def laplace(cls, self, parameters, variable, x='x', s='t'):
r"""
Returns the Laplace transform of self with respect to the variable
var.
INPUT:
- ``x`` - variable of self
- ``s`` - variable of Laplace transform.
We assume that a piecewise function is 0 outside of its domain and
that the left-most endpoint of the domain is 0.
EXAMPLES::
sage: x, s, w = var('x, s, w')
sage: f = piecewise([[(0,1),1],[[1,2], 1-x]])
sage: f.laplace(x, s)
-e^(-s)/s + (s + 1)*e^(-2*s)/s^2 + 1/s - e^(-s)/s^2
sage: f.laplace(x, w)
-e^(-w)/w + (w + 1)*e^(-2*w)/w^2 + 1/w - e^(-w)/w^2
::
sage: y, t = var('y, t')
sage: f = piecewise([[[1,2], 1-y]])
sage: f.laplace(y, t)
(t + 1)*e^(-2*t)/t^2 - e^(-t)/t^2
::
sage: s = var('s')
sage: t = var('t')
sage: f1(t) = -t
sage: f2(t) = 2
sage: f = piecewise([[[0,1],f1],[(1,infinity),f2]])
sage: f.laplace(t,s)
(s + 1)*e^(-s)/s^2 + 2*e^(-s)/s - 1/s^2
"""
from sage.all import assume, exp, forget
x = SR.var(x)
s = SR.var(s)
assume(s>0)
result = 0
for domain, f in parameters:
for interval in domain:
a = interval.lower()
b = interval.upper()
result += (SR(f)*exp(-s*x)).integral(x,a,b)
forget(s>0)
return result
def laplace(cls, self, parameters, variable, x='x', s='t'):
r"""
Returns the Laplace transform of self with respect to the variable
var.
INPUT:
- ``x`` - variable of self
- ``s`` - variable of Laplace transform.
We assume that a piecewise function is 0 outside of its domain and
that the left-most endpoint of the domain is 0.
EXAMPLES::
sage: x, s, w = var('x, s, w')
sage: f = piecewise([[(0,1),1],[[1,2], 1-x]])
sage: f.laplace(x, s)
-e^(-s)/s + (s + 1)*e^(-2*s)/s^2 + 1/s - e^(-s)/s^2
sage: f.laplace(x, w)
-e^(-w)/w + (w + 1)*e^(-2*w)/w^2 + 1/w - e^(-w)/w^2
::
sage: y, t = var('y, t')
sage: f = piecewise([[[1,2], 1-y]])
sage: f.laplace(y, t)
(t + 1)*e^(-2*t)/t^2 - e^(-t)/t^2
::
sage: s = var('s')
sage: t = var('t')
sage: f1(t) = -t
sage: f2(t) = 2
sage: f = piecewise([[[0,1],f1],[(1,infinity),f2]])
sage: f.laplace(t,s)
(s + 1)*e^(-s)/s^2 + 2*e^(-s)/s - 1/s^2
"""
from sage.all import assume, exp, forget
x = SR.var(x)
s = SR.var(s)
assume(s > 0)
result = 0
for domain, f in parameters:
for interval in domain:
a = interval.lower()
b = interval.upper()
result += (SR(f) * exp(-s * x)).integral(x, a, b)
forget(s > 0)
return result
def integral(cls, self, parameters, variable, x=None, a=None, b=None, definite=False):
r"""
By default, return the indefinite integral of the function.
If definite=True is given, returns the definite integral.
AUTHOR:
- Paul Butler
EXAMPLES::
sage: f1(x) = 1-x
sage: f = piecewise([((0,1),1), ((1,2),f1)])
sage: f.integral(definite=True)
1/2
::
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = piecewise([((0,pi/2),f1), ((pi/2,pi),f2)])
sage: f.integral(definite=True)
1/2*pi
sage: f1(x) = 2
sage: f2(x) = 3 - x
sage: f = piecewise([[(-2, 0), f1], [(0, 3), f2]])
sage: f.integral()
piecewise(x|-->2*x + 4 on (-2, 0), x|-->-1/2*x^2 + 3*x + 4 on (0, 3); x)
sage: f1(y) = -1
sage: f2(y) = y + 3
sage: f3(y) = -y - 1
sage: f4(y) = y^2 - 1
sage: f5(y) = 3
sage: f = piecewise([[[-4,-3],f1],[(-3,-2),f2],[[-2,0],f3],[(0,2),f4],[[2,3],f5]])
sage: F = f.integral(y)
sage: F
piecewise(y|-->-y - 4 on [-4, -3], y|-->1/2*y^2 + 3*y + 7/2 on (-3, -2), y|-->-1/2*y^2 - y - 1/2 on [-2, 0], y|-->1/3*y^3 - y - 1/2 on (0, 2), y|-->3*y - 35/6 on [2, 3]; y)
Ensure results are consistent with FTC::
sage: F(-3) - F(-4)
-1
sage: F(-1) - F(-3)
1
sage: F(2) - F(0)
2/3
sage: f.integral(y, 0, 2)
2/3
sage: F(3) - F(-4)
19/6
sage: f.integral(y, -4, 3)
19/6
sage: f.integral(definite=True)
19/6
::
sage: f1(y) = (y+3)^2
sage: f2(y) = y+3
sage: f3(y) = 3
sage: f = piecewise([[(-infinity, -3), f1], [(-3, 0), f2], [(0, infinity), f3]])
sage: f.integral()
piecewise(y|-->1/3*y^3 + 3*y^2 + 9*y + 9 on (-oo, -3), y|-->1/2*y^2 + 3*y + 9/2 on (-3, 0), y|-->3*y + 9/2 on (0, +oo); y)
::
sage: f1(x) = e^(-abs(x))
sage: f = piecewise([[(-infinity, infinity), f1]])
sage: f.integral(definite=True)
2
sage: f.integral()
piecewise(x|-->-1/2*((sgn(x) - 1)*e^(2*x) - 2*e^x*sgn(x) + sgn(x) + 1)*e^(-x) - 1 on (-oo, +oo); x)
::
sage: f = piecewise([((0, 5), cos(x))])
sage: f.integral()
piecewise(x|-->sin(x) on (0, 5); x)
TESTS:
Verify that piecewise integrals of zero work (trac #10841)::
sage: f0(x) = 0
sage: f = piecewise([[[0,1],f0]])
sage: f.integral(x,0,1)
0
sage: f = piecewise([[[0,1], 0]])
sage: f.integral(x,0,1)
0
sage: f = piecewise([[[0,1], SR(0)]])
sage: f.integral(x,0,1)
0
"""
if a != None and b != None:
F = self.integral(x)
return F(b) - F(a)
if a != None or b != None:
raise TypeError('only one endpoint given')
area = 0
new_pieces = []
if x == None:
x = self.default_variable()
# The integral is computed by iterating over the pieces in order.
# The definite integral for each piece is calculated and accumulated in `area`.
# The indefinite integral of each piece is also calculated,
# and the `area` before each piece is added to the piece.
#
# If a definite integral is requested, `area` is returned.
# Otherwise, a piecewise function is constructed from the indefinite integrals
# and returned.
#
# An exception is made if integral is called on a piecewise function
# that starts at -infinity. In this case, we do not try to calculate the
# definite integral of the first piece, and the value of `area` remains 0
# after the first piece.
from sage.symbolic.assumptions import assume, forget
for domain, fun in parameters:
for interval in domain:
start = interval.lower()
end = interval.upper()
if start == -infinity and not definite:
fun_integrated = fun.integral(x, end, x)
else:
try:
assume(start < x)
except ValueError: # Assumption is redundant
pass
fun_integrated = fun.integral(x, start, x) + area
forget(start < x)
if definite or end != infinity:
area += fun.integral(x, start, end)
new_pieces.append([interval, SR(fun_integrated).function(x)])
if definite:
return SR(area)
else:
return piecewise(new_pieces)
def curve(nb_equipes, max_points=100, K=1, R=2, base=2, verbose=False):
r"""
INPUT:
- ``nb_equipes`` -- integer
- ``max_points`` -- integer
- ``K`` -- the value at ``p = nb_equipes``
- ``R`` -- real (default: ``2``), curve parameter
- ``base`` -- 2
- ``verbose`` - bool
EXEMPLES::
sage: from slabbe.ranking_scale import curve
sage: curve(20, 100)
-99*(p*(log(40) + 1) - p*log(p) - 20*log(40) + 20*log(20) -
20)/(19*log(40) - 20*log(20) + 19) + 1
sage: curve(64, 100)
99*(p*(7*log(2) + 1) - p*log(p) + 64*log(64) - 448*log(2) -
64)/(64*log(64) - 441*log(2) - 63) + 1
::
sage: curve(64, 100)(p=64)
1
sage: curve(64, 100)(p=1)
100
sage: curve(64, 100)(p=2)
198*(32*log(64) - 218*log(2) - 31)/(64*log(64) - 441*log(2) - 63) + 1
sage: n(curve(64, 100)(p=2)) # abs tol 1e-10
95.6871477097753
::
sage: curve(64, 100, verbose=True)
fn = -(p*(7*log(2) + 1) - p*log(p) + 64*log(64) - 448*log(2) - 64)/log(2)
aire = 147.889787576005
fn normalise = 99*(p*(7*log(2) + 1) - p*log(p) + 64*log(64) - 448*log(2) - 64)/(64*log(64) - 441*log(2) - 63) + 1
99*(p*(7*log(2) + 1) - p*log(p) + 64*log(64) - 448*log(2) - 64)/(64*log(64) - 441*log(2) - 63) + 1
The base argument seems to be useless (why?)::
sage: curve(100,100,base=3)
-99*(p*(log(200) + 1) - p*log(p) - 100*log(200) + 100*log(100) -
100)/(99*log(200) - 100*log(100) + 99) + 1
sage: curve(100,100,base=2)
-99*(p*(log(200) + 1) - p*log(p) - 100*log(200) + 100*log(100) -
100)/(99*log(200) - 100*log(100) + 99) + 1
"""
from sage.symbolic.assumptions import forget, assume
from sage.misc.functional import integrate, n
from sage.functions.log import log
from sage.calculus.var import var
x, p = var('x,p')
forget()
assume(p - 1 > 0)
assume(p - nb_equipes < 0)
fn = integrate(
log(R * nb_equipes, base=base) - log(x, base=base), x, p, nb_equipes)
if verbose: print "fn = %s" % fn
aire = fn(p=1)
if verbose: print "aire = %s" % n(aire)
fn_normalise = fn / aire * (max_points - K) + K
if verbose: print "fn normalise = %s" % fn_normalise
return fn_normalise
def integral(cls,
self,
parameters,
variable,
x=None,
a=None,
b=None,
definite=False):
r"""
By default, return the indefinite integral of the function.
If definite=True is given, returns the definite integral.
AUTHOR:
- Paul Butler
EXAMPLES::
sage: f1(x) = 1-x
sage: f = piecewise([((0,1),1), ((1,2),f1)])
sage: f.integral(definite=True)
1/2
::
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = piecewise([((0,pi/2),f1), ((pi/2,pi),f2)])
sage: f.integral(definite=True)
1/2*pi
sage: f1(x) = 2
sage: f2(x) = 3 - x
sage: f = piecewise([[(-2, 0), f1], [(0, 3), f2]])
sage: f.integral()
piecewise(x|-->2*x + 4 on (-2, 0), x|-->-1/2*x^2 + 3*x + 4 on (0, 3); x)
sage: f1(y) = -1
sage: f2(y) = y + 3
sage: f3(y) = -y - 1
sage: f4(y) = y^2 - 1
sage: f5(y) = 3
sage: f = piecewise([[[-4,-3],f1],[(-3,-2),f2],[[-2,0],f3],[(0,2),f4],[[2,3],f5]])
sage: F = f.integral(y)
sage: F
piecewise(y|-->-y - 4 on [-4, -3], y|-->1/2*y^2 + 3*y + 7/2 on (-3, -2), y|-->-1/2*y^2 - y - 1/2 on [-2, 0], y|-->1/3*y^3 - y - 1/2 on (0, 2), y|-->3*y - 35/6 on [2, 3]; y)
Ensure results are consistent with FTC::
sage: F(-3) - F(-4)
-1
sage: F(-1) - F(-3)
1
sage: F(2) - F(0)
2/3
sage: f.integral(y, 0, 2)
2/3
sage: F(3) - F(-4)
19/6
sage: f.integral(y, -4, 3)
19/6
sage: f.integral(definite=True)
19/6
::
sage: f1(y) = (y+3)^2
sage: f2(y) = y+3
sage: f3(y) = 3
sage: f = piecewise([[(-infinity, -3), f1], [(-3, 0), f2], [(0, infinity), f3]])
sage: f.integral()
piecewise(y|-->1/3*y^3 + 3*y^2 + 9*y + 9 on (-oo, -3), y|-->1/2*y^2 + 3*y + 9/2 on (-3, 0), y|-->3*y + 9/2 on (0, +oo); y)
::
sage: f1(x) = e^(-abs(x))
sage: f = piecewise([[(-infinity, infinity), f1]])
sage: f.integral(definite=True)
2
sage: f.integral()
piecewise(x|-->-1/2*((sgn(x) - 1)*e^(2*x) - 2*e^x*sgn(x) + sgn(x) + 1)*e^(-x) - 1 on (-oo, +oo); x)
::
sage: f = piecewise([((0, 5), cos(x))])
sage: f.integral()
piecewise(x|-->sin(x) on (0, 5); x)
TESTS:
Verify that piecewise integrals of zero work (trac #10841)::
sage: f0(x) = 0
sage: f = piecewise([[[0,1],f0]])
sage: f.integral(x,0,1)
0
sage: f = piecewise([[[0,1], 0]])
sage: f.integral(x,0,1)
0
sage: f = piecewise([[[0,1], SR(0)]])
sage: f.integral(x,0,1)
0
"""
if a != None and b != None:
F = self.integral(x)
return F(b) - F(a)
if a != None or b != None:
raise TypeError('only one endpoint given')
area = 0
new_pieces = []
if x == None:
x = self.default_variable()
# The integral is computed by iterating over the pieces in order.
# The definite integral for each piece is calculated and accumulated in `area`.
# The indefinite integral of each piece is also calculated,
# and the `area` before each piece is added to the piece.
#
# If a definite integral is requested, `area` is returned.
# Otherwise, a piecewise function is constructed from the indefinite integrals
# and returned.
#
# An exception is made if integral is called on a piecewise function
# that starts at -infinity. In this case, we do not try to calculate the
# definite integral of the first piece, and the value of `area` remains 0
# after the first piece.
from sage.symbolic.assumptions import assume, forget
for domain, fun in parameters:
for interval in domain:
start = interval.lower()
end = interval.upper()
if start == -infinity and not definite:
fun_integrated = fun.integral(x, end, x)
else:
try:
assume(start < x)
except ValueError: # Assumption is redundant
pass
fun_integrated = fun.integral(x, start, x) + area
forget(start < x)
if definite or end != infinity:
area += fun.integral(x, start, end)
new_pieces.append(
[interval, SR(fun_integrated).function(x)])
if definite:
return SR(area)
else:
return piecewise(new_pieces)
def curve(nb_equipes, max_points=100, K=1, R=2, base=2, verbose=False):
r"""
INPUT:
- ``nb_equipes`` -- integer
- ``max_points`` -- integer
- ``K`` -- the value at ``p = nb_equipes``
- ``R`` -- real (default: ``2``), curve parameter
- ``base`` -- 2
- ``verbose`` - bool
EXEMPLES::
sage: from slabbe.ranking_scale import curve
sage: curve(20, 100)
-99*(p*(log(40) + 1) - p*log(p) - 20*log(40) + 20*log(20) -
20)/(19*log(40) - 20*log(20) + 19) + 1
sage: curve(64, 100)
-33*(p*(7*log(2) + 1) - p*log(p) - 64*log(2) - 64)/(19*log(2) + 21) + 1
::
sage: curve(64, 100)(p=64)
1
sage: curve(64, 100)(p=1)
100
sage: curve(64, 100)(p=2)
66*(26*log(2) + 31)/(19*log(2) + 21) + 1
sage: n(curve(64, 100)(p=2)) # abs tol 1e-10
95.6871477097753
::
sage: curve(64, 100, verbose=True)
fn = -(p*(7*log(2) + 1) - p*log(p) - 64*log(2) - 64)/log(2)
aire = 147.889787576005
fn normalise = -33*(p*(7*log(2) + 1) - p*log(p) - 64*log(2) - 64)/(19*log(2) + 21) + 1
-33*(p*(7*log(2) + 1) - p*log(p) - 64*log(2) - 64)/(19*log(2) + 21) + 1
The base argument seems to be useless (why?)::
sage: curve(100,100,base=3)
-99*(p*(log(200) + 1) - p*log(p) - 100*log(200) + 200*log(10) -
100)/(99*log(200) - 200*log(10) + 99) + 1
sage: curve(100,100,base=2)
-99*(p*(log(200) + 1) - p*log(p) - 100*log(200) + 200*log(10) -
100)/(99*log(200) - 200*log(10) + 99) + 1
"""
from sage.symbolic.assumptions import forget, assume
from sage.misc.functional import integrate, n
from sage.functions.log import log
from sage.calculus.var import var
x,p = var('x,p')
forget()
assume(p - 1 > 0)
assume(p-nb_equipes < 0)
fn = integrate(log(R*nb_equipes, base=base) - log(x, base=base), x, p, nb_equipes)
if verbose: print("fn = %s" % fn)
aire = fn(p=1)
if verbose: print("aire = %s" % n(aire))
fn_normalise = fn / aire * (max_points - K) + K
if verbose: print("fn normalise = %s" % fn_normalise)
return fn_normalise
