def _eval_(self, n, z):
"""
EXAMPLES::
sage: lambert_w(6.0)
1.43240477589830
sage: lambert_w(1)
lambert_w(1)
sage: lambert_w(x+1)
lambert_w(x + 1)
There are three special values which are automatically simplified::
sage: lambert_w(0)
0
sage: lambert_w(e)
1
sage: lambert_w(-1/e)
-1
sage: lambert_w(SR(0))
0
The special values only hold on the principal branch::
sage: lambert_w(1,e)
lambert_w(1, e)
sage: lambert_w(1, e.n())
-0.532092121986380 + 4.59715801330257*I
TESTS:
When automatic simplification occurs, the parent of the output
value should be either the same as the parent of the input, or
a Sage type::
sage: parent(lambert_w(int(0)))
<... 'int'>
sage: parent(lambert_w(Integer(0)))
Integer Ring
sage: parent(lambert_w(e))
Symbolic Ring
"""
if not isinstance(z, Expression):
if n == 0 and z == 0:
return s_parent(z)(0)
elif n == 0:
if z.is_trivial_zero():
return s_parent(z)(Integer(0))
elif (z-const_e).is_trivial_zero():
return s_parent(z)(Integer(1))
elif (z+1/const_e).is_trivial_zero():
return s_parent(z)(Integer(-1))
def _eval_(self, n, z):
"""
EXAMPLES::
sage: lambert_w(6.0)
1.43240477589830
sage: lambert_w(1)
lambert_w(1)
sage: lambert_w(x+1)
lambert_w(x + 1)
There are three special values which are automatically simplified::
sage: lambert_w(0)
0
sage: lambert_w(e)
1
sage: lambert_w(-1/e)
-1
sage: lambert_w(SR(0))
0
The special values only hold on the principal branch::
sage: lambert_w(1,e)
lambert_w(1, e)
sage: lambert_w(1, e.n())
-0.532092121986380 + 4.59715801330257*I
TESTS:
When automatic simplification occurs, the parent of the output
value should be either the same as the parent of the input, or
a Sage type::
sage: parent(lambert_w(int(0)))
<... 'int'>
sage: parent(lambert_w(Integer(0)))
Integer Ring
sage: parent(lambert_w(e))
Symbolic Ring
"""
if not isinstance(z, Expression):
if n == 0 and z == 0:
return s_parent(z)(0)
elif n == 0:
if z.is_trivial_zero():
return s_parent(z)(Integer(0))
elif (z - const_e).is_trivial_zero():
return s_parent(z)(Integer(1))
elif (z + 1 / const_e).is_trivial_zero():
return s_parent(z)(Integer(-1))
def _evalf_(self, x, parent=None, algorithm=None):
"""
EXAMPLES::
sage: erf(2).n()
0.995322265018953
sage: erf(2).n(200)
0.99532226501895273416206925636725292861089179704006007673835
sage: erf(pi - 1/2*I).n(100)
1.0000111669099367825726058952 + 1.6332655417638522934072124547e-6*I
TESTS:
Check that PARI/GP through the GP interface gives the same answer::
sage: gp.set_real_precision(59) # random
38
sage: print(gp.eval("1 - erfc(1)")); print(erf(1).n(200))
0.84270079294971486934122063508260925929606699796630290845994
0.84270079294971486934122063508260925929606699796630290845994
Check that for an imaginary input, the output is also imaginary, see
:trac:`13193`::
sage: erf(3.0*I)
1629.99462260157*I
sage: erf(33.0*I)
1.51286977510409e471*I
"""
R = parent or s_parent(x)
import mpmath
return mpmath_utils.call(mpmath.erf, x, parent=R)
def _evalf_(self, x, parent=None, algorithm=None):
"""
EXAMPLES::
sage: erf(2).n()
0.995322265018953
sage: erf(2).n(200)
0.99532226501895273416206925636725292861089179704006007673835
sage: erf(pi - 1/2*I).n(100)
1.0000111669099367825726058952 + 1.6332655417638522934072124547e-6*I
TESTS:
Check that PARI/GP through the GP interface gives the same answer::
sage: gp.set_real_precision(59) # random
38
sage: print(gp.eval("1 - erfc(1)")); print(erf(1).n(200))
0.84270079294971486934122063508260925929606699796630290845994
0.84270079294971486934122063508260925929606699796630290845994
Check that for an imaginary input, the output is also imaginary, see
:trac:`13193`::
sage: erf(3.0*I)
1629.99462260157*I
sage: erf(33.0*I)
1.51286977510409e471*I
"""
R = parent or s_parent(x)
import mpmath
return mpmath_utils.call(mpmath.erf, x, parent=R)
def _evalf_(self, x, parent=None, algorithm=None):
"""
EXAMPLES::
sage: erfinv(0.2)
0.179143454621292
sage: erfinv(1/5).n(100)
0.17914345462129167649274901663
"""
R = parent or s_parent(x)
import mpmath
return mpmath_utils.call(mpmath.erfinv, x, parent=R)
def _evalf_(self, x, parent=None, algorithm=None):
"""
EXAMPLES::
sage: erfinv(0.2)
0.179143454621292
sage: erfinv(1/5).n(100)
0.17914345462129167649274901663
"""
R = parent or s_parent(x)
import mpmath
return mpmath_utils.call(mpmath.erfinv, x, parent=R)
def _evalf_(self, x, parent=None, algorithm=None):
"""
EXAMPLES::
sage: erfc(4).n()
1.54172579002800e-8
sage: erfc(4).n(100)
1.5417257900280018852159673487e-8
sage: erfc(4*I).n(100)
1.0000000000000000000000000000 - 1.2969597307176392315279409506e6*I
"""
R = parent or s_parent(x)
import mpmath
return mpmath_utils.call(mpmath.erfc, x, parent=R)
def _evalf_(self, x, y, parent=None, algorithm='mpmath'):
"""
EXAMPLES::
sage: gamma_inc_lower(3,2.)
0.646647167633873
sage: gamma_inc_lower(3,2).n(200)
0.646647167633873081060005050275155...
sage: gamma_inc_lower(0,2.)
+infinity
"""
R = parent or s_parent(x)
# C is the complex version of R
# prec is the precision of R
if R is float:
prec = 53
C = complex
else:
try:
prec = R.precision()
except AttributeError:
prec = 53
try:
C = R.complex_field()
except AttributeError:
C = R
if algorithm == 'pari':
try:
v = ComplexField(prec)(x).gamma() - ComplexField(prec)(
x).gamma_inc(y)
except AttributeError:
if not (is_ComplexNumber(x)):
if is_ComplexNumber(y):
C = y.parent()
else:
C = ComplexField()
x = C(x)
v = ComplexField(prec)(x).gamma() - ComplexField(prec)(
x).gamma_inc(y)
else:
import mpmath
v = ComplexField(prec)(mpmath_utils.call(mpmath.gammainc,
x,
0,
y,
parent=R))
if v.is_real():
return R(v)
else:
return C(v)
def _evalf_(self, x, parent=None, algorithm=None):
"""
EXAMPLES::
sage: erfi(2.)
18.5648024145756
sage: erfi(2).n(100)
18.564802414575552598704291913
sage: erfi(-2*I).n(100)
-0.99532226501895273416206925637*I
"""
R = parent or s_parent(x)
import mpmath
return mpmath_utils.call(mpmath.erfi, x, parent=R)
def _evalf_(self, x, parent=None, algorithm=None):
"""
EXAMPLES::
sage: erfc(4).n()
1.54172579002800e-8
sage: erfc(4).n(100)
1.5417257900280018852159673487e-8
sage: erfc(4*I).n(100)
1.0000000000000000000000000000 - 1.2969597307176392315279409506e6*I
"""
R = parent or s_parent(x)
import mpmath
return mpmath_utils.call(mpmath.erfc, x, parent=R)
def _evalf_(self, x, parent=None, algorithm=None):
"""
EXAMPLES::
sage: erfi(2.)
18.5648024145756
sage: erfi(2).n(100)
18.564802414575552598704291913
sage: erfi(-2*I).n(100)
-0.99532226501895273416206925637*I
"""
R = parent or s_parent(x)
import mpmath
return mpmath_utils.call(mpmath.erfi, x, parent=R)
def _evalf_(self, n, z, parent=None, algorithm=None):
"""
EXAMPLES::
sage: N(lambert_w(1))
0.567143290409784
sage: lambert_w(RealField(100)(1))
0.56714329040978387299996866221
SciPy is used to evaluate for float, RDF, and CDF inputs::
sage: lambert_w(RDF(1))
0.5671432904097838
sage: lambert_w(float(1))
0.5671432904097838
sage: lambert_w(CDF(1))
0.5671432904097838
sage: lambert_w(complex(1))
(0.5671432904097838+0j)
sage: lambert_w(RDF(-1)) # abs tol 2e-16
-0.31813150520476413 + 1.3372357014306895*I
sage: lambert_w(float(-1)) # abs tol 2e-16
(-0.31813150520476413+1.3372357014306895j)
"""
R = parent or s_parent(z)
if R is float or R is RDF:
from scipy.special import lambertw
res = lambertw(z, n)
# SciPy always returns a complex value, make it real if possible
if not res.imag:
return R(res.real)
elif R is float:
return complex(res)
else:
return CDF(res)
elif R is complex or R is CDF:
from scipy.special import lambertw
return R(lambertw(z, n))
else:
import mpmath
return mpmath_utils.call(mpmath.lambertw, z, n, parent=R)
def _evalf_(self, x, y, parent=None, algorithm='mpmath'):
"""
EXAMPLES::
sage: gamma_inc_lower(3,2.)
0.646647167633873
sage: gamma_inc_lower(3,2).n(200)
0.646647167633873081060005050275155...
sage: gamma_inc_lower(0,2.)
+infinity
"""
R = parent or s_parent(x)
# C is the complex version of R
# prec is the precision of R
if R is float:
prec = 53
C = complex
else:
try:
prec = R.precision()
except AttributeError:
prec = 53
try:
C = R.complex_field()
except AttributeError:
C = R
if algorithm == 'pari':
try:
v = ComplexField(prec)(x).gamma() - ComplexField(prec)(x).gamma_inc(y)
except AttributeError:
if not (is_ComplexNumber(x)):
if is_ComplexNumber(y):
C = y.parent()
else:
C = ComplexField()
x = C(x)
v = ComplexField(prec)(x).gamma() - ComplexField(prec)(x).gamma_inc(y)
else:
import mpmath
v = ComplexField(prec)(mpmath_utils.call(mpmath.gammainc, x, 0, y, parent=R))
if v.is_real():
return R(v)
else:
return C(v)
def _evalf_(self, x, parent=None, algorithm=None):
"""
EXAMPLES::
sage: erf(2).n()
0.995322265018953
sage: erf(2).n(200)
0.99532226501895273416206925636725292861089179704006007673835
sage: erf(pi - 1/2*I).n(100)
1.0000111669099367825726058952 + 1.6332655417638522934072124547e-6*I
TESTS:
Check that PARI/GP through the GP interface gives the same answer::
sage: gp.set_real_precision(59) # random
38
sage: print(gp.eval("1 - erfc(1)")); print(erf(1).n(200))
0.84270079294971486934122063508260925929606699796630290845994
0.84270079294971486934122063508260925929606699796630290845994
Check that for an imaginary input, the output is also imaginary, see
:trac:`13193`::
sage: erf(3.0*I)
1629.99462260157*I
sage: erf(33.0*I)
1.51286977510409e471*I
Check that real ball evaluation is fixed :trac:`28061`::
sage: RealBallField(128)(erf(5))
[0.99999999999846254020557196514981165651 +/- 7.33e-39]
"""
R = parent or s_parent(x)
if isinstance(R, RealBallField):
# NOTE: erf exists on complex balls but not on real ones
C = R.complex_field()
return C(x).erf().real()
else:
import mpmath
y = mpmath_utils.call(mpmath.erf, x, parent=R)
return y
def _evalf_(self, n, z, parent=None, algorithm=None):
"""
EXAMPLES::
sage: N(lambert_w(1))
0.567143290409784
sage: lambert_w(RealField(100)(1))
0.56714329040978387299996866221
SciPy is used to evaluate for float, RDF, and CDF inputs::
sage: lambert_w(RDF(1))
0.5671432904097838
sage: lambert_w(float(1))
0.5671432904097838
sage: lambert_w(CDF(1))
0.5671432904097838
sage: lambert_w(complex(1))
(0.5671432904097838+0j)
sage: lambert_w(RDF(-1)) # abs tol 2e-16
-0.31813150520476413 + 1.3372357014306895*I
sage: lambert_w(float(-1)) # abs tol 2e-16
(-0.31813150520476413+1.3372357014306895j)
"""
R = parent or s_parent(z)
if R is float or R is RDF:
from scipy.special import lambertw
res = lambertw(z, n)
# SciPy always returns a complex value, make it real if possible
if not res.imag:
return R(res.real)
elif R is float:
return complex(res)
else:
return CDF(res)
elif R is complex or R is CDF:
from scipy.special import lambertw
return R(lambertw(z, n))
else:
import mpmath
return mpmath_utils.call(mpmath.lambertw, z, n, parent=R)
def _evalf_(self, x, y, parent=None, algorithm='pari'):
"""
EXAMPLES::
sage: gamma_inc(0,2)
-Ei(-2)
sage: gamma_inc(0,2.)
0.0489005107080611
sage: gamma_inc(0,2).n(algorithm='pari')
0.0489005107080611
sage: gamma_inc(0,2).n(200)
0.048900510708061119567239835228...
sage: gamma_inc(3,2).n()
1.35335283236613
TESTS:
Check that :trac:`7099` is fixed::
sage: R = RealField(1024)
sage: gamma(R(9), R(10^-3)) # rel tol 1e-308
40319.99999999999999999999999999988898884344822911869926361916294165058203634104838326009191542490601781777105678829520585311300510347676330951251563007679436243294653538925717144381702105700908686088851362675381239820118402497959018315224423868693918493033078310647199219674433536605771315869983788442389633
sage: numerical_approx(gamma(9, 10^(-3)) - gamma(9), digits=40) # abs tol 1e-36
-1.110111598370794007949063502542063148294e-28
Check that :trac:`17328` is fixed::
sage: gamma_inc(float(-1), float(-1))
(-0.8231640121031085+3.141592653589793j)
sage: gamma_inc(RR(-1), RR(-1))
-0.823164012103109 + 3.14159265358979*I
sage: gamma_inc(-1, float(-log(3))) - gamma_inc(-1, float(-log(2))) # abs tol 1e-15
(1.2730972164471142+0j)
Check that :trac:`17130` is fixed::
sage: r = gamma_inc(float(0), float(1)); r
0.21938393439552029
sage: type(r)
<... 'float'>
"""
R = parent or s_parent(x)
# C is the complex version of R
# prec is the precision of R
if R is float:
prec = 53
C = complex
else:
try:
prec = R.precision()
except AttributeError:
prec = 53
try:
C = R.complex_field()
except AttributeError:
C = R
if algorithm == 'pari':
v = ComplexField(prec)(x).gamma_inc(y)
else:
import mpmath
v = ComplexField(prec)(mpmath_utils.call(mpmath.gammainc,
x,
y,
parent=R))
if v.is_real():
return R(v)
else:
return C(v)
def _evalf_(self, x, y, parent=None, algorithm='pari'):
"""
EXAMPLES::
sage: gamma_inc(0,2)
-Ei(-2)
sage: gamma_inc(0,2.)
0.0489005107080611
sage: gamma_inc(0,2).n(algorithm='pari')
0.0489005107080611
sage: gamma_inc(0,2).n(200)
0.048900510708061119567239835228...
sage: gamma_inc(3,2).n()
1.35335283236613
TESTS:
Check that :trac:`7099` is fixed::
sage: R = RealField(1024)
sage: gamma(R(9), R(10^-3)) # rel tol 1e-308
40319.99999999999999999999999999988898884344822911869926361916294165058203634104838326009191542490601781777105678829520585311300510347676330951251563007679436243294653538925717144381702105700908686088851362675381239820118402497959018315224423868693918493033078310647199219674433536605771315869983788442389633
sage: numerical_approx(gamma(9, 10^(-3)) - gamma(9), digits=40) # abs tol 1e-36
-1.110111598370794007949063502542063148294e-28
Check that :trac:`17328` is fixed::
sage: gamma_inc(float(-1), float(-1))
(-0.8231640121031085+3.141592653589793j)
sage: gamma_inc(RR(-1), RR(-1))
-0.823164012103109 + 3.14159265358979*I
sage: gamma_inc(-1, float(-log(3))) - gamma_inc(-1, float(-log(2))) # abs tol 1e-15
(1.2730972164471142+0j)
Check that :trac:`17130` is fixed::
sage: r = gamma_inc(float(0), float(1)); r
0.21938393439552029
sage: type(r)
<... 'float'>
"""
R = parent or s_parent(x)
# C is the complex version of R
# prec is the precision of R
if R is float:
prec = 53
C = complex
else:
try:
prec = R.precision()
except AttributeError:
prec = 53
try:
C = R.complex_field()
except AttributeError:
C = R
if algorithm == 'pari':
v = ComplexField(prec)(x).gamma_inc(y)
else:
import mpmath
v = ComplexField(prec)(mpmath_utils.call(mpmath.gammainc, x, y, parent=R))
if v.is_real():
return R(v)
else:
return C(v)
