def __str__(self): # pragma: no cover
from sympy.printing.str import sstr
if self.dom is None:
return "%s does not divide %s" % (sstr(self.g), sstr(self.f))
else:
return "%s does not divide %s in %s" % (sstr(self.g), sstr(self.f), sstr(self.dom))
def test_extended_domain_in_expr_to_holonomic():
x = symbols('x')
p = expr_to_holonomic(1.2 * cos(3.1 * x))
assert p.to_expr() == 1.2 * cos(3.1 * x)
assert sstr(p.integrate(x).to_expr()) == '0.387096774193548*sin(3.1*x)'
_, Dx = DifferentialOperators(RR.old_poly_ring(x), 'Dx')
p = expr_to_holonomic(1.1329138213 * x)
q = HolonomicFunction((-1.1329138213) + (1.1329138213 * x) * Dx, x, 0,
{1: [1.1329138213]})
assert p == q
assert p.to_expr() == 1.1329138213 * x
assert sstr(p.integrate((x, 1, 2))) == sstr((1.1329138213 * x).integrate(
(x, 1, 2)))
y, z = symbols('y, z')
p = expr_to_holonomic(sin(x * y * z), x=x)
assert p.to_expr() == sin(x * y * z)
assert p.integrate(x).to_expr() == (-cos(x * y * z) + 1) / (y * z)
p = expr_to_holonomic(sin(x * y + z), x=x).integrate(x).to_expr()
q = (cos(z) - cos(x * y + z)) / y
assert p == q
a = symbols('a')
p = expr_to_holonomic(a * x, x)
assert p.to_expr() == a * x
assert p.integrate(x).to_expr() == a * x**2 / 2
D_2, C_1 = symbols("D_2, C_1")
p = expr_to_holonomic(x) + expr_to_holonomic(1.2 * cos(x))
p = p.to_expr().subs(D_2, 0)
assert p - x - 1.2 * cos(1.0 * x) == 0
p = expr_to_holonomic(x) * expr_to_holonomic(1.2 * cos(x))
p = p.to_expr().subs(C_1, 0)
assert p - 1.2 * x * cos(1.0 * x) == 0
def __str__(self): # pragma: no cover
from sympy.printing.str import sstr
if self.dom is None:
return "%s does not divide %s" % (sstr(self.g), sstr(self.f))
else:
return "%s does not divide %s in %s" % (sstr(self.g), sstr(self.f), sstr(self.dom))
def test_complex_space():
c1 = ComplexSpace(2)
assert isinstance(c1, ComplexSpace)
assert c1.dimension == 2
assert sstr(c1) == 'C(2)'
assert srepr(c1) == 'ComplexSpace(Integer(2))'
n = Symbol('n')
c2 = ComplexSpace(n)
assert isinstance(c2, ComplexSpace)
assert c2.dimension == n
assert sstr(c2) == 'C(n)'
assert srepr(c2) == "ComplexSpace(Symbol('n'))"
assert c2.subs(n, 2) == ComplexSpace(2)
def _print_Float(self, e):
# we will use StrPrinter's Float printer, but we need to handle the
# full_prec ourselves, according to the self._print_level
full_prec = self._settings["full_prec"]
if full_prec == "auto":
full_prec = self._print_level == 1
return prettyForm(sstr(e, full_prec=full_prec))
def _print_Float(self, e):
# we will use StrPrinter's Float printer, but we need to handle the
# full_prec ourselves, according to the self._print_level
full_prec = self._settings["full_prec"]
if full_prec == "auto":
full_prec = self._print_level == 1
return prettyForm(sstr(e, full_prec=full_prec))
def test_integrate():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = expr_to_holonomic(sin(x)**2 / x, x0=1).integrate((x, 2, 3))
q = '0.166270406994788'
assert sstr(p) == q
p = expr_to_holonomic(sin(x)).integrate((x, 0, x)).to_expr()
q = 1 - cos(x)
assert p == q
p = expr_to_holonomic(sin(x)).integrate((x, 0, 3))
q = 1 - cos(3)
assert p == q
p = expr_to_holonomic(sin(x) / x, x0=1).integrate((x, 1, 2))
q = '0.659329913368450'
assert sstr(p) == q
p = expr_to_holonomic(sin(x)**2 / x, x0=1).integrate((x, 1, 0))
q = '-0.423690480850035'
assert sstr(p) == q
p = expr_to_holonomic(sin(x) / x)
assert p.integrate(x).to_expr() == Si(x)
assert p.integrate((x, 0, 2)) == Si(2)
p = expr_to_holonomic(sin(x)**2 / x)
q = p.to_expr()
assert p.integrate(x).to_expr() == q.integrate((x, 0, x))
assert p.integrate((x, 0, 1)) == q.integrate((x, 0, 1))
assert expr_to_holonomic(1 / x, x0=1).integrate(x).to_expr() == log(x)
p = expr_to_holonomic((x + 1)**3 * exp(-x), x0=-1).integrate(x).to_expr()
q = (-x**3 - 6 * x**2 - 15 * x + 6 * exp(x + 1) - 16) * exp(-x)
assert p == q
p = expr_to_holonomic(cos(x)**2 / x**2, y0={
-2: [1, 0, -1]
}).integrate(x).to_expr()
q = -Si(2 * x) - cos(x)**2 / x
assert p == q
p = expr_to_holonomic(sqrt(x**2 + x)).integrate(x).to_expr()
q = (x**Rational(3, 2) * (2 * x**2 + 3 * x + 1) -
x * sqrt(x + 1) * asinh(sqrt(x))) / (4 * x * sqrt(x + 1))
assert p == q
p = expr_to_holonomic(sqrt(x**2 + 1)).integrate(x).to_expr()
q = (sqrt(x**2 + 1)).integrate(x)
assert (p - q).simplify() == 0
p = expr_to_holonomic(1 / x**2, y0={-2: [1, 0, 0]})
r = expr_to_holonomic(1 / x**2, lenics=3)
assert p == r
q = expr_to_holonomic(cos(x)**2)
assert (r * q).integrate(x).to_expr() == -Si(2 * x) - cos(x)**2 / x
def __call__(self, a):
"""Construct an element of `self` domain from `a`. """
res = self.dtype(a, self.dom, len(self.gens) - 1, ring=self)
# make sure res is actually in our ring
if res.denom().terms(order=self.order)[0][0] != (0,)*len(self.gens):
from sympy.printing.str import sstr
raise CoercionFailed("denominator %s not allowed in %s"
% (sstr(res), self))
return res
def new(self, a):
"""Construct an element of `self` domain from `a`. """
res = self.dtype(a, self.dom, len(self.gens) - 1, ring=self)
# make sure res is actually in our ring
if res.denom().terms(order=self.order)[0][0] != (0,)*len(self.gens):
from sympy.printing.str import sstr
raise CoercionFailed("denominator %s not allowed in %s"
% (sstr(res), self))
return res
def _sympystr(self, printer, *args):
from sympy.printing.str import sstr
length = len(self.args)
s = ''
for i in range(length):
if isinstance(self.args[i], (Add, Pow, Mul)):
s = s + '('
s = s + sstr(self.args[i])
if isinstance(self.args[i], (Add, Pow, Mul)):
s = s + ')'
if i != length-1:
s = s + 'x'
return s
def _sympystr(self, printer, *args):
from sympy.printing.str import sstr
length = len(self.args)
s = ''
for i in range(length):
if isinstance(self.args[i], (Add, Pow, Mul)):
s = s + '('
s = s + sstr(self.args[i])
if isinstance(self.args[i], (Add, Pow, Mul)):
s = s + ')'
if i != length - 1:
s = s + 'x'
return s
def test_from_hyper():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = hyper([1, 1], [Rational(3, 2)], x**2 / 4)
q = HolonomicFunction(
(4 * x) + (5 * x**2 - 8) * Dx + (x**3 - 4 * x) * Dx**2, x, 1,
[2 * sqrt(3) * pi / 9, -4 * sqrt(3) * pi / 27 + Rational(4, 3)])
r = from_hyper(p)
assert r == q
p = from_hyper(hyper([1], [Rational(3, 2)], x**2 / 4))
q = HolonomicFunction(-x + (-x**2 / 2 + 2) * Dx + x * Dx**2, x)
# x0 = 1
y0 = '[sqrt(pi)*exp(1/4)*erf(1/2), -sqrt(pi)*exp(1/4)*erf(1/2)/2 + 1]'
assert sstr(p.y0) == y0
assert q.annihilator == p.annihilator
def test_evalf_rk4():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
# log(1+x)
p = HolonomicFunction((1 + x) * Dx**2 + Dx, x, 0, [0, 1])
# path taken is a straight line from 0 to 1, on the real axis
r = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1]
s = '0.693146363174626' # approx. equal to log(2) i.e. 0.693147180559945
assert sstr(p.evalf(r)[-1]) == s
# path taken is a traingle 0-->1+i-->2
r = [0.1 + 0.1 * I]
for i in range(9):
r.append(r[-1] + 0.1 + 0.1 * I)
for i in range(10):
r.append(r[-1] + 0.1 - 0.1 * I)
# close to the exact solution 1.09861228866811
# imaginary part also close to zero
s = '1.098616 + 1.36083e-7*I'
assert sstr(p.evalf(r)[-1].n(7)) == s
# sin(x)
p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1])
s = '0.90929463522785 + 1.52655665885959e-16*I'
assert sstr(p.evalf(r)[-1]) == s
# computing sin(pi/2) using this method
# using a linear path from 0 to pi/2
r = [0.1]
for i in range(14):
r.append(r[-1] + 0.1)
r.append(pi / 2)
s = '0.999999895088917' # close to 1.0 (exact solution)
assert sstr(p.evalf(r)[-1]) == s
# trying different path, a rectangle (0-->i-->pi/2 + i-->pi/2)
# computing the same value sin(pi/2) using different path
r = [0.1 * I]
for i in range(9):
r.append(r[-1] + 0.1 * I)
for i in range(15):
r.append(r[-1] + 0.1)
r.append(pi / 2 + I)
for i in range(10):
r.append(r[-1] - 0.1 * I)
# close to 1.0
s = '1.00000003415141 + 6.11940487991086e-16*I'
assert sstr(p.evalf(r)[-1]) == s
# cos(x)
p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0])
# compute cos(pi) along 0-->pi
r = [0.05]
for i in range(61):
r.append(r[-1] + 0.05)
r.append(pi)
# close to -1 (exact answer)
s = '-0.999999993238714'
assert sstr(p.evalf(r)[-1]) == s
# a rectangular path (0 -> i -> 2+i -> 2)
r = [0.1 * I]
for i in range(9):
r.append(r[-1] + 0.1 * I)
for i in range(20):
r.append(r[-1] + 0.1)
for i in range(10):
r.append(r[-1] - 0.1 * I)
p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1]).evalf(r)
s = '0.493152791638442 - 1.41553435639707e-15*I'
assert sstr(p[-1]) == s
def __str__(f):
from sympy.printing.str import sstr
return sstr(f.rep)
def __str__(self):
from sympy.printing.str import sstr
return sstr(self, order=None)
def __repr__(self):
from sympy.printing.str import sstr
return '[' + ', '.join(sstr(x) for x in self.data) + ']'
def test_hilbert_space():
hs = HilbertSpace()
assert isinstance(hs, HilbertSpace)
assert sstr(hs) == 'H'
assert srepr(hs) == 'HilbertSpace()'
def __str__(self):
from sympy.printing.str import sstr
return sstr(self, order=None)
def __repr__(self):
from sympy.printing.str import sstr
return '<' + ','.join(sstr(x) for [x] in self._module.gens) + '>'
def NS(e, n=15, **options):
return sstr(sympify(e).evalf(n, **options), full_prec=True)
def test_evalf_euler():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
# log(1+x)
p = HolonomicFunction((1 + x) * Dx**2 + Dx, x, 0, [0, 1])
# path taken is a straight line from 0 to 1, on the real axis
r = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1]
s = '0.699525841805253' # approx. equal to log(2) i.e. 0.693147180559945
assert sstr(p.evalf(r, method='Euler')[-1]) == s
# path taken is a traingle 0-->1+i-->2
r = [0.1 + 0.1 * I]
for i in range(9):
r.append(r[-1] + 0.1 + 0.1 * I)
for i in range(10):
r.append(r[-1] + 0.1 - 0.1 * I)
# close to the exact solution 1.09861228866811
# imaginary part also close to zero
s = '1.07530466271334 - 0.0251200594793912*I'
assert sstr(p.evalf(r, method='Euler')[-1]) == s
# sin(x)
p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1])
s = '0.905546532085401 - 6.93889390390723e-18*I'
assert sstr(p.evalf(r, method='Euler')[-1]) == s
# computing sin(pi/2) using this method
# using a linear path from 0 to pi/2
r = [0.1]
for i in range(14):
r.append(r[-1] + 0.1)
r.append(pi / 2)
s = '1.08016557252834' # close to 1.0 (exact solution)
assert sstr(p.evalf(r, method='Euler')[-1]) == s
# trying different path, a rectangle (0-->i-->pi/2 + i-->pi/2)
# computing the same value sin(pi/2) using different path
r = [0.1 * I]
for i in range(9):
r.append(r[-1] + 0.1 * I)
for i in range(15):
r.append(r[-1] + 0.1)
r.append(pi / 2 + I)
for i in range(10):
r.append(r[-1] - 0.1 * I)
# close to 1.0
s = '0.976882381836257 - 1.65557671738537e-16*I'
assert sstr(p.evalf(r, method='Euler')[-1]) == s
# cos(x)
p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0])
# compute cos(pi) along 0-->pi
r = [0.05]
for i in range(61):
r.append(r[-1] + 0.05)
r.append(pi)
# close to -1 (exact answer)
s = '-1.08140824719196'
assert sstr(p.evalf(r, method='Euler')[-1]) == s
# a rectangular path (0 -> i -> 2+i -> 2)
r = [0.1 * I]
for i in range(9):
r.append(r[-1] + 0.1 * I)
for i in range(20):
r.append(r[-1] + 0.1)
for i in range(10):
r.append(r[-1] - 0.1 * I)
p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1]).evalf(r, method='Euler')
s = '0.501421652861245 - 3.88578058618805e-16*I'
assert sstr(p[-1]) == s
def __str__(f):
from sympy.printing.str import sstr
return sstr(f.rep)
def __str__(self):
from sympy.printing.str import sstr
return sstr(self.data) + " + " + str(self.ring.base_ideal)
