def _eval_evalf(self, prec):
"""Evaluate this complex root to the given precision. """
_prec, mp.prec = mp.prec, prec
try:
func = lambdify(self.poly.gen, self.expr)
interval = self._get_interval()
refined = False
while True:
if self.is_real:
x0 = mpf(str(interval.center))
else:
x0 = mpc(*map(str, interval.center))
try:
root = findroot(func, x0)
except ValueError:
interval = interval.refine()
refined = True
continue
else:
if refined:
self._set_interval(interval)
break
finally:
mp.prec = _prec
return Float._new(root.real._mpf_, prec) + I*Float._new(root.imag._mpf_, prec)
def _eval_evalf(self, prec):
"""Evaluate this complex root to the given precision. """
_prec, mp.prec = mp.prec, prec
try:
func = lambdify(self.poly.gen, self.expr)
interval = self._get_interval()
refined = False
while True:
if self.is_real:
x0 = mpf(str(interval.center))
else:
x0 = mpc(*map(str, interval.center))
try:
root = findroot(func, x0)
except ValueError:
interval = interval.refine()
refined = True
continue
else:
if refined:
self._set_interval(interval)
break
finally:
mp.prec = _prec
return Float._new(root.real._mpf_, prec) + I*Float._new(root.imag._mpf_, prec)
def _eval_evalf(self, prec):
"""Evaluate this complex root to the given precision. """
with workprec(prec):
g = self.poly.gen
if not g.is_Symbol:
d = Dummy('x')
func = lambdify(d, self.expr.subs(g, d))
else:
func = lambdify(g, self.expr)
interval = self._get_interval()
if not self.is_real:
# For complex intervals, we need to keep refining until the
# imaginary interval is disjunct with other roots, that is,
# until both ends get refined.
ay = interval.ay
by = interval.by
while interval.ay == ay or interval.by == by:
interval = interval.refine()
while True:
if self.is_real:
x0 = mpf(str(interval.center))
else:
x0 = mpc(*map(str, interval.center))
try:
root = findroot(func, x0, verify=False)
# If the (real or complex) root is not in the 'interval',
# then keep refining the interval. This happens if findroot
# accidentally finds a different root outside of this
# interval because our initial estimate 'x0' was not close
# enough.
if self.is_real:
a = mpf(str(interval.a))
b = mpf(str(interval.b))
if a == b:
root = a
break
if not (a < root < b):
raise ValueError("Root not in the interval.")
else:
ax = mpf(str(interval.ax))
bx = mpf(str(interval.bx))
ay = mpf(str(interval.ay))
by = mpf(str(interval.by))
if ax == bx and ay == by:
root = ax + S.ImaginaryUnit * by
break
if not (ax < root.real < bx and ay < root.imag < by):
raise ValueError("Root not in the interval.")
except ValueError:
interval = interval.refine()
continue
else:
break
return Float._new(root.real._mpf_,
prec) + I * Float._new(root.imag._mpf_, prec)
def _eval_evalf(self, prec):
"""Evaluate this complex root to the given precision. """
with workprec(prec):
g = self.poly.gen
if not g.is_Symbol:
d = Dummy('x')
func = lambdify(d, self.expr.subs(g, d))
else:
func = lambdify(g, self.expr)
interval = self._get_interval()
if not self.is_real:
# For complex intervals, we need to keep refining until the
# imaginary interval is disjunct with other roots, that is,
# until both ends get refined.
ay = interval.ay
by = interval.by
while interval.ay == ay or interval.by == by:
interval = interval.refine()
while True:
if self.is_real:
x0 = mpf(str(interval.center))
else:
x0 = mpc(*map(str, interval.center))
try:
root = findroot(func, x0, verify=False)
# If the (real or complex) root is not in the 'interval',
# then keep refining the interval. This happens if findroot
# accidentally finds a different root outside of this
# interval because our initial estimate 'x0' was not close
# enough.
if self.is_real:
a = mpf(str(interval.a))
b = mpf(str(interval.b))
if a == b:
root = a
break
if not (a < root < b):
raise ValueError("Root not in the interval.")
else:
ax = mpf(str(interval.ax))
bx = mpf(str(interval.bx))
ay = mpf(str(interval.ay))
by = mpf(str(interval.by))
if ax == bx and ay == by:
root = ax + S.ImaginaryUnit*by
break
if not (ax < root.real < bx and ay < root.imag < by):
raise ValueError("Root not in the interval.")
except ValueError:
interval = interval.refine()
continue
else:
break
return Float._new(root.real._mpf_, prec) + I*Float._new(root.imag._mpf_, prec)
def _eval_evalf(self, prec):
"""Evaluate this complex root to the given precision. """
_prec, mp.prec = mp.prec, prec
try:
func = lambdify(self.poly.gen, self.expr)
interval = self._get_interval()
if not self.is_real:
# For complex intervals, we need to keep refining until the
# imaginary interval is disjunct with other roots, that is,
# until both ends get refined.
ay = interval.ay
by = interval.by
while interval.ay == ay or interval.by == by:
interval = interval.refine()
while True:
if self.is_real:
x0 = mpf(str(interval.center))
else:
x0 = mpc(*map(str, interval.center))
try:
root = findroot(func, x0)
# If the (real or complex) root is not in the 'interval',
# then keep refining the interval. This happens if findroot
# accidentally finds a different root outside of this
# interval because our initial estimate 'x0' was not close
# enough.
if self.is_real:
a = mpf(str(interval.a))
b = mpf(str(interval.b))
# This is needed due to the bug #3364:
a, b = min(a, b), max(a, b)
if not (a < root < b):
raise ValueError("Root not in the interval.")
else:
ax = mpf(str(interval.ax))
bx = mpf(str(interval.bx))
ay = mpf(str(interval.ay))
by = mpf(str(interval.by))
# This is needed due to the bug #3364:
ax, bx = min(ax, bx), max(ax, bx)
ay, by = min(ay, by), max(ay, by)
if not (ax < root.real < bx and ay < root.imag < by):
raise ValueError("Root not in the interval.")
except ValueError:
interval = interval.refine()
continue
else:
break
finally:
mp.prec = _prec
return Float._new(root.real._mpf_,
prec) + I * Float._new(root.imag._mpf_, prec)
def _eval_evalf(self, prec):
"""Evaluate this complex root to the given precision. """
_prec, mp.prec = mp.prec, prec
try:
func = lambdify(self.poly.gen, self.expr)
interval = self._get_interval()
if not self.is_real:
# For complex intervals, we need to keep refining until the
# imaginary interval is disjunct with other roots, that is,
# until both ends get refined.
ay = interval.ay
by = interval.by
while interval.ay == ay or interval.by == by:
interval = interval.refine()
while True:
if self.is_real:
x0 = mpf(str(interval.center))
else:
x0 = mpc(*map(str, interval.center))
try:
root = findroot(func, x0)
# If the (real or complex) root is not in the 'interval',
# then keep refining the interval. This happens if findroot
# accidentally finds a different root outside of this
# interval because our initial estimate 'x0' was not close
# enough.
if self.is_real:
a = mpf(str(interval.a))
b = mpf(str(interval.b))
# This is needed due to the bug #3364:
a, b = min(a, b), max(a, b)
if not (a < root < b):
raise ValueError("Root not in the interval.")
else:
ax = mpf(str(interval.ax))
bx = mpf(str(interval.bx))
ay = mpf(str(interval.ay))
by = mpf(str(interval.by))
# This is needed due to the bug #3364:
ax, bx = min(ax, bx), max(ax, bx)
ay, by = min(ay, by), max(ay, by)
if not (ax < root.real < bx and ay < root.imag < by):
raise ValueError("Root not in the interval.")
except ValueError:
interval = interval.refine()
continue
else:
break
finally:
mp.prec = _prec
return Float._new(root.real._mpf_, prec) + I*Float._new(root.imag._mpf_, prec)
def sympy2mpmath(value, prec=None):
if prec is None:
from mathics.builtin.numeric import machine_precision
prec = machine_precision
value = value.n(dps(prec))
if value.is_real:
return mpmath.mpf(value)
elif value.is_number:
return mpmath.mpc(*value.as_real_imag())
else:
return None
def sympy2mpmath(value, prec=None):
if prec is None:
from mathics.builtin.numeric import machine_precision
prec = machine_precision
value = value.n(dps(prec))
if value.is_real:
return mpmath.mpf(value)
elif value.is_number:
return mpmath.mpc(*value.as_real_imag())
else:
return None
def f(t):
re, im, re_acc, im_acc = evalf(func, mp.prec, {'subs': {x: t}})
have_part[0] = re or have_part[0]
have_part[1] = im or have_part[1]
max_real_term[0] = max(max_real_term[0], fastlog(re))
max_imag_term[0] = max(max_imag_term[0], fastlog(im))
if im:
return mpc(re or fzero, im)
return mpf(re or fzero)
def f(t):
re, im, re_acc, im_acc = evalf(func, mp.prec, {"subs": {x: t}})
have_part[0] = re or have_part[0]
have_part[1] = im or have_part[1]
max_real_term[0] = max(max_real_term[0], fastlog(re))
max_imag_term[0] = max(max_imag_term[0], fastlog(im))
if im:
return mpc(re or fzero, im)
return mpf(re or fzero)
def test_sympify_mpmath():
value = sympify(mpmath.mpf(1.0))
assert value == Float(1.0) and type(value) is Float
mpmath.mp.dps = 12
assert sympify(mpmath.pi).epsilon_eq(Float("3.14159265359"), Float("1e-12")) is True
assert sympify(mpmath.pi).epsilon_eq(Float("3.14159265359"), Float("1e-13")) is False
mpmath.mp.dps = 6
assert sympify(mpmath.pi).epsilon_eq(Float("3.14159"), Float("1e-5")) is True
assert sympify(mpmath.pi).epsilon_eq(Float("3.14159"), Float("1e-6")) is False
assert sympify(mpmath.mpc(1.0 + 2.0j)) == Float(1.0) + Float(2.0)*I
def test_sympify_mpmath():
value = sympify(mpmath.mpf(1.0))
assert value == Float(1.0) and type(value) is Float
mpmath.mp.dps = 12
assert sympify(mpmath.pi).epsilon_eq(Float("3.14159265359"), Float("1e-12")) is True
assert sympify(mpmath.pi).epsilon_eq(Float("3.14159265359"), Float("1e-13")) is False
mpmath.mp.dps = 6
assert sympify(mpmath.pi).epsilon_eq(Float("3.14159"), Float("1e-5")) is True
assert sympify(mpmath.pi).epsilon_eq(Float("3.14159"), Float("1e-6")) is False
assert sympify(mpmath.mpc(1.0 + 2.0j)) == Float(1.0) + Float(2.0)*I
def as_mpmath(x, prec, options):
x = sympify(x)
if isinstance(x, C.Zero):
return mpf(0)
if isinstance(x, C.Infinity):
return mpf("inf")
if isinstance(x, C.NegativeInfinity):
return mpf("-inf")
# XXX
re, im, _, _ = evalf(x, prec, options)
if im:
return mpc(re or fzero, im)
return mpf(re)
def as_mpmath(x, prec, options):
x = sympify(x)
if isinstance(x, C.Zero):
return mpf(0)
if isinstance(x, C.Infinity):
return mpf('inf')
if isinstance(x, C.NegativeInfinity):
return mpf('-inf')
# XXX
re, im, _, _ = evalf(x, prec, options)
if im:
return mpc(re or fzero, im)
return mpf(re)
def test_poly():
output("""\
poly_:{
r:{$[prec>=abs(x:reim x)1;x 0;x]} each .qml.poly x;
r iasc {x:reim x;(abs x 1;neg x 1;x 0)} each r};""")
for A in poly_subjects:
A = sp.Poly(A, sp.Symbol("x"))
if A.degree() <= 3 and all(a.is_real for a in A.all_coeffs()):
R = []
for r in sp.roots(A, multiple=True):
r = sp.simplify(sp.expand_complex(r))
R.append(r)
R.sort(key=lambda x: (abs(sp.im(x)), -sp.im(x), sp.re(x)))
else:
R = mp.polyroots([mp.mpc(*(a.evalf(mp.mp.dps)).as_real_imag())
for a in A.all_coeffs()])
R.sort(key=lambda x: (abs(x.imag), -x.imag, x.real))
assert len(R) == A.degree()
test("poly_", A.all_coeffs(), R, complex_pair=True)
def _eval_evalf(self, prec):
"""Evaluate this complex root to the given precision. """
_prec, mp.prec = mp.prec, prec
try:
func = lambdify(self.poly.gen, self.expr)
interval, refined = self._get_interval(), False
while True:
if self.is_real:
x0 = mpf(str(interval.center))
else:
re, im = interval.center
re = mpf(str(re))
im = mpf(str(im))
x0 = mpc(re, im)
try:
root = findroot(func, x0)
except ValueError:
interval = interval.refine()
refined = True
continue
else:
if refined:
self._set_interval(interval)
if self.is_conjugate:
root = root.conjugate()
break
finally:
mp.prec = _prec
return Real._new(root.real._mpf_,
prec) + I * Real._new(root.imag._mpf_, prec)
def _eval_evalf(self, prec):
"""Evaluate this complex root to the given precision. """
_prec, mp.prec = mp.prec, prec
try:
func = lambdify(self.poly.gen, self.expr)
interval, refined = self._get_interval(), False
while True:
if self.is_real:
x0 = mpf(str(interval.center))
else:
re, im = interval.center
re = mpf(str(re))
im = mpf(str(im))
x0 = mpc(re, im)
try:
root = findroot(func, x0)
except ValueError:
interval = interval.refine()
refined = True
continue
else:
if refined:
self._set_interval(interval)
if self.is_conjugate:
root = root.conjugate()
break
finally:
mp.prec = _prec
return Real._new(root.real._mpf_, prec) + I*Real._new(root.imag._mpf_, prec)
def voigt_slow(a, u):
with mp.workdps(581):
z = mp.mpc(u, a)
result = mp.exp(-z * z) * mp.erfc(-1j * z)
return result.real
def test_J6():
assert mpmath.besselj(2, 1 + 1j).ae(
mpc('0.04157988694396212', '0.24739764151330632'))
def test_J6():
assert mpmath.besselj(2, 1 + 1j).ae(mpc('0.04157988694396212', '0.24739764151330632'))
