def _evalf_(self, f, x, a, b, parent=None, algorithm=None):
"""
Returns numerical approximation of the integral
EXAMPLES::
sage: from sage.symbolic.integration.integral import definite_integral
sage: h = definite_integral(sin(x)*log(x)/x^2, x, 1, 2); h
integrate(log(x)*sin(x)/x^2, x, 1, 2)
sage: h.n() # indirect doctest
0.14839875208053...
TESTS:
Check if #3863 is fixed::
sage: integrate(x^2.7 * e^(-2.4*x), x, 0, 3).n()
0.154572952320790
"""
from sage.gsl.integration import numerical_integral
# The gsl routine returns a tuple, which also contains the error.
# We only return the result.
return numerical_integral(f, a, b)[0]
def _evalf_(self, f, x, a, b, parent=None):
"""
Returns numerical approximation of the integral
EXAMPLES::
sage: from sage.symbolic.integration.integral import definite_integral
sage: h = definite_integral(sin(x)*log(x)/x^2, x, 1, 2); h
integrate(log(x)*sin(x)/x^2, x, 1, 2)
sage: h.n() # indirect doctest
0.14839875208053...
TESTS:
Check if #3863 is fixed::
sage: integrate(x^2.7 * e^(-2.4*x), x, 0, 3).n()
0.154572952320790
"""
from sage.gsl.integration import numerical_integral
# The gsl routine returns a tuple, which also contains the error.
# We only return the result.
return numerical_integral(f, a, b)[0]
def Li(x, eps_rel=None, err_bound=False):
r"""
Return value of the function Li(x) as a real double field element.
This is the function
.. math::
\int_2^{x} dt / \log(t).
The function Li(x) is an approximation for the number of primes up
to `x`. In fact, the famous Riemann Hypothesis is
equivalent to the statement that for `x \geq 2.01` we have
.. math::
|\pi(x) - Li(x)| \leq \sqrt{x} \log(x).
For "small" `x`, `Li(x)` is always slightly bigger
than `\pi(x)`. However it is a theorem that there are (very
large, e.g., around `10^{316}`) values of `x` so
that `\pi(x) > Li(x)`. See
"A new bound for the smallest x with `\pi(x) > li(x)`",
Bays and Hudson, Mathematics of Computation, 69 (2000) 1285-1296.
ALGORITHM: Computed numerically using GSL.
INPUT:
- ``x`` - a real number = 2.
OUTPUT:
- ``x`` - a real double
EXAMPLES::
sage: Li(2)
0.0
sage: Li(5)
2.58942452992
sage: Li(1000)
176.56449421
sage: Li(10^5)
9628.76383727
sage: prime_pi(10^5)
9592
sage: Li(1)
Traceback (most recent call last):
...
ValueError: Li only defined for x at least 2.
::
sage: for n in range(1,7):
... print '%-10s%-10s%-20s'%(10^n, prime_pi(10^n), Li(10^n))
10 4 5.12043572467
100 25 29.080977804
1000 168 176.56449421
10000 1229 1245.09205212
100000 9592 9628.76383727
1000000 78498 78626.5039957
"""
x = float(x)
if x < 2:
raise ValueError, "Li only defined for x at least 2."
if eps_rel:
ans = numerical_integral(_one_over_log, 2, float(x), eps_rel=eps_rel)
else:
ans = numerical_integral(_one_over_log, 2, float(x))
if err_bound:
return real_double.RDF(ans[0]), ans[1]
else:
return real_double.RDF(ans[0])
def Li(x, eps_rel=None, err_bound=False):
r"""
Return value of the function Li(x) as a real double field element.
This is the function
.. math::
\int_2^{x} dt / \log(t).
The function Li(x) is an approximation for the number of primes up
to `x`. In fact, the famous Riemann Hypothesis is
equivalent to the statement that for `x \geq 2.01` we have
.. math::
|\pi(x) - Li(x)| \leq \sqrt{x} \log(x).
For "small" `x`, `Li(x)` is always slightly bigger
than `\pi(x)`. However it is a theorem that there are (very
large, e.g., around `10^{316}`) values of `x` so
that `\pi(x) > Li(x)`. See
"A new bound for the smallest x with `\pi(x) > li(x)`",
Bays and Hudson, Mathematics of Computation, 69 (2000) 1285-1296.
ALGORITHM: Computed numerically using GSL.
INPUT:
- ``x`` - a real number = 2.
OUTPUT:
- ``x`` - a real double
EXAMPLES::
sage: Li(2)
0.0
sage: Li(5)
2.58942452992
sage: Li(1000)
176.56449421
sage: Li(10^5)
9628.76383727
sage: prime_pi(10^5)
9592
sage: Li(1)
Traceback (most recent call last):
...
ValueError: Li only defined for x at least 2.
::
sage: for n in range(1,7):
... print '%-10s%-10s%-20s'%(10^n, prime_pi(10^n), Li(10^n))
10 4 5.12043572467
100 25 29.080977804
1000 168 176.56449421
10000 1229 1245.09205212
100000 9592 9628.76383727
1000000 78498 78626.5039957
"""
x = float(x)
if x < 2:
raise ValueError, "Li only defined for x at least 2."
if eps_rel:
ans = numerical_integral(_one_over_log, 2, float(x),
eps_rel=eps_rel)
else:
ans = numerical_integral(_one_over_log, 2, float(x))
if err_bound:
return real_double.RDF(ans[0]), ans[1]
else:
return real_double.RDF(ans[0])
