def _get_general_solution(self, *, simplify: bool = True):
x = self.ode_problem.sym
(C1,) = self.ode_problem.get_numbered_constants(num=1)
gensol = Eq(self.ly, (((C1 + Integral((self.cx/self.ax)*exp(Integral(self.bx/self.ax, x)),x))
* exp(-Integral(self.bx/self.ax, x)))))
return [gensol]
def test_Integral():
assert mcode(Integral(sin(sin(x)), x)) == "Hold[Integrate[Sin[Sin[x]], x]]"
assert mcode(Integral(exp(-x**2 - y**2),
(x, -oo, oo),
(y, -oo, oo))) == \
"Hold[Integrate[Exp[-x^2 - y^2], {x, -Infinity, Infinity}, " \
"{y, -Infinity, Infinity}]]"
def _get_general_solution(self, *, simplify: bool = True):
P, Q = self.wilds_match()
fx = self.ode_problem.func
x = self.ode_problem.sym
(C1,) = self.ode_problem.get_numbered_constants(num=1)
gensol = Eq(fx, (((C1 + Integral(Q*exp(Integral(P, x)),x))
* exp(-Integral(P, x)))))
return [gensol]
def _get_general_solution(self, *, simplify: bool = True):
d, e, k = self.wilds_match()
fx = self.ode_problem.func
x = self.ode_problem.sym
C1, C2 = self.ode_problem.get_numbered_constants(num=2)
int = Integral(exp(Integral(self.g, self.y)), (self.y, None, fx))
gen_sol = Eq(int + C1 * Integral(exp(-Integral(self.h, x)), x) + C2, 0)
return [gen_sol]
def _get_general_solution(self, *, simplify: bool = True):
fx = self.ode_problem.func
x = self.ode_problem.sym
(C1, ) = self.ode_problem.get_numbered_constants(num=1)
int = Integral(self.m2['coeff'] * self.m2[self.y] / self.m1[self.y],
(self.y, None, fx))
gen_sol = Eq(
int,
Integral(-self.m1['coeff'] * self.m1[x] / self.m2[x], x) + C1)
return [gen_sol]
def _get_general_solution(self, *, simplify: bool = True):
g = self.wilds_match()[0]
fx = self.ode_problem.func
x = self.ode_problem.sym
u = Dummy('u')
g = g.subs(fx, u)
C1, C2 = self.ode_problem.get_numbered_constants(num=2)
inside = -2 * Integral(g, u) + C1
lhs = Integral(1 / sqrt(inside), (u, fx))
return [Eq(lhs, C2 + x), Eq(lhs, C2 - x)]
def _get_general_solution(self, *, simplify: bool = True):
P, Q, n = self.wilds_match()
fx = self.ode_problem.func
x = self.ode_problem.sym
(C1, ) = self.ode_problem.get_numbered_constants(num=1)
if n == 1:
gensol = Eq(log(fx), ((C1 + Integral((-P + Q), x))))
else:
gensol = Eq(fx**(1 - n), ((C1 - (n - 1) * Integral(
Q * exp(-n * Integral(P, x)) * exp(Integral(P, x)), x)) *
exp(-(1 - n) * Integral(P, x))))
return [gensol]
def _get_general_solution(self, *, simplify: bool = True):
m, n = self.wilds_match()
fx = self.ode_problem.func
x = self.ode_problem.sym
(C1,) = self.ode_problem.get_numbered_constants(num=1)
y = Dummy('y')
m = m.subs(fx, y)
n = n.subs(fx, y)
gen_sol = Eq(Subs(Integral(m, x)
+ Integral(n - Integral(m, x).diff(y), y), y, fx), C1)
return gen_sol
def _verify(self, fx) -> bool:
P, Q = self.wilds()
x = self.ode_problem.sym
y = Dummy('y')
m, n = self.wilds_match()
try:
if simplify(m) != 0:
m = m.subs(fx,y)
n = n.subs(fx,y)
numerator= simplify(m.diff(y)-n.diff(x))
# The following few conditions try to convert a non-exact
# differential equation into an exact one.
# References :
#1. Differential equations with applications
# and historical notes - George E. Simmons
#2. https://math.okstate.edu/people/binegar/2233-S99/2233-l12.pdf
if numerator:
# If (dP/dy - dQ/dx) / Q = f(x)
# then exp(integral(f(x))*equation becomes exact
factor = simplify(numerator/n)
variables = factor.free_symbols
if len(variables) == 1 and x == variables.pop():
factor = exp(Integral(factor).doit())
m *= factor
n *= factor
self._wilds_match[P] = m.subs(y, fx)
self._wilds_match[Q] = n.subs(y, fx)
return True
else:
# If (dP/dy - dQ/dx) / -P = f(y)
# then exp(integral(f(y))*equation becomes exact
factor = simplify(-numerator / m)
variables = factor.free_symbols
if len(variables) == 1 and y == variables.pop():
factor = exp(Integral(factor).doit())
m *= factor
n *= factor
self._wilds_match[P] = m.subs(y, fx)
self._wilds_match[Q] = n.subs(y, fx)
return True
else:
return True
except NotImplementedError:
# Differentiating the coefficients might fail because of things
# like f(2*x).diff(x). See issue 4624 and issue 4719.
pass
return False
def __init__(self):
if Parser.__instance is not None:
raise Exception("Invalid initialistion of Parser.")
Parser.__instance = self
self.__transformations = standard_transformations \
+ (function_exponentiation, convert_xor, convert_equals_signs)
self.__global_dict = {}
exec_('from sympy.core import *', self.__global_dict)
exec_('from sympy.functions import *', self.__global_dict)
exec_('from sympy.integrals import *', self.__global_dict)
exclusions = ['sympify', 'SympifyError', 'Subs', 'evalf', 'evaluate']
for e in exclusions:
self.__global_dict.pop(e)
e, pi = sy.symbols('e pi')
def func_check(name):
return lambda e : isinstance(e, sy.Function) \
and getattr(e, 'name', None) == name
self.__default_subs = [(e, np.exp(1)), (pi, np.pi)]
self.__default_repl = [
(func_check('der'), lambda e: Derivative(*e.args)),
(func_check('int'), lambda e: Integral(*e.args))
]
self.__default_trans = [(lambda x: isinstance(x,
(Derivative, Integral)),
lambda e: e.doit())]
self.__invalid_atoms = (Derivative, Integral)
def _solve_variation_of_parameters(eq,
func,
roots,
homogen_sol,
order,
match_obj,
simplify_flag=True):
r"""
Helper function for the method of variation of parameters and nonhomogeneous euler eq.
See the
:py:meth:`~sympy.solvers.ode.single.NthLinearConstantCoeffVariationOfParameters`
docstring for more information on this method.
The parameter are ``match_obj`` should be a dictionary that has the following
keys:
``list``
A list of solutions to the homogeneous equation.
``sol``
The general solution.
"""
f = func.func
x = func.args[0]
r = match_obj
psol = 0
wr = wronskian(roots, x)
if simplify_flag:
wr = simplify(wr) # We need much better simplification for
# some ODEs. See issue 4662, for example.
# To reduce commonly occurring sin(x)**2 + cos(x)**2 to 1
wr = trigsimp(wr, deep=True, recursive=True)
if not wr:
# The wronskian will be 0 iff the solutions are not linearly
# independent.
raise NotImplementedError(
"Cannot find " + str(order) +
" solutions to the homogeneous equation necessary to apply " +
"variation of parameters to " + str(eq) + " (Wronskian == 0)")
if len(roots) != order:
raise NotImplementedError(
"Cannot find " + str(order) +
" solutions to the homogeneous equation necessary to apply " +
"variation of parameters to " + str(eq) +
" (number of terms != order)")
negoneterm = (-1)**(order)
for i in roots:
psol += negoneterm * Integral(
wronskian([sol for sol in roots if sol != i], x) * r[-1] / wr,
x) * i / r[order]
negoneterm *= -1
if simplify_flag:
psol = simplify(psol)
psol = trigsimp(psol, deep=True)
return Eq(f(x), homogen_sol.rhs + psol)
def get_sol_2F1_hypergeometric(eq, func, match_object):
x = func.args[0]
from sympy.simplify.hyperexpand import hyperexpand
from sympy.polys.polytools import factor
C0, C1 = get_numbered_constants(eq, num=2)
a = match_object['a']
b = match_object['b']
c = match_object['c']
A = match_object['A']
sol = None
if c.is_integer == False:
sol = C0 * hyper([a, b], [c], x) + C1 * hyper([a - c + 1, b - c + 1],
[2 - c], x) * x**(1 - c)
elif c == 1:
y2 = Integral(
exp(Integral((-(a + b + 1) * x + c) / (x**2 - x), x)) /
(hyperexpand(hyper([a, b], [c], x))**2), x) * hyper([a, b], [c], x)
sol = C0 * hyper([a, b], [c], x) + C1 * y2
elif (c - a - b).is_integer == False:
sol = C0 * hyper([a, b], [1 + a + b - c], 1 - x) + C1 * hyper(
[c - a, c - b], [1 + c - a - b], 1 - x) * (1 - x)**(c - a - b)
if sol:
# applying transformation in the solution
subs = match_object['mobius']
dtdx = simplify(1 / (subs.diff(x)))
_B = ((a + b + 1) * x - c).subs(x, subs) * dtdx
_B = factor(_B + ((x**2 - x).subs(x, subs)) * (dtdx.diff(x) * dtdx))
_A = factor((x**2 - x).subs(x, subs) * (dtdx**2))
e = exp(logcombine(Integral(cancel(_B / (2 * _A)), x), force=True))
sol = sol.subs(x, match_object['mobius'])
sol = sol.subs(x, x**match_object['k'])
e = e.subs(x, x**match_object['k'])
if not A.is_zero:
e1 = Integral(A / 2, x)
e1 = exp(logcombine(e1, force=True))
sol = cancel((e / e1) * x**((-match_object['k'] + 1) / 2)) * sol
sol = Eq(func, sol)
return sol
sol = cancel((e) * x**((-match_object['k'] + 1) / 2)) * sol
sol = Eq(func, sol)
return sol
def _sympy_(self, f, x, a, b):
"""
Convert this integral to the equivalent SymPy object
The resulting SymPy integral can be evaluated using ``doit()``.
EXAMPLES::
sage: integral(x, x, 0, 1, hold=True)._sympy_()
Integral(x, (x, 0, 1))
sage: _.doit()
1/2
"""
from sympy.integrals import Integral
return Integral(f, (x, a, b))
def _verify(self, fx) -> bool:
P, Q = self.wilds()
x = self.ode_problem.sym
y = Dummy('y')
m, n = self.wilds_match()
m = m.subs(fx, y)
n = n.subs(fx, y)
numerator = cancel(m.diff(y) - n.diff(x))
if numerator.is_zero:
# Is exact
return True
else:
# The following few conditions try to convert a non-exact
# differential equation into an exact one.
# References:
# 1. Differential equations with applications
# and historical notes - George E. Simmons
# 2. https://math.okstate.edu/people/binegar/2233-S99/2233-l12.pdf
factor_n = cancel(numerator / n)
factor_m = cancel(-numerator / m)
if y not in factor_n.free_symbols:
# If (dP/dy - dQ/dx) / Q = f(x)
# then exp(integral(f(x))*equation becomes exact
factor = factor_n
integration_variable = x
elif x not in factor_m.free_symbols:
# If (dP/dy - dQ/dx) / -P = f(y)
# then exp(integral(f(y))*equation becomes exact
factor = factor_m
integration_variable = y
else:
# Couldn't convert to exact
return False
factor = exp(Integral(factor, integration_variable))
m *= factor
n *= factor
self._wilds_match[P] = m.subs(y, fx)
self._wilds_match[Q] = n.subs(y, fx)
return True
def _get_general_solution(self, *, simplify: bool = True):
P, Q = self.wilds_match()
fx = self.ode_problem.func
x = self.ode_problem.sym
(C1, ) = self.ode_problem.get_numbered_constants(num=1)
if self.fxx is None or self.gx is None or self.kx is None:
gensol = Eq(fx, (((C1 + Integral(Q * exp(Integral(P, x)), x)) *
exp(-Integral(P, x)))))
else:
gensol = Eq(self.substituting, (((C1 + Integral(
(self.gx / self.fxx) * exp(Integral(self.kx / self.fxx, x)),
x)) * exp(-Integral(self.kx / self.fxx, x)))))
return [gensol]
return components[:self.dim]
class VectorMagnitude(Basic):
def __init__(self, arg):
self._args = Tuple(arg)
def decompose(self):
comps = self.args[0].components()
return sqrt(sum([x*x for x in comps]))
if __name__ == '__main__':
from sympy import Function, exp
from sympy.functions import Abs
from sympy.integrals import Integral
v1 = Vector('v1',dim=2)
v2 = Vector('v2',dim=2)
i = Symbol('i',integer=True)
a = Symbol('a')
print v1,type(v1)
#e = 2*Integral(Abs(v1-v2),(v1,0.2))
V = Function('V')
#V2 = V(i)
e = Integral(exp(-a*V(v1)),v1)
print e,e.subs(V(v1),V(Abs(v1)))
from sympy import *
from sympy import sin, sqrt
from sympy.abc import x, n
from sympy.integrals import Integral
x = Symbol('x')
# example 1: finding the indefinite integral for function: x**2+8 = x**4 + 7*x**(3 + 8)
# reminder: Indefinite integral is an integral with no lower or upper limit specified.
integralex = Integral((x**2) + 8, x)
print(integralex.doit()) # x**3/3 + 8*x
# example 2: integrating the same function above (x**2+8), but we will perform a definite integral with
# respect to a lower limit of 2 and an upper limit of 4.
x = Symbol('x')
integralex = Integral(
(x**2) + 8, (x, 2, 4)
) # the second argument reads as: wrt x variable, with lower limit 2, and upper limit 4
print(integralex.doit())
def inverse(self):
# don't use integrate here because fx has been replaced by _t
# in the equation; integrals will not be correct while solve
# is at work.
return lambda expr: Integral(expr, var) + Dummy('C')
def _as_integral(self, f, t, s):
from sympy import Integral, exp
return Integral(f * exp(-s * t), (t, 0, oo))
def _as_integral(self, F, s, x):
from sympy import Integral, I, oo
c = self.__class__._c
return Integral(F * x**(-s), (s, c - I * oo, c + I * oo))
def _as_integral(self, f, x, s):
from sympy import Integral
return Integral(f * x**(s - 1), (x, 0, oo))
def _as_integral(self, f, x, k):
from sympy import Integral, exp, I
a = self.__class__._a
b = self.__class__._b
return Integral(a * f * exp(b * I * x * k), (x, -oo, oo))
def _as_integral(self, F, s, t):
from sympy import I, Integral, exp
c = self.__class__._c
return Integral(exp(s * t) * F, (s, c - I * oo, c + I * oo))
def euler_maclaurin(self, m=0, n=0, eps=0, eval_integral=True):
"""
Return an Euler-Maclaurin approximation of self, where m is the
number of leading terms to sum directly and n is the number of
terms in the tail.
With m = n = 0, this is simply the corresponding integral
plus a first-order endpoint correction.
Returns (s, e) where s is the Euler-Maclaurin approximation
and e is the estimated error (taken to be the magnitude of
the first omitted term in the tail):
>>> from sympy.abc import k, a, b
>>> from sympy import Sum
>>> Sum(1/k, (k, 2, 5)).doit().evalf()
1.28333333333333
>>> s, e = Sum(1/k, (k, 2, 5)).euler_maclaurin()
>>> s
-log(2) + 7/20 + log(5)
>>> from sympy import sstr
>>> print(sstr((s.evalf(), e.evalf()), full_prec=True))
(1.26629073187415, 0.0175000000000000)
The endpoints may be symbolic:
>>> s, e = Sum(1/k, (k, a, b)).euler_maclaurin()
>>> s
-log(a) + log(b) + 1/(2*b) + 1/(2*a)
>>> e
Abs(1/(12*b**2) - 1/(12*a**2))
If the function is a polynomial of degree at most 2n+1, the
Euler-Maclaurin formula becomes exact (and e = 0 is returned):
>>> Sum(k, (k, 2, b)).euler_maclaurin()
(b**2/2 + b/2 - 1, 0)
>>> Sum(k, (k, 2, b)).doit()
b**2/2 + b/2 - 1
With a nonzero eps specified, the summation is ended
as soon as the remainder term is less than the epsilon.
"""
from sympy.functions import bernoulli, factorial
from sympy.integrals import Integral
m = int(m)
n = int(n)
f = self.function
if len(self.limits) != 1:
raise ValueError("More than 1 limit")
i, a, b = self.limits[0]
if (a > b) == True:
if a - b == 1:
return S.Zero, S.Zero
a, b = b + 1, a - 1
f = -f
s = S.Zero
if m:
if b.is_Integer and a.is_Integer:
m = min(m, b - a + 1)
if not eps or f.is_polynomial(i):
for k in range(m):
s += f.subs(i, a + k)
else:
term = f.subs(i, a)
if term:
test = abs(term.evalf(3)) < eps
if test == True:
return s, abs(term)
elif not (test == False):
# a symbolic Relational class, can't go further
return term, S.Zero
s += term
for k in range(1, m):
term = f.subs(i, a + k)
if abs(term.evalf(3)) < eps and term != 0:
return s, abs(term)
s += term
if b - a + 1 == m:
return s, S.Zero
a += m
x = Dummy('x')
I = Integral(f.subs(i, x), (x, a, b))
if eval_integral:
I = I.doit()
s += I
def fpoint(expr):
if b is S.Infinity:
return expr.subs(i, a), 0
return expr.subs(i, a), expr.subs(i, b)
fa, fb = fpoint(f)
iterm = (fa + fb) / 2
g = f.diff(i)
for k in range(1, n + 2):
ga, gb = fpoint(g)
term = bernoulli(2 * k) / factorial(2 * k) * (gb - ga)
if (eps and term and abs(term.evalf(3)) < eps) or (k > n):
break
s += term
g = g.diff(i, 2, simplify=False)
return s + iterm, abs(term)
def approximate(func, from_val, to_val, degree):
"""
Approximate given continuous function over real numbers, using a polynomial of given degree, with inner product
defined as <f, g> = INTEGRATE f(x) * g(x) dx from a to b.
:param func: Function to approximate represented in a string, with variable as 'x'.
:param from_val: a as a string.
:param to_val: b as a string.
:param degree: Highest degree of result polynomial, as an integer.
:return: Approximated polynomial function in string format.
"""
print('------ Started Calculating Approximation ------')
print()
print('f(x) = {0}'.format(func))
print()
start_time = time.time()
orth_basis = orthonormal_basis(_get_float_value(from_val),
_get_float_value(to_val), degree)
print('Orthonormal basis:')
for idx, ele in enumerate(orth_basis):
print('e{0} = {1}'.format(idx + 1, ele))
print()
x = sp.Symbol('x')
res = 0
func_str = '({0})'.format(func)
for idx, e_j in enumerate(orth_basis):
start_time_e_j = time.time()
if idx is 0:
print('Calculating projection on span(e{0}) --- '.format(idx + 1),
end='')
else:
print(
'Calculating projection on span(e1,...,e{0}) --- '.format(idx +
1),
end='')
e_j_str = '({0})'.format(
e_j.standard_coeff_rep(show_mul_op=True, double_stars=True))
product_str = '{0} * {1}'.format(func_str, e_j_str)
func_product = parse_expr(product_str)
tmp = Integral(func_product,
(x, from_val, to_val)).as_sum(100,
method="midpoint").n()
tmp *= parse_expr(e_j_str)
res += tmp
end_time_e_j = time.time()
print('%.2fs' % (end_time_e_j - start_time_e_j))
# Print the current result
tmp_res = str(res)
tmp_res = tmp_res.replace('**', '^')
tmp_res = tmp_res.replace('*', '')
tmp_res = Polynomial(tmp_res)
print()
print(' f{0}(x) = {1}'.format(idx + 1, tmp_res))
print()
end_time = time.time()
print()
print("Duration: %.2fs" % (end_time - start_time))
print()
print('------ Finished Calculating Approximation ------')
res = str(res)
res = res.replace('**', '^')
res = res.replace('*', '')
res = Polynomial(res)
return res
def euler_maclaurin(self, m=0, n=0, eps=0, eval_integral=True):
"""
Return an Euler-Maclaurin approximation of self, where m is the
number of leading terms to sum directly and n is the number of
terms in the tail.
With m = n = 0, this is simply the corresponding integral
plus a first-order endpoint correction.
Returns (s, e) where s is the Euler-Maclaurin approximation
and e is the estimated error (taken to be the magnitude of
the first omitted term in the tail):
>>> from sympy.abc import k, a, b
>>> from sympy import Sum
>>> Sum(1/k, (k, 2, 5)).doit().evalf()
1.28333333333333
>>> s, e = Sum(1/k, (k, 2, 5)).euler_maclaurin()
>>> s
-log(2) + 7/20 + log(5)
>>> from sympy import sstr
>>> print(sstr((s.evalf(), e.evalf()), full_prec=True))
(1.26629073187415, 0.0175000000000000)
The endpoints may be symbolic:
>>> s, e = Sum(1/k, (k, a, b)).euler_maclaurin()
>>> s
-log(a) + log(b) + 1/(2*b) + 1/(2*a)
>>> e
Abs(1/(12*b**2) - 1/(12*a**2))
If the function is a polynomial of degree at most 2n+1, the
Euler-Maclaurin formula becomes exact (and e = 0 is returned):
>>> Sum(k, (k, 2, b)).euler_maclaurin()
(b**2/2 + b/2 - 1, 0)
>>> Sum(k, (k, 2, b)).doit()
b**2/2 + b/2 - 1
With a nonzero eps specified, the summation is ended
as soon as the remainder term is less than the epsilon.
"""
from sympy.functions import bernoulli, factorial
from sympy.integrals import Integral
m = int(m)
n = int(n)
f = self.function
if len(self.limits) != 1:
raise ValueError("More than 1 limit")
i, a, b = self.limits[0]
if (a > b) == True:
if a - b == 1:
return S.Zero,S.Zero
a, b = b + 1, a - 1
f = -f
s = S.Zero
if m:
if b.is_Integer and a.is_Integer:
m = min(m, b - a + 1)
if not eps or f.is_polynomial(i):
for k in range(m):
s += f.subs(i, a + k)
else:
term = f.subs(i, a)
if term:
test = abs(term.evalf(3)) < eps
if test == True:
return s, abs(term)
elif not (test == False):
# a symbolic Relational class, can't go further
return term, S.Zero
s += term
for k in range(1, m):
term = f.subs(i, a + k)
if abs(term.evalf(3)) < eps and term != 0:
return s, abs(term)
s += term
if b - a + 1 == m:
return s, S.Zero
a += m
x = Dummy('x')
I = Integral(f.subs(i, x), (x, a, b))
if eval_integral:
I = I.doit()
s += I
def fpoint(expr):
if b is S.Infinity:
return expr.subs(i, a), 0
return expr.subs(i, a), expr.subs(i, b)
fa, fb = fpoint(f)
iterm = (fa + fb)/2
g = f.diff(i)
for k in range(1, n + 2):
ga, gb = fpoint(g)
term = bernoulli(2*k)/factorial(2*k)*(gb - ga)
if (eps and term and abs(term.evalf(3)) < eps) or (k > n):
break
s += term
g = g.diff(i, 2, simplify=False)
return s + iterm, abs(term)