def _evalf_(self, x, **kwds):
"""
TESTS:
Check that :trac:`16587` is fixed::
sage: M = sgn(3/2, hold=True); M
sgn(3/2)
sage: M.n()
1
sage: h(x) = sgn(x)
sage: h(pi).numerical_approx()
1.00000000000000
"""
if hasattr(x, 'sign'): # First check if x has a sign method
return x.sign()
if hasattr(x, 'sgn'): # or a sgn method
return x.sgn()
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return ZZ(0)
# Now we have a non-zero real
if bool((approx_x**(0.5)).imag() == 0): # Check: x > 0
return ZZ(1)
else:
return ZZ(-1)
raise ValueError("Numeric evaluation of symbolic expression")
def _evalf_(self, x, **kwds):
"""
TESTS:
Check that :trac:`16587` is fixed::
sage: M = sgn(3/2, hold=True); M
sgn(3/2)
sage: M.n()
1
sage: h(x) = sgn(x)
sage: h(pi).numerical_approx()
1.00000000000000
"""
if hasattr(x,'sign'): # First check if x has a sign method
return x.sign()
if hasattr(x,'sgn'): # or a sgn method
return x.sgn()
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return ZZ(0)
# Now we have a non-zero real
if bool((approx_x**(0.5)).imag() == 0): # Check: x > 0
return ZZ(1)
else:
return ZZ(-1)
raise ValueError("Numeric evaluation of symbolic expression")
def _eval_(self, x):
"""
INPUT:
- ``x`` - a real number or a symbolic expression
EXAMPLES::
sage: dirac_delta(1)
0
sage: dirac_delta(0)
dirac_delta(0)
sage: dirac_delta(x)
dirac_delta(x)
sage: dirac_delta(exp(-10000000000000000000))
0
Evaluation test::
sage: dirac_delta(x).subs(x=1)
0
"""
try:
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return None
else:
return 0
except StandardError: # x is symbolic
pass
return None
def _eval_(self, x):
"""
INPUT:
- ``x`` - a real number or a symbolic expression
EXAMPLES::
sage: dirac_delta(1)
0
sage: dirac_delta(0)
dirac_delta(0)
sage: dirac_delta(x)
dirac_delta(x)
sage: dirac_delta(exp(-10000000000000000000))
0
Evaluation test::
sage: dirac_delta(x).subs(x=1)
0
"""
try:
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return None
else:
return 0
except Exception: # x is symbolic
pass
return None
def _is_numerically_zero_CIF(x):
from sage.rings.all import ComplexIntervalField
try:
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0) and bool(approx_x.real() == 0):
return True
except:
return False
def _eval_(self, x):
"""
EXAMPLES::
sage: sgn(-1)
-1
sage: sgn(1)
1
sage: sgn(0)
0
sage: sgn(x)
sgn(x)
sage: sgn(-exp(-10000000000000000000))
-1
Evaluation test::
sage: sgn(x).subs(x=1)
1
sage: sgn(x).subs(x=0)
0
sage: sgn(x).subs(x=-1)
-1
More tests::
sage: sign(RR(2))
1
sage: sign(RDF(2))
1
sage: sign(AA(-2))
-1
sage: sign(AA(0))
0
"""
if hasattr(x, 'sign'): # First check if x has a sign method
return x.sign()
if hasattr(x, 'sgn'): # or a sgn method
return x.sgn()
try:
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return ZZ(0)
# Now we have a non-zero real
if bool((approx_x**(0.5)).imag() == 0): # Check: x > 0
return ZZ(1)
else:
return ZZ(-1)
except StandardError: # x is symbolic
pass
return None
def _eval_(self, x):
"""
EXAMPLES::
sage: sgn(-1)
-1
sage: sgn(1)
1
sage: sgn(0)
0
sage: sgn(x)
sgn(x)
sage: sgn(-exp(-10000000000000000000))
-1
Evaluation test::
sage: sgn(x).subs(x=1)
1
sage: sgn(x).subs(x=0)
0
sage: sgn(x).subs(x=-1)
-1
More tests::
sage: sign(RR(2))
1
sage: sign(RDF(2))
1
sage: sign(AA(-2))
-1
sage: sign(AA(0))
0
"""
if hasattr(x,'sign'): # First check if x has a sign method
return x.sign()
if hasattr(x,'sgn'): # or a sgn method
return x.sgn()
try:
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return ZZ(0)
# Now we have a non-zero real
if bool((approx_x**(0.5)).imag() == 0): # Check: x > 0
return ZZ(1)
else:
return ZZ(-1)
except Exception: # x is symbolic
pass
return None
def _eval_(self, x):
"""
INPUT:
- ``x`` - a real number or a symbolic expression
EXAMPLES::
sage: heaviside(-1/2)
0
sage: heaviside(1)
1
sage: heaviside(0)
heaviside(0)
sage: heaviside(x)
heaviside(x)
sage: heaviside(exp(-1000000000000000000000))
1
Evaluation test::
sage: heaviside(x).subs(x=1)
1
sage: heaviside(x).subs(x=-1)
0
::
sage: ex = heaviside(x)+1
sage: t = loads(dumps(ex)); t
heaviside(x) + 1
sage: bool(t == ex)
True
sage: t.subs(x=1)
2
"""
try:
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return None
# Now we have a non-zero real
if bool((approx_x**(0.5)).imag() == 0): # Check: x > 0
return 1
else:
return 0
except Exception: # x is symbolic
pass
return None
def _eval_(self, x):
"""
INPUT:
- ``x`` - a real number or a symbolic expression
EXAMPLES::
sage: heaviside(-1/2)
0
sage: heaviside(1)
1
sage: heaviside(0)
heaviside(0)
sage: heaviside(x)
heaviside(x)
sage: heaviside(exp(-1000000000000000000000))
1
Evaluation test::
sage: heaviside(x).subs(x=1)
1
sage: heaviside(x).subs(x=-1)
0
::
sage: ex = heaviside(x)+1
sage: t = loads(dumps(ex)); t
heaviside(x) + 1
sage: bool(t == ex)
True
sage: t.subs(x=1)
2
"""
try:
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return None
# Now we have a non-zero real
if bool((approx_x**(0.5)).imag() == 0): # Check: x > 0
return 1
else:
return 0
except StandardError: # x is symbolic
pass
return None
def _evalf_(self, x, **kwds):
"""
TESTS::
sage: h(x) = dirac_delta(x)
sage: h(pi).numerical_approx()
0.000000000000000
"""
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return None
else:
return 0
raise ValueError("Numeric evaluation of symbolic expression")
def _evalf_(self, x, **kwds):
"""
TESTS::
sage: h(x) = dirac_delta(x)
sage: h(pi).numerical_approx()
0.000000000000000
"""
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return None
else:
return 0
raise ValueError("Numeric evaluation of symbolic expression")
def _evalf_(self, m, n, **kwds):
"""
TESTS::
sage: h(x) = kronecker_delta(3,x)
sage: h(pi).numerical_approx()
0.000000000000000
"""
if bool(repr(m) > repr(n)):
return kronecker_delta(n, m)
x = m - n
approx_x = ComplexIntervalField()(x)
if approx_x.imag() == 0: # x is real
if approx_x.real() == 0: # x is zero
return 1
else:
return 0
return 0 # x is complex
def _evalf_(self, x, **kwds):
"""
TESTS::
sage: h(x) = unit_step(x)
sage: h(pi).numerical_approx()
1.00000000000000
"""
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return 1
# Now we have a non-zero real
if bool((approx_x**(0.5)).imag() == 0): # Check: x > 0
return 1
else:
return 0
raise ValueError("Numeric evaluation of symbolic expression")
def _evalf_(self, x, **kwds):
"""
TESTS::
sage: h(x) = unit_step(x)
sage: h(pi).numerical_approx()
1.00000000000000
"""
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return 1
# Now we have a non-zero real
if bool((approx_x**(0.5)).imag() == 0): # Check: x > 0
return 1
else:
return 0
raise ValueError("Numeric evaluation of symbolic expression")
def _eval_(self, m, n):
"""
The Kronecker delta function.
EXAMPLES::
sage: kronecker_delta(1,2)
0
sage: kronecker_delta(1,1)
1
Kronecker delta is a symmetric function. We keep arguments sorted to
ensure that k_d(m, n) - k_d(n, m) cancels automatically::
sage: x,y=var('x,y')
sage: kronecker_delta(x, y)
kronecker_delta(x, y)
sage: kronecker_delta(y, x)
kronecker_delta(x, y)
sage: kronecker_delta(x,2*x)
kronecker_delta(2*x, x)
Evaluation test::
sage: kronecker_delta(1,x).subs(x=1)
1
"""
if bool(repr(m) > repr(n)):
return kronecker_delta(n, m)
x = m - n
try:
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return 1
else:
return 0
else:
return 0 # x is complex
except Exception: # x is symbolic
pass
return None
def _eval_(self, m, n):
"""
The Kronecker delta function.
EXAMPLES::
sage: kronecker_delta(1,2)
0
sage: kronecker_delta(1,1)
1
Kronecker delta is a symmetric function. We keep arguments sorted to
ensure that (k_d(m, n) - k_d(n, m) cancels automatically::
sage: x,y=var('x,y')
sage: kronecker_delta(x, y)
kronecker_delta(x, y)
sage: kronecker_delta(y, x)
kronecker_delta(x, y)
sage: kronecker_delta(x,2*x)
kronecker_delta(2*x, x)
Evaluation test::
sage: kronecker_delta(1,x).subs(x=1)
1
"""
if bool(repr(m) > repr(n)):
return kronecker_delta(n, m)
x = m - n
try:
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return 1
else:
return 0
else:
return 0 # x is complex
except StandardError: # x is symbolic
pass
return None
def _evalf_(self, m, n, **kwds):
"""
TESTS::
sage: h(x) = kronecker_delta(3,x)
sage: h(pi).numerical_approx()
0.000000000000000
"""
if bool(repr(m) > repr(n)):
return kronecker_delta(n, m)
x = m - n
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return 1
else:
return 0
else:
return 0 # x is complex
raise ValueError("Numeric evaluation of symbolic expression")
def _evalf_(self, m, n, **kwds):
"""
TESTS::
sage: h(x) = kronecker_delta(3,x)
sage: h(pi).numerical_approx()
0.000000000000000
"""
if bool(repr(m) > repr(n)):
return kronecker_delta(n, m)
x = m - n
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return 1
else:
return 0
else:
return 0 # x is complex
raise ValueError("Numeric evaluation of symbolic expression")
def _eval_(self, x):
"""
INPUT:
- ``x`` - a real number or a symbolic expression
EXAMPLES::
sage: unit_step(-1)
0
sage: unit_step(1)
1
sage: unit_step(0)
1
sage: unit_step(x)
unit_step(x)
sage: unit_step(-exp(-10000000000000000000))
0
Evaluation test::
sage: unit_step(x).subs(x=1)
1
sage: unit_step(x).subs(x=0)
1
"""
try:
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return 1
# Now we have a non-zero real
if bool((approx_x**(0.5)).imag() == 0): # Check: x > 0
return 1
else:
return 0
except Exception: # x is symbolic
pass
return None
def _eval_(self, x):
"""
INPUT:
- ``x`` - a real number or a symbolic expression
EXAMPLES::
sage: unit_step(-1)
0
sage: unit_step(1)
1
sage: unit_step(0)
1
sage: unit_step(x)
unit_step(x)
sage: unit_step(-exp(-10000000000000000000))
0
Evaluation test::
sage: unit_step(x).subs(x=1)
1
sage: unit_step(x).subs(x=0)
1
"""
try:
approx_x = ComplexIntervalField()(x)
if bool(approx_x.imag() == 0): # x is real
if bool(approx_x.real() == 0): # x is zero
return 1
# Now we have a non-zero real
if bool((approx_x**(0.5)).imag() == 0): # Check: x > 0
return 1
else:
return 0
except StandardError: # x is symbolic
pass
return None