi)[0]/kkk * fun_wovar
Lambda_i+=itgd
return Lambda_i
def thomson():
var("theta,phi,beta,S0,S1,S2,S3,a", domain="real")
S = vector([S0, S1, S2, S3])
M1 = MuellerMatrixRotation(beta)
M2 = jones_to_mueller(JonesMatrixThomson(theta))
# print mueller_to_jones(M2)
M2 = M2.substitute(theta=acos(sqrt(_sage_const_2 * a -
_sage_const_1))).simplify()
M = (M2 * M1).simplify()
print "\nMueller matrix:"
print M
S_scat = (M * S).simplify()
I_scat = S_scat[_sage_const_0].substitute(beta=pi / _sage_const_2 -
phi).simplify()
print "\nScattered intensity:"
print I_scat
It_scat = I_scat.substitute(a=(_sage_const_1 + cos(theta)**_sage_const_2) /
_sage_const_2)
print "\nScattered intensity (unpol):"
print It_scat.substitute(S1=_sage_const_0).substitute(
S2=_sage_const_0).simplify_full()
print "\nScattered intensity (horizontal linear pol.):"
print It_scat.substitute(S1=S0).substitute(
S2=_sage_const_0).simplify_full()
It_scat = definite_integral(It_scat * sin(theta), theta, _sage_const_0, pi)
It_scat = definite_integral(It_scat, phi, _sage_const_0,
_sage_const_2 * pi)
print "\nThomson cross-section (units of r_e^2):"
print It_scat / S0
def convolution(cls, self, parameters, variable, other):
"""
Return the convolution function,
`f*g(t)=\int_{-\infty}^\infty f(u)g(t-u)du`, for compactly
supported `f,g`.
EXAMPLES::
sage: x = PolynomialRing(QQ,'x').gen()
sage: f = piecewise([[[0,1],1]]) ## example 0
sage: g = f.convolution(f); g
piecewise(x|-->x on (0, 1], x|-->-x + 2 on (1, 2]; x)
sage: h = f.convolution(g); h
piecewise(x|-->1/2*x^2 on (0, 1], x|-->-x^2 + 3*x - 3/2 on (1, 2], x|-->1/2*x^2 - 3*x + 9/2 on (2, 3]; x)
sage: f = piecewise([[(0,1),1],[(1,2),2],[(2,3),1]]) ## example 1
sage: g = f.convolution(f)
sage: h = f.convolution(g); h
piecewise(x|-->1/2*x^2 on (0, 1], x|-->2*x^2 - 3*x + 3/2 on (1, 3], x|-->-2*x^2 + 21*x - 69/2 on (3, 4], x|-->-5*x^2 + 45*x - 165/2 on (4, 5], x|-->-2*x^2 + 15*x - 15/2 on (5, 6], x|-->2*x^2 - 33*x + 273/2 on (6, 8], x|-->1/2*x^2 - 9*x + 81/2 on (8, 9]; x)
sage: f = piecewise([[(-1,1),1]]) ## example 2
sage: g = piecewise([[(0,3),x]])
sage: f.convolution(g)
piecewise(x|-->1/2*x^2 + x + 1/2 on (-1, 1], x|-->2*x on (1, 2], x|-->-1/2*x^2 + x + 4 on (2, 4]; x)
sage: g = piecewise([[(0,3),1],[(3,4),2]])
sage: f.convolution(g)
piecewise(x|-->x + 1 on (-1, 1], x|-->2 on (1, 2], x|-->x on (2, 3], x|-->-x + 6 on (3, 4], x|-->-2*x + 10 on (4, 5]; x)
Check that the bugs raised in :trac:`12123` are fixed::
sage: f = piecewise([[(-2, 2), 2]])
sage: g = piecewise([[(0, 2), 3/4]])
sage: f.convolution(g)
piecewise(x|-->3/2*x + 3 on (-2, 0], x|-->3 on (0, 2], x|-->-3/2*x + 6 on (2, 4]; x)
sage: f = piecewise([[(-1, 1), 1]])
sage: g = piecewise([[(0, 1), x], [(1, 2), -x + 2]])
sage: f.convolution(g)
piecewise(x|-->1/2*x^2 + x + 1/2 on (-1, 0], x|-->-1/2*x^2 + x + 1/2 on (0, 2], x|-->1/2*x^2 - 3*x + 9/2 on (2, 3]; x)
"""
from sage.symbolic.integration.integral import definite_integral
f = self
g = other
if len(f.end_points())*len(g.end_points()) == 0:
raise ValueError('one of the piecewise functions is nowhere defined')
M = min(min(f.end_points()),min(g.end_points()))
N = max(max(f.end_points()),max(g.end_points()))
tt = SR.var('tt')
uu = SR.var('uu')
conv = 0
fd,f0 = parameters[0]
gd,g0 = next(other.items())
if len(f)==1 and len(g)==1:
f = f.unextend_zero()
g = g.unextend_zero()
a1 = fd[0].lower()
a2 = fd[0].upper()
b1 = gd[0].lower()
b2 = gd[0].upper()
i1 = f0.subs({variable: uu})
i2 = g0.subs({variable: tt-uu})
fg1 = definite_integral(i1*i2, uu, a1, tt-b1).subs(tt = variable)
fg2 = definite_integral(i1*i2, uu, tt-b2, tt-b1).subs(tt = variable)
fg3 = definite_integral(i1*i2, uu, tt-b2, a2).subs(tt = variable)
fg4 = definite_integral(i1*i2, uu, a1, a2).subs(tt = variable)
if a1-b1<a2-b2:
if a2+b1!=a1+b2:
h = piecewise([[(a1+b1,a1+b2),fg1],[(a1+b2,a2+b1),fg2],[(a2+b1,a2+b2),fg3]])
else:
h = piecewise([[(a1+b1,a1+b2),fg1],[(a1+b2,a2+b2),fg3]])
else:
if a1+b2!=a2+b1:
h = piecewise([[(a1+b1,a2+b1),fg1],[(a2+b1,a1+b2),fg4],[(a1+b2,a2+b2),fg3]])
else:
h = piecewise([[(a1+b1,a2+b1),fg1],[(a2+b1,a2+b2),fg3]])
return (piecewise([[(minus_infinity,infinity),0]]).piecewise_add(h)).unextend_zero()
if len(f)>1 or len(g)>1:
z = piecewise([[(0,0),0]])
for fpiece in f.pieces():
for gpiece in g.pieces():
h = gpiece.convolution(fpiece)
z = z.piecewise_add(h)
return z.unextend_zero()
def compton():
var("theta,phi,beta,S0,S1,S2,S3,a,c,costheta,s,E,Esc", domain="real")
assume(E > _sage_const_0)
assume(Esc > _sage_const_0)
S = vector([S0, S1, S2, S3])
M1 = MuellerMatrixRotation(beta)
M2 = (matrix(
SR,
_sage_const_4,
_sage_const_4,
[
a + c,
_sage_const_1 - a,
_sage_const_0,
_sage_const_0,
_sage_const_1 - a,
a,
_sage_const_0,
_sage_const_0,
_sage_const_0,
_sage_const_0,
cos(theta),
_sage_const_0,
_sage_const_0,
_sage_const_0,
_sage_const_0,
cos(theta) * (_sage_const_1 + c),
],
) * s)
# print mueller_to_jones(M2)
M2 = M2.substitute(theta=acos(sqrt(_sage_const_2 * a -
_sage_const_1))).simplify()
M = (M2 * M1).simplify()
print "\nMueller matrix:"
print M
S_scat = (M * S).simplify()
I_scat = S_scat[_sage_const_0].substitute(beta=pi / _sage_const_2 -
phi).simplify()
print "\nScattered intensity:"
print I_scat
# Energy in units of m_e*c^2
It_scat = I_scat.substitute(a=(_sage_const_1 + cos(theta)**_sage_const_2) /
_sage_const_2)
It_scat = It_scat.substitute(s=Esc**_sage_const_2 / E**_sage_const_2)
It_scat = It_scat.substitute(c=(E - Esc) / _sage_const_2 *
(_sage_const_1 - cos(theta)))
It_scat = It_scat.simplify_full()
print "\nScattered intensity (unpol):"
print It_scat.substitute(S1=_sage_const_0).substitute(
S2=_sage_const_0).simplify_full()
It_scatu1 = It_scat.substitute(S1=_sage_const_0).substitute(
S2=_sage_const_0)
It_scatu2 = (Esc**_sage_const_2 / E**_sage_const_2 * S0 *
(E / Esc + Esc / E - sin(theta)**_sage_const_2) /
_sage_const_2)
It_scatu1 = (It_scatu1.substitute(
Esc=E /
(_sage_const_1 + E *
(_sage_const_1 - cos(theta)))).simplify_full().simplify_trig())
It_scatu2 = (It_scatu2.substitute(
Esc=E /
(_sage_const_1 + E *
(_sage_const_1 - cos(theta)))).simplify_full().simplify_trig())
print bool(It_scatu1 == It_scatu2)
It_scatu1 = It_scat.substitute(S1=S0).substitute(S2=_sage_const_0)
It_scatu2 = (
Esc**_sage_const_2 / E**_sage_const_2 * S0 *
(E / Esc + Esc / E -
_sage_const_2 * sin(theta)**_sage_const_2 * cos(phi)**_sage_const_2) /
_sage_const_2)
It_scatu1 = (It_scatu1.substitute(
Esc=E /
(_sage_const_1 + E *
(_sage_const_1 - cos(theta)))).simplify_full().simplify_trig())
It_scatu2 = (It_scatu2.substitute(
Esc=E /
(_sage_const_1 + E *
(_sage_const_1 - cos(theta)))).simplify_full().simplify_trig())
print bool(It_scatu1 == It_scatu2)
print "\nScattered intensity (horizontal linear pol.):"
print It_scat.substitute(S1=S0).substitute(
S2=_sage_const_0).simplify_full()
It_scat = It_scat.substitute(Esc=E / (_sage_const_1 + E *
(_sage_const_1 - cos(theta))))
It_scat = It_scat.simplify_full()
It_scat = definite_integral(It_scat * sin(theta), theta, _sage_const_0, pi)
It_scat = definite_integral(It_scat, phi, _sage_const_0,
_sage_const_2 * pi)
print "\nKlein-Nishina cross-section (units of r_e^2):"
print(It_scat / S0).simplify_full()
def convolution(cls, self, parameters, variable, other):
"""
Return the convolution function,
`f*g(t)=\int_{-\infty}^\infty f(u)g(t-u)du`, for compactly
supported `f,g`.
EXAMPLES::
sage: x = PolynomialRing(QQ,'x').gen()
sage: f = piecewise([[[0,1],1]]) ## example 0
sage: g = f.convolution(f); g
piecewise(x|-->x on (0, 1], x|-->-x + 2 on (1, 2]; x)
sage: h = f.convolution(g); h
piecewise(x|-->1/2*x^2 on (0, 1], x|-->-x^2 + 3*x - 3/2 on (1, 2], x|-->1/2*x^2 - 3*x + 9/2 on (2, 3]; x)
sage: f = piecewise([[(0,1),1],[(1,2),2],[(2,3),1]]) ## example 1
sage: g = f.convolution(f)
sage: h = f.convolution(g); h
piecewise(x|-->1/2*x^2 on (0, 1], x|-->2*x^2 - 3*x + 3/2 on (1, 3], x|-->-2*x^2 + 21*x - 69/2 on (3, 4], x|-->-5*x^2 + 45*x - 165/2 on (4, 5], x|-->-2*x^2 + 15*x - 15/2 on (5, 6], x|-->2*x^2 - 33*x + 273/2 on (6, 8], x|-->1/2*x^2 - 9*x + 81/2 on (8, 9]; x)
sage: f = piecewise([[(-1,1),1]]) ## example 2
sage: g = piecewise([[(0,3),x]])
sage: f.convolution(g)
piecewise(x|-->1/2*x^2 + x + 1/2 on (-1, 1], x|-->2*x on (1, 2], x|-->-1/2*x^2 + x + 4 on (2, 4]; x)
sage: g = piecewise([[(0,3),1],[(3,4),2]])
sage: f.convolution(g)
piecewise(x|-->x + 1 on (-1, 1], x|-->2 on (1, 2], x|-->x on (2, 3], x|-->-x + 6 on (3, 4], x|-->-2*x + 10 on (4, 5]; x)
Check that the bugs raised in :trac:`12123` are fixed::
sage: f = piecewise([[(-2, 2), 2]])
sage: g = piecewise([[(0, 2), 3/4]])
sage: f.convolution(g)
piecewise(x|-->3/2*x + 3 on (-2, 0], x|-->3 on (0, 2], x|-->-3/2*x + 6 on (2, 4]; x)
sage: f = piecewise([[(-1, 1), 1]])
sage: g = piecewise([[(0, 1), x], [(1, 2), -x + 2]])
sage: f.convolution(g)
piecewise(x|-->1/2*x^2 + x + 1/2 on (-1, 0], x|-->-1/2*x^2 + x + 1/2 on (0, 2], x|-->1/2*x^2 - 3*x + 9/2 on (2, 3]; x)
"""
from sage.symbolic.integration.integral import definite_integral
f = self
g = other
if len(f.end_points()) * len(g.end_points()) == 0:
raise ValueError(
'one of the piecewise functions is nowhere defined')
M = min(min(f.end_points()), min(g.end_points()))
N = max(max(f.end_points()), max(g.end_points()))
tt = SR.var('tt')
uu = SR.var('uu')
conv = 0
fd, f0 = parameters[0]
gd, g0 = next(other.items())
if len(f) == 1 and len(g) == 1:
f = f.unextend_zero()
g = g.unextend_zero()
a1 = fd[0].lower()
a2 = fd[0].upper()
b1 = gd[0].lower()
b2 = gd[0].upper()
i1 = f0.subs({variable: uu})
i2 = g0.subs({variable: tt - uu})
fg1 = definite_integral(i1 * i2, uu, a1,
tt - b1).subs(tt=variable)
fg2 = definite_integral(i1 * i2, uu, tt - b2,
tt - b1).subs(tt=variable)
fg3 = definite_integral(i1 * i2, uu, tt - b2,
a2).subs(tt=variable)
fg4 = definite_integral(i1 * i2, uu, a1, a2).subs(tt=variable)
if a1 - b1 < a2 - b2:
if a2 + b1 != a1 + b2:
h = piecewise([[(a1 + b1, a1 + b2), fg1],
[(a1 + b2, a2 + b1), fg2],
[(a2 + b1, a2 + b2), fg3]])
else:
h = piecewise([[(a1 + b1, a1 + b2), fg1],
[(a1 + b2, a2 + b2), fg3]])
else:
if a1 + b2 != a2 + b1:
h = piecewise([[(a1 + b1, a2 + b1), fg1],
[(a2 + b1, a1 + b2), fg4],
[(a1 + b2, a2 + b2), fg3]])
else:
h = piecewise([[(a1 + b1, a2 + b1), fg1],
[(a2 + b1, a2 + b2), fg3]])
return (piecewise([[(minus_infinity, infinity),
0]]).piecewise_add(h)).unextend_zero()
if len(f) > 1 or len(g) > 1:
z = piecewise([[(0, 0), 0]])
for fpiece in f.pieces():
for gpiece in g.pieces():
h = gpiece.convolution(fpiece)
z = z.piecewise_add(h)
return z.unextend_zero()
def cdf_from_pdf(self, x):
return definite_integral(self.pdf(self.tmpvar), self.tmpvar, self.a,
x).simplify_full()
var("R")
var("A")
var("B")
var("beta")
var("delta")
# Elastic scattering
Kunpol = (_sage_const_1 + (cos(theta))**_sage_const_2) / _sage_const_2
Kpol = Kunpol - (sin(theta))**_sage_const_2 / _sage_const_2 * (
P * cos(_sage_const_2 * phi) + sqrt(_sage_const_1 - P**_sage_const_2) *
cos(delta) * sin(_sage_const_2 * phi))
__tmp__ = var("theta,phi")
ElasticDiffPol2 = symbolic_expression(Kpol).function(theta, phi)
__tmp__ = var("theta")
ElasticDiffPol1 = symbolic_expression(
definite_integral(ElasticDiffPol2(theta, phi), phi, _sage_const_0,
_sage_const_2 * pi)).function(theta)
__tmp__ = var("theta,phi")
ElasticDiffUnPol2 = symbolic_expression(Kunpol).function(theta, phi)
__tmp__ = var("theta")
ElasticDiffUnPol1 = symbolic_expression(
definite_integral(ElasticDiffUnPol2(theta, phi), phi, _sage_const_0,
_sage_const_2 * pi)).function(theta)
assert ElasticDiffPol1(theta) == ElasticDiffUnPol1(theta)
assert (definite_integral(
ElasticDiffUnPol1(theta) * sin(theta), theta, _sage_const_0,
pi) == _sage_const_8 * pi / _sage_const_3)
# Inelastic scattering
__tmp__ = var("theta,phi")
InelasticDiffPol2 = symbolic_expression(