def tractionMax(mass, g, wtRearFrac, wheelbase, cgh, driveWheel, muTire):
Fr = sym.Symbol('Fr')
Ff = sym.Symbol('Ff')
Fa = sym.Symbol('Fa')
if driveWheel == 'AWD':
print('AWD')
wheelForceMax = mass*g*muTire
elif driveWheel == 'FWD':
print('FWD')
eq1 = Fr + Ff - mass*g
eq2 = Fr*(1-wtRearFrac)*wheelbase - Ff*wtRearFrac*wheelbase - Fa*cgh
eq3 = Fa - Ff*muTire
X = linsolve([eq1, eq2, eq3], (Fr, Ff, Fa))
# extract the solution for eq3
wheelForceMax = X.args[0][2]
else: #RWD case
eq1 = Fr + Ff - mass*g
eq2 = Fr*(1-wtRearFrac)*wheelbase - Ff*wtRearFrac*wheelbase - Fa*cgh
eq3 = Fa - Fr*muTire
X = linsolve([eq1, eq2, eq3], (Fr, Ff, Fa))
# extract the solution for eq3
wheelForceMax = X.args[0][2]
# cast to float for speed
return float(wheelForceMax)
def test_linsolve():
x, y, z, u, v, w = symbols("x, y, z, u, v, w")
x1, x2, x3, x4 = symbols('x1, x2, x3, x4')
# Test for different input forms
M = Matrix([[1, 2, 1, 1, 7], [1, 2, 2, -1, 12], [2, 4, 0, 6, 4]])
system1 = A, b = M[:, :-1], M[:, -1]
Eqns = [
x1 + 2 * x2 + x3 + x4 - 7, x1 + 2 * x2 + 2 * x3 - x4 - 12,
2 * x1 + 4 * x2 + 6 * x4 - 4
]
sol = FiniteSet((-2 * x2 - 3 * x4 + 2, x2, 2 * x4 + 5, x4))
assert linsolve(M, (x1, x2, x3, x4)) == sol
assert linsolve(Eqns, (x1, x2, x3, x4)) == sol
assert linsolve(system1, (x1, x2, x3, x4)) == sol
# raise ValueError if no symbols are given
raises(ValueError, lambda: linsolve(system1))
# raise ValueError if, A & b is not given as tuple
raises(ValueError, lambda: linsolve(A, b, x1, x2, x3, x4))
# raise ValueError for garbage value
raises(ValueError, lambda: linsolve(Eqns[0], x1, x2, x3, x4))
# Fully symbolic test
a, b, c, d, e, f = symbols('a, b, c, d, e, f')
A = Matrix([[a, b], [c, d]])
B = Matrix([[e], [f]])
system2 = (A, B)
sol = FiniteSet((-b * (f - c * e / a) / (a * (d - b * c / a)) + e / a,
(f - c * e / a) / (d - b * c / a)))
assert linsolve(system2, [x, y]) == sol
# Test for Dummy Symbols issue #9667
x1 = Dummy('x1')
x2 = Dummy('x2')
x3 = Dummy('x3')
x4 = Dummy('x4')
assert linsolve(system1, x1, x2, x3, x4) == FiniteSet(
(-2 * x2 - 3 * x4 + 2, x2, 2 * x4 + 5, x4))
# No solution
A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
b = Matrix([0, 0, 1])
assert linsolve((A, b), (x, y, z)) == EmptySet()
# Issue #10121 - Assignment of free variables
a, b, c, d, e = symbols('a, b, c, d, e')
Augmatrix = Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]])
assert linsolve(Augmatrix, a, b, c, d, e) == FiniteSet((a, 0, c, 0, e))
def test_linsolve():
x, y, z, u, v, w = symbols("x, y, z, u, v, w")
x1, x2, x3, x4 = symbols('x1, x2, x3, x4')
# Test for different input forms
M = Matrix([[1, 2, 1, 1, 7], [1, 2, 2, -1, 12], [2, 4, 0, 6, 4]])
system1 = A, b = M[:, :-1], M[:, -1]
Eqns = [x1 + 2*x2 + x3 + x4 - 7, x1 + 2*x2 + 2*x3 - x4 - 12,
2*x1 + 4*x2 + 6*x4 - 4]
sol = FiniteSet((-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4))
assert linsolve(M, (x1, x2, x3, x4)) == sol
assert linsolve(Eqns, (x1, x2, x3, x4)) == sol
assert linsolve(system1, (x1, x2, x3, x4)) == sol
# raise ValueError if no symbols are given
raises(ValueError, lambda: linsolve(system1))
# raise ValueError if, A & b is not given as tuple
raises(ValueError, lambda: linsolve(A, b, x1, x2, x3, x4))
# raise ValueError for garbage value
raises(ValueError, lambda: linsolve(Eqns[0], x1, x2, x3, x4))
# Fully symbolic test
a, b, c, d, e, f = symbols('a, b, c, d, e, f')
A = Matrix([[a, b], [c, d]])
B = Matrix([[e], [f]])
system2 = (A, B)
sol = FiniteSet((-b*(f - c*e/a)/(a*(d - b*c/a)) + e/a,
(f - c*e/a)/(d - b*c/a)))
assert linsolve(system2, [x, y]) == sol
# Test for Dummy Symbols issue #9667
x1 = Dummy('x1')
x2 = Dummy('x2')
x3 = Dummy('x3')
x4 = Dummy('x4')
assert linsolve(system1, x1, x2, x3, x4) == FiniteSet((-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4))
# No solution
A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
b = Matrix([0, 0, 1])
assert linsolve((A, b), (x, y, z)) == EmptySet()
# Issue #10121 - Assignment of free variables
a, b, c, d, e = symbols('a, b, c, d, e')
Augmatrix = Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]])
assert linsolve(Augmatrix, a, b, c, d, e) == FiniteSet((a, 0, c, 0, e))
def testAll(chosen,numchose,poss,dctCov,cnt,A,numEq):
ret=[];
if numchose<len(poss):
for i in range(0,len(poss[numchose])):
next=[c for c in chosen];
next.append(poss[numchose][i]);
ret.extend(testAll(next,numchose+1,poss,dctCov,cnt,A,numEq))
return ret;
b=[0 for i in range(0,numEq)];
num=0;
##add eqations from histogram
for d in dctCov:
b[num]=cnt[d]
num=num+1;
for i in range(0,len(chosen)):
for k in range(0,6):
b[num+k]=chosen[i][k]
num=num+6;
##want all nice solutions to Ax=b!!
b=Matrix(b);
symb=[symbols("x"+str(i)) for i in range(0,3*len(dctCov))]
##
##FINISH!!
##
solution=linsolve((A,b),symb);
if solution==EmptySet():
return [];
else:
return [solution];
def symbolCIP7():
# Set symbolic parameters
x, a, b, c, d, e, f, g, l, h, ui, ui1, gi, gi1, ggi, ggi1, gggi, gggi1 \
= symbols('x, a, b, c, d, e, f, g, l, h, ui, ui1, gi, gi1, ggi, ggi1, gggi, gggi1')
# Set interpolation polynomial
func = l + g * x + f * x**2 + e * x**3 + d * x**4 + c * x**5 + b * x**6 + a * x**7
df = func.diff(x)
ddf = df.diff(x)
dddf = ddf.diff(x)
eq01 = func.subs(x, 0)
eq02 = df.subs(x, 0)
eq03 = func.subs(x, -h)
eq04 = df.subs(x, -h)
eq05 = ddf.subs(x, 0)
eq06 = ddf.subs(x, -h)
eq07 = dddf.subs(x, 0)
eq08 = dddf.subs(x, -h)
# Solve system under constrains
sol = linsolve([
eq01 - ui, eq02 - gi, eq03 - ui1, eq05 - ggi, eq04 - gi1, eq07 - gggi,
eq06 - ggi1, eq08 - gggi1
], (a, b, c, d, e, f, g, l))
sol_get = next(iter(sol))
symbol_expr = ["a", "b", "c", "d", "e", "f", "g", "l"]
for i in range(0, 8):
print("{0} = {1}\n".format(symbol_expr[i], sol_get[i]))
def _transform_explike_DE(DE, g, x, order, syms):
"""Converts DE with free parameters into DE with constant coefficients."""
from sympy.solvers.solveset import linsolve
eq = []
highest_coeff = DE.coeff(Derivative(g(x), x, order))
for i in range(order):
coeff = DE.coeff(Derivative(g(x), x, i))
coeff = (coeff / highest_coeff).expand().collect(x)
for t in Add.make_args(coeff):
eq.append(t)
temp = []
for e in eq:
if e.has(x):
break
elif e.has(Symbol):
temp.append(e)
else:
eq = temp
if eq:
sol = dict(zip(syms, (i for s in linsolve(eq, list(syms)) for i in s)))
if sol:
DE = DE.subs(sol)
DE = DE.factor().as_coeff_mul(Derivative)[1][0]
DE = DE.collect(Derivative(g(x)))
return DE
def solveSystemOfEqs(eqs):
x = Symbol("x")
y = Symbol("y")
sol = linsolve([parse_expr(eqs[0],evaluate=0), parse_expr(eqs[1], evaluate=0)],(x,y))
x_sol,y_sol = next(iter(sol))
print ("x = " + str(x_sol) + ", y = " + str(y_sol))
return (x_sol, y_sol)
def weingarten_transform(self, vec):
"""
Args:
v: planar vector in tangent plane, which would be decomposed into r_u and r_v.
"""
from sympy.solvers.solveset import linsolve
from sympy import Matrix, symbols
r_u, r_v = self.r_u, self.r_v
c_ru, c_rv = symbols('c_ru, c_rv', real=True)
solset = linsolve(
Matrix(((r_u[0], r_v[0], vec[0]), (r_u[1], r_v[1], vec[1]),
(r_u[2], r_v[2], vec[2]))), (c_ru, c_rv))
try:
if len(solset) != 1:
raise RuntimeError(
f"Sympy is not smart enough to decompose the v vector with r_u, r_v as the basis.\
It found these solutions: {solset}.\
Users need to choose from them or deduce manually, and then set it by arguments."
)
except:
raise RuntimeError(
"We failed to decompose the input vec into r_u and r_v")
else:
c_ru, c_rv = next(iter(solset))
omega = self.weingarten_matrix
W_r_u = omega[0, 0] * r_u + omega[1, 0] * r_v
W_r_v = omega[0, 1] * r_u + omega[1, 1] * r_v
return c_ru * W_r_u + c_rv * W_r_v
def eval_constraints(tf_constraints, solution_dict={}):
# var dict: string variables names -> sympy variables
# sol'n dict: string variable snames -> real values
vardict = {}
for i in range(1, tf_constraints.numVars + 1):
vardict['x' + str(i)] = symbols('x' + str(i))
for c in tf_constraints.constraints:
if not isinstance(c, ReluConstraint):
varsineq = [vardict[v] for v in c.x.flatten().tolist()]
A = c.A
b = c.b
for string_varname in solution_dict.keys():
# if appears in solution dict already, add row to matrix
if string_varname in c.x:
Aaddition = np.array([
1 if e == string_varname else 0 for e in c.x
]).reshape(1, -1)
A = Matrix(np.vstack((Aaddition, A)))
b = Matrix(
np.vstack(([[solution_dict[string_varname]]], b)))
solns = linsolve((A, b), varsineq)
sol_list = list([i for i in solns][0])
for i in range(len(c.x)):
solution_dict[c.x[i][0]] = sol_list[i]
else:
# if isa ReluConstraint
for i in range(len(c.varout)):
solution_dict[c.varout[i]] = max(0, solution_dict[c.varin[i]])
return solution_dict
def ExactPressure(p_, lamb, mu, sx):
lamb1, lamb2, lamb3, lamb4 = lamb
mu1, mu2, mu3, mu4 = mu
sx0, sx1, sx2, sx3, sx4 = sx
x = sp.symbols('x')
#b1=0
##only compression, epsz =0
a1, a2, a3, a4, b2, b3, b4 = sp.symbols(
'a_1, a_2, a_3, a_4, b_2, b_3, b_4')
p = p_
cof= linsolve([(a1-a2)*sx1 -b2/sx1, (a2-a3)*sx2 -b3/sx2 + b2/sx2, (a3-a4)*sx3 -b4/sx3 + b3/sx3,\
2*(lamb1+mu1)*(a1) - 2*(lamb2+mu2)*(a2) + 2*mu2/sx1**2*b2,\
2*(lamb2+mu2)*(a2) - 2*(lamb3+mu3)*(a3) -2*mu2/sx2**2*b2 + 2*mu3/sx2**2*b3, \
2*(lamb3+mu3)*(a3) - 2*(lamb4+mu4)*(a4) -2*mu3/sx3**2*b3 + 2*mu4/sx3**2*b4,\
2*(lamb4+mu4)*(a4) -2*mu4/sx4**2*b4 -p ], (a1,a2,a3,a4,b2,b3,b4))
a1_, a2_, a3_, a4_, b2_, b3_, b4_ = tuple(*cof)
ans = sp.Piecewise((a1_ * x, x <= sx1),
(a2_ * x + b2_ / x, (x <= sx2) & (x > sx1)),
(a3_ * x + b3_ / x, (x <= sx3) & (x > sx2)),
(a4_ * x + b4_ / x, (x <= sx4) & (x > sx3)), (0, True))
msh = np.linspace(sx0, sx4, 400)
f_ans = sp.lambdify(x, ans)
#print(tuple(*cof))
return msh, f_ans(msh)
def collocations_method_modifyed_points(basis,
points,
func_for_substantiation,
c,
k,
q):
# unite basis functions to one
func = build_function_from_basis(basis)
# create from base func our psi func
psi_func_1 = func_for_substantiation(func, k[0], q[0])
psi_func_2 = func_for_substantiation(func, k[1], q[1])
# generate variables for linsolve
symbols = [sympy.Symbol('a' + str(i)) for i in range(len(points))]
lin_system = []
# substitude variables for linear system and simplify to linear with evalf
for point in points:
if point < c:
psi_func = psi_func_1
else:
psi_func = psi_func_2
lin_system.append(psi_func.subs(x, point).evalf())
# solving our system
answer = list(list(linsolve(lin_system, *symbols))[0])
return answer
def solve_final_solution(init_conditions, a_n, c):
debug_print('Equation to solve values for a_(i):', a_n)
# -- step 5 (step 7 if nonhomogeneous) -- determine values for a_i
eqs = []
symbols_dict = dict(('a_%d' % k, symbols('a_%d' % k)) for k in range(1, c))
locals().update(symbols_dict)
for i in init_conditions:
a = a_n.replace('n', str(i))
a = a.split(' = ')
eqs.append(parse_expr(a[1] + ' - ' + init_conditions.get(i)))
#for eq in eqs:
# print('---->',eq)
var = [symbols_dict.get(sym) for sym in symbols_dict]
#for i in eqs:
# print(i)
#print('------>',var)
#solved = list(linsolve(eqs, var))
solved = list(linsolve(eqs, var))[0]
#print('---->',solved)
#alpha = sorted([[var[key],solved[key]] for key in range(len(solved))])
var, solved = zip(*sorted(zip([str(v) for v in var], solved))[::-1])
#print('---->',var1)
debug_print('Solved values for a_(i):', solved)
for i in range(c - 1):
a_n = a_n.replace(str(var[i]), str(solved[i]))
a_n = a_n.split(' = ')[1]
result = str(simplify(a_n))
return result
def ExactEigenstrain(eig, lamb, mu, sx):
lamb1, lamb2, lamb3, lamb4 = lamb
mu1, mu2, mu3, mu4 = mu
sx0, sx1, sx2, sx3, sx4 = sx
eig_12, eig_23 = eig
x = sp.symbols('x')
#b1=0
##, epsz =0
a1, a2, a3, a4, b2, b3, b4 = sp.symbols(
'a_1, a_2, a_3, a_4, b_2, b_3, b_4')
cof= linsolve([(a1-a2)*sx1 -b2/sx1, (a2-a3)*sx2 -b3/sx2 + b2/sx2, (a3-a4)*sx3 -b4/sx3 + b3/sx3,\
2*(lamb1+mu1)*(a1) - 2*(lamb2+mu2)*(a2) + 2*mu2/sx1**2*b2 + (3*lamb2+2*mu2)*eig_12,\
2*(lamb2+mu2)*(a2) - 2*(lamb3+mu3)*(a3) -2*mu2/sx2**2*b2 + 2*mu3/sx2**2*b3 -(3*lamb2+2*mu2)*eig_12 + (3*lamb3+2*mu3)*eig_23, \
2*(lamb3+mu3)*(a3) - 2*(lamb4+mu4)*(a4) -2*mu3/sx3**2*b3 + 2*mu4/sx3**2*b4 - (3*lamb3+2*mu3)*eig_23,\
2*(lamb4+mu4)*(a4) -2*mu4/sx4**2*b4], (a1,a2,a3,a4,b2,b3,b4))
a1_, a2_, a3_, a4_, b2_, b3_, b4_ = tuple(*cof)
ans = sp.Piecewise((a1_ * x, x <= sx1),
(a2_ * x + b2_ / x, (x <= sx2) & (x > sx1)),
(a3_ * x + b3_ / x, (x <= sx3) & (x > sx2)),
(a4_ * x + b4_ / x, (x <= sx4) & (x > sx3)), (0, True))
msh = np.linspace(sx0, sx4, 400)
f_ans = sp.lambdify(x, ans)
return msh, f_ans(msh)
def remove_redundant_sols(sol1, sol2, x):
"""
Helper function to remove redundant
solutions to the differential equation.
"""
# If y1 and y2 are redundant solutions, there is
# some value of the arbitrary constant for which
# they will be equal
syms1 = sol1.atoms(Symbol, Dummy)
syms2 = sol2.atoms(Symbol, Dummy)
num1, den1 = [Poly(e, x, extension=True) for e in sol1.together().as_numer_denom()]
num2, den2 = [Poly(e, x, extension=True) for e in sol2.together().as_numer_denom()]
# Cross multiply
e = num1*den2 - den1*num2
# Check if there are any constants
syms = list(e.atoms(Symbol, Dummy))
if len(syms):
# Find values of constants for which solutions are equal
redn = linsolve(e.all_coeffs(), syms)
if len(redn):
# Return the general solution over a particular solution
if len(syms1) > len(syms2):
return sol2
# If both have constants, return the lesser complex solution
elif len(syms1) == len(syms2):
return sol1 if count_ops(syms1) >= count_ops(syms2) else sol2
else:
return sol1
def _transform_explike_DE(DE, g, x, order, syms):
"""Converts DE with free parameters into DE with constant coefficients."""
from sympy.solvers.solveset import linsolve
eq = []
highest_coeff = DE.coeff(Derivative(g(x), x, order))
for i in range(order):
coeff = DE.coeff(Derivative(g(x), x, i))
coeff = (coeff / highest_coeff).expand().collect(x)
for t in Add.make_args(coeff):
eq.append(t)
temp = []
for e in eq:
if e.has(x):
break
elif e.has(Symbol):
temp.append(e)
else:
eq = temp
if eq:
sol = dict(zip(syms, (i for s in linsolve(eq, list(syms)) for i in s)))
if sol:
DE = DE.subs(sol)
DE = DE.factor().as_coeff_mul(Derivative)[1][0]
DE = DE.collect(Derivative(g(x)))
return DE
def symbolCIP9():
# Set symbolic parameters
x, a, b, c, d, e, f, g, l, k, q, h, ui, ui1, gi, gi1, ggi, ggi1, g3i, g3i1, g4i, g4i1 \
= symbols('x, a, b, c, d, e, f, g, l, k, q, h, ui, ui1, gi, gi1, ggi, ggi1, g3i, g3i1, g4i, g4i1')
# Set interpolation polynomial
func = q + k * x + l * x**2 + g * x**3 + f * x**4 + e * x**5 + d * x**6 + c * x**7 + b * x**8 + a * x**9
df = func.diff(x)
ddf = df.diff(x)
d3f = ddf.diff(x)
d4f = d3f.diff(x)
eq01 = func.subs(x, 0)
eq02 = df.subs(x, 0)
eq03 = func.subs(x, -h)
eq04 = df.subs(x, -h)
eq05 = ddf.subs(x, 0)
eq06 = ddf.subs(x, -h)
eq07 = d3f.subs(x, 0)
eq08 = d3f.subs(x, -h)
eq09 = d4f.subs(x, 0)
eq10 = d4f.subs(x, -h)
# Solve system under constrains
sol = linsolve([
eq01 - ui, eq02 - gi, eq03 - ui1, eq05 - ggi, eq04 - gi1, eq07 - g3i,
eq06 - ggi1, eq08 - g3i1, eq09 - g4i, eq10 - g4i1
], (a, b, c, d, e, f, g, l, k, q))
sol_get = next(iter(sol))
symbol_expr = ["a", "b", "c", "d", "e", "f", "g", "l", "k", "q"]
for i in range(0, 10):
print("{0} = {1}\n".format(symbol_expr[i], sol_get[i]))
def combineDct(dctt1, dct2, cnt):
if len(dct1) == 0 or len(dct2) == 0:
return []
ret = []
b = [0 for i in range(0, 8 * len(dct1))]
A = [[0 for i in range(0, 6 * len(dct1))] for j in range(0, 8 * len(dct1))]
cnt_keys = [c for c in cnt]
cnt_dct = {}
for i in range(0, len(cnt)):
cnt_dct[cnt_keys[i]] = i
num = 0
for j in range(0, 3):
for d in dct1:
val1 = d + (1)
val2 = d + (0)
b[num] = dct1[d][j]
A[num][3 * cnt_dct[val1] + j] = 1
A[num][3 * cnt_dct[val2] + j] = 1
num = num + 1
val1 = (1) + d
val2 = (0) + d
b[num] = dct2[d][j]
A[num][3 * cnt_dct[val1] + j] = 1
A[num][3 * cnt_dct[val2] + j] = 1
num = num + 1
for d in dct:
val1 = d + (1)
val2 = d + (0)
b[num] = cnt[val1]
for j in range(0, 3):
A[num][3 * cnt_dct[val1] + j] = 1
num = num + 1
b[num] = cnt[val2]
for j in range(0, 3):
A[num][3 * cnt_dct[val2] + j] = 1
num = num + 1
#A=np.asarray();
poss = []
A = Matrix(A)
b = Matrix(b)
linsolve()
def test_linsolve():
x, y, z, u, v, w = symbols("x, y, z, u, v, w")
x1, x2, x3, x4 = symbols('x1, x2, x3, x4')
# Test for different input forms
M = Matrix([[1, 2, 1, 1, 7], [1, 2, 2, -1, 12], [2, 4, 0, 6, 4]])
system = A, b = M[:, :-1], M[:, -1]
Eqns = [x1 + 2*x2 + x3 + x4 - 7, x1 + 2*x2 + 2*x3 - x4 - 12,
2*x1 + 4*x2 + 6*x4 - 4]
sol = FiniteSet((-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4))
assert linsolve(M, (x1, x2, x3, x4)) == sol
assert linsolve(Eqns, (x1, x2, x3, x4)) == sol
assert linsolve(system, (x1, x2, x3, x4)) == sol
# raise ValueError if no symbols are given
raises(ValueError, lambda: linsolve(system))
# raise ValueError if, A & b is not given as tuple
raises(ValueError, lambda: linsolve(A, b, x1, x2, x3, x4))
# raise ValueError for garbage value
raises(ValueError, lambda: linsolve(Eqns[0], x1, x2, x3, x4))
# Fully symbolic test
a, b, c, d, e, f = symbols('a, b, c, d, e, f')
A = Matrix([[a, b], [c, d]])
B = Matrix([[e], [f]])
system = (A, B)
sol = FiniteSet((-b*(f - c*e/a)/(a*(d - b*c/a)) + e/a,
(f - c*e/a)/(d - b*c/a)))
assert linsolve(system, [x, y]) == sol
def test_linsolve():
x, y, z, u, v, w = symbols("x, y, z, u, v, w")
x1, x2, x3, x4 = symbols("x1, x2, x3, x4")
# Test for different input forms
M = Matrix([[1, 2, 1, 1, 7], [1, 2, 2, -1, 12], [2, 4, 0, 6, 4]])
system = A, b = M[:, :-1], M[:, -1]
Eqns = [x1 + 2 * x2 + x3 + x4 - 7, x1 + 2 * x2 + 2 * x3 - x4 - 12, 2 * x1 + 4 * x2 + 6 * x4 - 4]
sol = FiniteSet((-2 * x2 - 3 * x4 + 2, x2, 2 * x4 + 5, x4))
assert linsolve(M, (x1, x2, x3, x4)) == sol
assert linsolve(Eqns, (x1, x2, x3, x4)) == sol
assert linsolve(system, (x1, x2, x3, x4)) == sol
# raise ValueError if no symbols are given
raises(ValueError, lambda: linsolve(system))
# raise ValueError if, A & b is not given as tuple
raises(ValueError, lambda: linsolve(A, b, x1, x2, x3, x4))
# raise ValueError for garbage value
raises(ValueError, lambda: linsolve(Eqns[0], x1, x2, x3, x4))
# Fully symbolic test
a, b, c, d, e, f = symbols("a, b, c, d, e, f")
A = Matrix([[a, b], [c, d]])
B = Matrix([[e], [f]])
system = (A, B)
sol = FiniteSet((-b * (f - c * e / a) / (a * (d - b * c / a)) + e / a, (f - c * e / a) / (d - b * c / a)))
assert linsolve(system, [x, y]) == sol
def _contains(self, other):
from sympy.matrices import Matrix
from sympy.solvers.solveset import solveset, linsolve
from sympy.utilities.iterables import iterable, cartes
L = self.lamda
if self._is_multivariate():
if not iterable(L.expr):
if iterable(other):
return S.false
return other.as_numer_denom() in self.func(
Lambda(L.variables, L.expr.as_numer_denom()), self.base_set)
if len(L.expr) != len(self.lamda.variables):
raise NotImplementedError(filldedent('''
Dimensions of input and output of Lambda are different.'''))
eqs = [expr - val for val, expr in zip(other, L.expr)]
variables = L.variables
free = set(variables)
if all(i.is_number for i in list(Matrix(eqs).jacobian(variables))):
solns = list(linsolve([e - val for e, val in
zip(L.expr, other)], variables))
else:
syms = [e.free_symbols & free for e in eqs]
solns = {}
for i, (e, s, v) in enumerate(zip(eqs, syms, other)):
if not s:
if e != v:
return S.false
solns[vars[i]] = [v]
continue
elif len(s) == 1:
sy = s.pop()
sol = solveset(e, sy)
if sol is S.EmptySet:
return S.false
elif isinstance(sol, FiniteSet):
solns[sy] = list(sol)
else:
raise NotImplementedError
else:
raise NotImplementedError
solns = cartes(*[solns[s] for s in variables])
else:
# assume scalar -> scalar mapping
solnsSet = solveset(L.expr - other, L.variables[0])
if solnsSet.is_FiniteSet:
solns = list(solnsSet)
else:
raise NotImplementedError(filldedent('''
Determining whether an ImageSet contains %s has not
been implemented.''' % func_name(other)))
for soln in solns:
try:
if soln in self.base_set:
return S.true
except TypeError:
return self.base_set.contains(soln.evalf())
return S.false
def symbolCIP(nOrder):
# Simple check
if nOrder % 2 == 0:
sys.exit("Order might be odd more than 3,\n"
"for issue 3, 5, 7, 9, etc.")
# Set values
coeff = symbols('a0:%d' % (nOrder + 1))
xi, h = symbols('xi, h')
ucval = symbols('uC:%d' % (nOrder - 1))
urval = symbols('uR:%d' % (nOrder - 1))
ulval = symbols('uL:%d' % (nOrder - 1))
# Set interpolation polynomials
fpoly = sum(coeff[k] * xi**k for k in range(nOrder + 1))
dfpoly = []
for k in range(np.int((nOrder + 1) / 2)):
dfpoly.append(fpoly.diff(xi, k))
# Set constrains
eqs_left = []
eqs_right = []
for k in range(np.int((nOrder + 1) / 2)):
eqs_right.append(dfpoly[k].subs(xi, 0))
eqs_right.append(dfpoly[k].subs(xi, -h))
eqs_left.append(dfpoly[k].subs(xi, 0))
eqs_left.append(dfpoly[k].subs(xi, h))
# Solve system under constrains
# Right wave
vals_right = ucval + ulval
sys_eqns_right = [
eqs_right[k] - vals_right[k] for k in range((nOrder + 1))
]
sol_right = linsolve(sys_eqns_right, coeff)
sol_right_list = list(sol_right)
print('Right wave:\n', sol_right_list[0])
# Left wave
vals_left = ucval + urval
sys_eqns_left = [eqs_left[k] - vals_left[k] for k in range((nOrder + 1))]
sol_left = linsolve(sys_eqns_left, coeff)
sol_left_list = list(sol_left)
print('Left wave:\n', sol_left_list[0])
def expectation(lam):
p = Matrix([[-prob(i, j, lam) for j in range(k)] for i in range(k)])
for i in range(k):
p[(k + 1) * i] = -sum(p.row(i)) + (Fraction(
lam, n))**(k - i) * (Fraction(n - lam, n))**(n - k + i)
return next(
iter(
linsolve(
(p, ones(k, 1)),
symbols(', '.join(['x_{}'.format(i) for i in range(k)])))))[0]
def linsolve_dict(eq, syms):
"""
Get the output of linsolve as a dict
"""
# Convert tuple type return value of linsolve
# to a dictionary for ease of use
sol = linsolve(eq, syms)
if not sol:
return {}
return {k:v for k, v in zip(syms, list(sol)[0])}
def symboldeltaP3():
x, a, h, t, nu, c1, c2, c3, c4, ui, ui1, ri, ri1, uxx, u3x, rx, rxx, _uni \
= symbols('x, a, h, t, nu, c1, c2, c3, c4, ui, ui1, ri, ri1, uxx, u3x, rx, rxx, _uni')
u = c1 + c2 * x + c3 * x**2 + c4 * x**3
du = u.diff(x)
# Set constrains
eq01 = u.subs(x, 0)
eq02 = du.subs(x, 0)
eq03 = u.subs(x, -h)
eq04 = du.subs(x, -h)
# Solve system under constrains
sol = linsolve([eq01 - ui, eq02 - ri, eq03 - ui1, eq04 - ri1],
(c1, c2, c3, c4))
# Get coefficients
sol_get = next(iter(sol))
c1_expr = sol_get[0].subs(ri, ri / h)
c2_expr = sol_get[1].subs(ri, ri / h)
c3_expr = sol_get[2].subs(ri, ri / h).subs(ri1, ri1 / h)
c4_expr = sol_get[3].subs(ri, ri / h).subs(ri1, ri1 / h)
print(("c1 = {0}\nc2 = {1}\nc3 = {2}\nc4 = {3}".format(
c1_expr, c2_expr, c3_expr, c4_expr)))
# Set constrains
ui = u.subs(x, 0)
ux = u.diff(x)
uxi = ux.subs(x, 0)
uxx = ux.diff(x)
uxxi = uxx.subs(x, 0)
u3x = uxx.diff(x)
u3xi = u3x.subs(x, 0)
r = h * u.diff(x)
ri = r.subs(x, 0)
rx = r.diff(x)
rxi = rx.subs(x, 0)
rxx = rx.diff(x)
rxxi = rxx.subs(x, 0)
uni = ui - a * t * uxi + a**2 * t**2 / 2 * uxxi - a**3 * t**3 / 6 * u3xi
_uni = uni.subs(c1, c1_expr).subs(c2, c2_expr).subs(c3, c3_expr).subs(
c4, c4_expr).subs(a * t / h, nu)
_uni = collect(expand(_uni), nu)
print(_uni)
rni = simplify(ri - a * t * rxi + a**2 * t**2 / 2 * rxxi)
_rni = rni.subs(c1, c1_expr).subs(c2, c2_expr).subs(c3, c3_expr).subs(
c4, c4_expr).subs(a * t / h, nu)
print(collect(expand(_rni), nu))
def find_point(p1, p2, r, oc):
p = p1[0]
q = p1[1]
a = p2[0]
b = p2[1]
x0 = oc[0]
y0 = oc[1]
# solution 1
cons11 = (p * q + r * (math.sqrt(p**2 + q**2 - r**2))) / (r**2 - p**2)
cons12 = (p * q - r * (math.sqrt(p**2 + q**2 - r**2))) / (r**2 - p**2)
cons21 = (a * b + r * (math.sqrt(a**2 + b**2 - r**2))) / (r**2 - a**2)
cons22 = (a * b - r * (math.sqrt(a**2 + b**2 - r**2))) / (r**2 - a**2)
x, y = symbols('x,y')
a1 = -(cons11) * (x - x0 - p) + q + y0 - y
a2 = -(cons12) * (x - x0 - p) + q + y0 - y
b1 = -(cons21) * (x - x0 - a) + b + y0 - y
b2 = -(cons22) * (x - x0 - a) + b + y0 - y
ans1 = linsolve([a1, b1], (x, y))
x1, y1 = next(iter(ans1))
ans2 = linsolve([a1, b2], (x, y))
x2, y2 = next(iter(ans2))
ans3 = linsolve([a2, b1], (x, y))
x3, y3 = next(iter(ans3))
ans4 = linsolve([a2, b2], (x, y))
x4, y4 = next(iter(ans4))
d1 = distance(p, q, x1, y1)
d2 = distance(p, q, x2, y2)
d3 = distance(p, q, x3, y3)
d4 = distance(p, q, x4, y4)
if (d1 <= min(d2, d3, d4)):
return x1, y1
elif (d2 <= min(d1, d3, d4)):
return x2, y2
elif (d3 <= min(d2, d1, d4)):
return x3, y3
elif (d4 <= min(d2, d3, d1)):
return x4, y4
def main():
equations = get_equations(layout)
symbols = set()
for eqn in equations:
symbols = symbols.union(eqn.atoms(Symbol))
symbols = list(symbols)
solved_equations = linsolve(equations, *symbols)
if not solved_equations:
raise OverconstrainedError("A solution could not be found.")
solve = list(solved_equations)[0]
print(list(zip(symbols, solve)))
def calc(self): # The function that calculates the expression of f(t)
f = Function('f')
t = symbols('t')
#The lagrangian
self.L = ode.kinetic(self) - ode.potential(self)
print('the lagrangian is : ')
display(self.L)
##The different placebos for the Lagrange equation##
de = diff(self.L, Derivative(f(t), t))
de = diff(de, t)
dw = diff(self.L, f(t))
da = 0.5 * self.alpha * (self.v**2) #Dissipation function
da = da.subs({sin(f(t)): f(t)})
da = da.subs({cos(f(t)): 1 - (f(t))**2 / 2})
da = diff(da, Derivative(f(t), t))
eq = de - dw + da #LAgrange Equation
##############----------##############
##Solving the equation##
print('The equation is : ')
eq = simplify(eq)
display(eq)
solution = dsolve(eq, f(t))
print('The solution is : ')
display(solution)
#######----------#######
#Initial conditions:
cnd0 = Eq(solution.subs({t: 0}), self.x0)
cnd1 = Eq(solution.diff(t).subs({t: 0}), self.xd0)
display(cnd0)
#Solve for C1 and C2:
C1, C2 = symbols('C1 C2') #Temporary symbols
C1C2_sl = linsolve([cnd0, cnd1], (C1, C2))
#Substitute into the Solution of the equation
solution2 = simplify(solution.subs(C1C2_sl))
if 'C1' in str(solution2):
solution3 = solution2.subs({C1: 1})
if 'C2' in str(solution2):
solution3 = solution2.subs({C2: 2})
print('The solution is :')
display(solution3) #The Solution
return solution3
def show_prices(sequence:str, price_vars:list, budget_eqs:list, switches:list):
switch_eqs = switch_equalities(switches, price_vars)
prices_sol = linsolve(budget_eqs + switch_eqs, price_vars).args
if len(prices_sol)==0:
logger.debug("{}, {}: No prices!".format(sequence, switches))
return
prices = prices_sol[0]
possible_budgets = budget_range(prices)
if possible_budgets.is_EmptySet:
logger.debug("{}, {}: {}, No budget!".format(sequence, switches, prices))
return
else:
switches_str = ", ".join([str(s) for s in switches])
logger.info("{}: handles {}. Requires a in {}".format(" ".join([str(p) for p in prices]), switches_str, possible_budgets))
def _contains(self, other):
from sympy.solvers.solveset import solveset, linsolve
L = self.lamda
if self._is_multivariate():
solns = list(linsolve([expr - val for val, expr in zip(other, L.expr)],
L.variables).args[0])
else:
solns = list(solveset(L.expr - other, L.variables[0]))
for soln in solns:
try:
if soln in self.base_set:
return S.true
except TypeError:
return self.base_set.contains(soln.evalf())
return S.false
def collocations_method(basis, points):
# unite basis functions to one
func = build_function_from_basis(basis)
# create from base func our psi func
psi_func = func_for_substantiation(func)
# generate variables for linsolve
symbols = [sympy.Symbol('a' + str(i)) for i in range(len(points))]
lin_system = []
# substitude variables for linear system and simplify to linear with evalf
for point in points:
lin_system.append(psi_func.subs(x, point).evalf())
# solving our system
answer = linsolve(lin_system, *symbols)
return answer
def onParabola(points):
'''
y = a*x**2 + b*x + c
'''
quadratic_eq = a * x**2 + b * x + c - y
linear_equations = []
for (i, j) in points:
linear_equations.append(quadratic_eq.subs({x: i, y: j}))
try:
ak, bk, ck = list(linsolve(linear_equations, (a, b, c)))[0]
except IndexError:
print('[!] Solution doesn\'t exist')
return {()}
return quadratic_eq.subs({a: ak, b: bk, c: ck, y: 0})
def _contains(self, other):
from sympy.solvers.solveset import solveset, linsolve
L = self.lamda
if self._is_multivariate():
solns = list(linsolve([expr - val for val, expr in zip(other, L.expr)],
L.variables).args[0])
else:
solns = list(solveset(L.expr - other, L.variables[0]))
for soln in solns:
try:
if soln in self.base_set:
return S.true
except TypeError:
return self.base_set.contains(soln.evalf())
return S.false
def differences_method(start_variables_count,
a,
b,
y_a,
y_b,
func_for_partition,
points_k,
func=None):
# generate variables for linsolve
start_variables_count += 2
symbols = [
sympy.Symbol('y' + str(i)) for i in range(start_variables_count)
]
# define our step
h = (b - a) / start_variables_count
# generate x points:
points = linspace(a + h, b - h, start_variables_count - 2).tolist()
# build system of equations
lin_system = []
selected_k = 0
for i in range(1, start_variables_count - 1):
if points[i - 1] > points_k[selected_k][0]:
selected_k += 1
lin_system.append(
func_for_partition(symbols[i - 1], symbols[i], symbols[i + 1], h,
points_k[selected_k][1], func).evalf())
for i in range(start_variables_count - 2):
lin_system[i] = lin_system[i].subs(x, points[i])
lin_system[0] = lin_system[0].subs(symbols[0], y_a)
lin_system[-1] = lin_system[-1].subs(symbols[-1], y_b)
del symbols[0], symbols[-1]
# solving our system and converting FiniteSet to simple list
answer = list(list(linsolve(lin_system, *symbols))[0])
data_type = namedtuple('data', ('points', 'answer', 'step'))
points.insert(0, a)
points.append(b)
answer.insert(0, y_a)
answer.append(y_b)
return data_type(points, answer, (b - a) / start_variables_count)
def galerkin_method(basis, a, b):
# unite basis functions to one
func = build_function_from_basis(basis)
# create from base func our psi func
psi_func = func_for_substantiation(func)
# generate variables for linsolve
symbols = [sympy.Symbol('a' + str(i)) for i in range(len(basis))]
lin_system = []
# getting our integral system and simplify equations to linear with evalf
for i in range(len(basis)):
lin_system.append(
sympy.integrate(psi_func * basis[i], (x, a, b)).evalf())
# solving our system
answer = linsolve(lin_system, *symbols)
return answer
def __solve_equations(equations, m_symbols, p_symbols, s_symbols):
all_symbols = list(p_symbols.values()) + list(
s_symbols.values()) + list(m_symbols.values())
raw_result = linsolve(equations, all_symbols)
if not isinstance(raw_result, sympy.FiniteSet):
raise ValueError()
result = list(raw_result)[0]
p_values = result[:len(p_symbols)]
s_values = result[len(p_symbols):len(p_symbols) + len(s_symbols)]
m_values = result[len(p_symbols) + len(s_symbols):len(p_symbols) +
len(s_symbols) + len(m_symbols)]
p_result = dict(list(zip(p_symbols.keys(), p_values)))
s_result = dict(list(zip(s_symbols.keys(), s_values)))
m_result = dict(list(zip(m_symbols.keys(), m_values)))
return p_result, s_result, m_result
def simpleDE(f, x, g, order=4):
r"""Generates simple DE.
DE is of the form
.. math::
f^k(x) + \sum\limits_{j=0}^{k-1} A_j f^j(x) = 0
where :math:`A_j` should be rational function in x.
Generates DE's upto order 4 (default). DE's can also have free parameters.
By increasing order, higher order DE's can be found.
Yields a tuple of (DE, order).
"""
from sympy.solvers.solveset import linsolve
a = symbols('a:%d' % (order))
def _makeDE(k):
eq = f.diff(x, k) + Add(*[a[i]*f.diff(x, i) for i in range(0, k)])
DE = g(x).diff(x, k) + Add(*[a[i]*g(x).diff(x, i) for i in range(0, k)])
return eq, DE
eq, DE = _makeDE(order)
found = False
for k in range(1, order + 1):
eq, DE = _makeDE(k)
eq = eq.expand()
terms = eq.as_ordered_terms()
ind = rational_independent(terms, x)
if found or len(ind) == k:
sol = dict(zip(a, (i for s in linsolve(ind, a[:k]) for i in s)))
if sol:
found = True
DE = DE.subs(sol)
DE = DE.as_numer_denom()[0]
DE = DE.factor().as_coeff_mul(Derivative)[1][0]
yield DE.collect(Derivative(g(x))), k
def _transform_DE_RE(DE, g, k, order, syms):
"""Converts DE with free parameters into RE of hypergeometric type."""
from sympy.solvers.solveset import linsolve
RE = hyper_re(DE, g, k)
eq = []
for i in range(1, order):
coeff = RE.coeff(g(k + i))
eq.append(coeff)
sol = dict(zip(syms, (i for s in linsolve(eq, list(syms)) for i in s)))
if sol:
m = Wild('m')
RE = RE.subs(sol)
RE = RE.factor().as_numer_denom()[0].collect(g(k + m))
RE = RE.as_coeff_mul(g)[1][0]
for i in range(order): # smallest order should be g(k)
if RE.coeff(g(k + i)) and i:
RE = RE.subs(k, k - i)
break
return RE
def _contains(self, other):
from sympy.solvers.solveset import solveset, linsolve
L = self.lamda
if self._is_multivariate():
solns = list(linsolve([expr - val for val, expr in
zip(other, L.expr)], L.variables).args[0])
else:
solnsSet = solveset(L.expr-other, L.variables[0])
if solnsSet.is_FiniteSet:
solns = list(solveset(L.expr - other, L.variables[0]))
else:
raise NotImplementedError(filldedent('''
Determining whether an ImageSet contains %s has not
been implemented.''' % func_name(other)))
for soln in solns:
try:
if soln in self.base_set:
return S.true
except TypeError:
return self.base_set.contains(soln.evalf())
return S.false
coeff = '1'
Q,rest = str_term.split('*',1)
else:
coeff,Q,rest = str_term.split('*',2)
if Q not in our_Qs:
our_Qs[Q] = [coeff + "*" + rest]
else:
our_Qs[Q].append(coeff + "*" + rest)
for key,value in our_Qs.items():
print("%s: %s" % (key,value))
variables = symbols(" ".join(list(our_Qs))) # assumes our_Qs is an ordered dictionary, not a standard one.
print(variables)
result = linsolve(system_of_eqns,variables)
print("result:",result)
Q_values = OrderedDict()
for k,Q in enumerate(our_Qs):
value = list(result)[0][k]
print("%s: %s" % (Q,value))
Q_values[Q] = value
sys.exit(0)
# bah this is ugly!!!
R_det = R.det()
#det_table = {}
det_table = OrderedDict()
def interpolating_spline(d, x, X, Y):
"""Return spline of degree ``d``, passing through the given ``X``
and ``Y`` values.
This function returns a piecewise function such that each part is
a polynomial of degree not greater than ``d``. The value of ``d``
must be 1 or greater and the values of ``X`` must be strictly
increasing.
Examples
========
>>> from sympy import interpolating_spline
>>> from sympy.abc import x
>>> interpolating_spline(1, x, [1, 2, 4, 7], [3, 6, 5, 7])
Piecewise((3*x, (x >= 1) & (x <= 2)),
(7 - x/2, (x >= 2) & (x <= 4)),
(2*x/3 + 7/3, (x >= 4) & (x <= 7)))
>>> interpolating_spline(3, x, [-2, 0, 1, 3, 4], [4, 2, 1, 1, 3])
Piecewise((-x**3/36 - x**2/36 - 17*x/18 + 2, (x >= -2) & (x <= 1)),
(5*x**3/36 - 13*x**2/36 - 11*x/18 + 7/3, (x >= 1) & (x <= 4)))
See Also
========
bsplines_basis_set, sympy.polys.specialpolys.interpolating_poly
"""
from sympy import symbols, Number, Dummy, Rational
from sympy.solvers.solveset import linsolve
from sympy.matrices.dense import Matrix
# Input sanitization
d = sympify(d)
if not(d.is_Integer and d.is_positive):
raise ValueError(
"Spline degree must be a positive integer, not %s." % d)
if len(X) != len(Y):
raise ValueError(
"Number of X and Y coordinates must be the same.")
if len(X) < d + 1:
raise ValueError(
"Degree must be less than the number of control points.")
if not all(a < b for a, b in zip(X, X[1:])):
raise ValueError(
"The x-coordinates must be strictly increasing.")
# Evaluating knots value
if d.is_odd:
j = (d + 1) // 2
interior_knots = X[j:-j]
else:
j = d // 2
interior_knots = [Rational(a + b, 2) for a, b in
zip(X[j:-j - 1], X[j + 1:-j])]
knots = [X[0]] * (d + 1) + list(interior_knots) + [X[-1]] * (d + 1)
basis = bspline_basis_set(d, knots, x)
A = [[b.subs(x, v) for b in basis] for v in X]
coeff = linsolve((Matrix(A), Matrix(Y)), symbols('c0:{}'.format(
len(X)), cls=Dummy))
coeff = list(coeff)[0]
intervals = set([c for b in basis for (e, c) in b.args
if c != True])
# Sorting the intervals
# ival contains the end-points of each interval
ival = [e.atoms(Number) for e in intervals]
ival = [list(sorted(e))[0] for e in ival]
com = zip(ival, intervals)
com = sorted(com, key=lambda x: x[0])
intervals = [y for x, y in com]
basis_dicts = [dict((c, e) for (e, c) in b.args) for b in basis]
spline = []
for i in intervals:
piece = sum([c*d.get(i, S.Zero) for (c, d) in
zip(coeff, basis_dicts)], S.Zero)
spline.append((piece, i))
return(Piecewise(*spline))
def test_issue_9953():
assert linsolve([ ], x) == S.EmptySet
def intersection(self, o):
"""The intersection with another geometrical entity.
Parameters
==========
o : Point or LinearEntity3D
Returns
=======
intersection : list of geometrical entities
See Also
========
sympy.geometry.point.Point3D
Examples
========
>>> from sympy import Point3D, Line3D, Segment3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(7, 7, 7)
>>> l1 = Line3D(p1, p2)
>>> l1.intersection(p3)
[Point3D(7, 7, 7)]
>>> l1 = Line3D(Point3D(4,19,12), Point3D(5,25,17))
>>> l2 = Line3D(Point3D(-3, -15, -19), direction_ratio=[2,8,8])
>>> l1.intersection(l2)
[Point3D(1, 1, -3)]
>>> p6, p7 = Point3D(0, 5, 2), Point3D(2, 6, 3)
>>> s1 = Segment3D(p6, p7)
>>> l1.intersection(s1)
[]
"""
if isinstance(o, Point3D):
if o in self:
return [o]
else:
return []
elif isinstance(o, LinearEntity3D):
if self == o:
return [self]
elif self.is_parallel(o):
if isinstance(self, Line3D):
if o.p1 in self:
return [o]
return []
elif isinstance(self, Ray3D):
if isinstance(o, Ray3D):
# case 1, rays in the same direction
if self.xdirection == o.xdirection and \
self.ydirection == o.ydirection and \
self.zdirection == o.zdirection:
return [self] if (self.source in o) else [o]
# case 2, rays in the opposite directions
else:
if o.source in self:
if self.source == o.source:
return [self.source]
return [Segment3D(o.source, self.source)]
return []
elif isinstance(o, Segment3D):
if o.p1 in self:
if o.p2 in self:
return [o]
return [Segment3D(o.p1, self.source)]
elif o.p2 in self:
return [Segment3D(o.p2, self.source)]
return []
elif isinstance(self, Segment3D):
if isinstance(o, Segment3D):
# A reminder that the points of Segments are ordered
# in such a way that the following works. See
# Segment3D.__new__ for details on the ordering.
if self.p1 not in o:
if self.p2 not in o:
# Neither of the endpoints are in o so either
# o is contained in this segment or it isn't
if o in self:
return [o]
return []
else:
# p1 not in o but p2 is. Either there is a
# segment as an intersection, or they only
# intersect at an endpoint
if self.p2 == o.p1:
return [o.p1]
return [Segment3D(o.p1, self.p2)]
elif self.p2 not in o:
# p2 not in o but p1 is. Either there is a
# segment as an intersection, or they only
# intersect at an endpoint
if self.p1 == o.p2:
return [o.p2]
return [Segment3D(o.p2, self.p1)]
# Both points of self in o so the whole segment
# is in o
return [self]
else: # unrecognized LinearEntity
raise NotImplementedError
else:
# If the lines are not parallel then solve their arbitrary points
# to obtain the point of intersection
t = t1, t2 = Dummy(), Dummy()
a = self.arbitrary_point(t1)
b = o.arbitrary_point(t2)
dx = a.x - b.x
c = linsolve([dx, a.y - b.y], t).args[0]
d = linsolve([dx, a.z - b.z], t).args[0]
if len(c.free_symbols) == 1 and len(d.free_symbols) == 1:
return []
e = a.subs(t1, c[0])
if e in self and e in o:
return [e]
else:
return []
return o.intersection(self)
def test_linsolve():
x, y, z, u, v, w = symbols("x, y, z, u, v, w")
x1, x2, x3, x4 = symbols("x1, x2, x3, x4")
# Test for different input forms
M = Matrix([[1, 2, 1, 1, 7], [1, 2, 2, -1, 12], [2, 4, 0, 6, 4]])
system1 = A, b = M[:, :-1], M[:, -1]
Eqns = [x1 + 2 * x2 + x3 + x4 - 7, x1 + 2 * x2 + 2 * x3 - x4 - 12, 2 * x1 + 4 * x2 + 6 * x4 - 4]
sol = FiniteSet((-2 * x2 - 3 * x4 + 2, x2, 2 * x4 + 5, x4))
assert linsolve(M, (x1, x2, x3, x4)) == sol
assert linsolve(Eqns, (x1, x2, x3, x4)) == sol
assert linsolve(system1, (x1, x2, x3, x4)) == sol
# raise ValueError if no symbols are given
raises(ValueError, lambda: linsolve(system1))
# raise ValueError if, A & b is not given as tuple
raises(ValueError, lambda: linsolve(A, b, x1, x2, x3, x4))
# raise ValueError for garbage value
raises(ValueError, lambda: linsolve(Eqns[0], x1, x2, x3, x4))
# Fully symbolic test
a, b, c, d, e, f = symbols("a, b, c, d, e, f")
A = Matrix([[a, b], [c, d]])
B = Matrix([[e], [f]])
system2 = (A, B)
sol = FiniteSet(((-b * f + d * e) / (a * d - b * c), (a * f - c * e) / (a * d - b * c)))
assert linsolve(system2, [x, y]) == sol
# Test for Dummy Symbols issue #9667
x1 = Dummy("x1")
x2 = Dummy("x2")
x3 = Dummy("x3")
x4 = Dummy("x4")
assert linsolve(system1, x1, x2, x3, x4) == FiniteSet((-2 * x2 - 3 * x4 + 2, x2, 2 * x4 + 5, x4))
# No solution
A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
b = Matrix([0, 0, 1])
assert linsolve((A, b), (x, y, z)) == EmptySet()
# Issue #10056
A, B, J1, J2 = symbols("A B J1 J2")
Augmatrix = Matrix(
[[2 * I * J1, 2 * I * J2, -2 / J1], [-2 * I * J2, -2 * I * J1, 2 / J2], [0, 2, 2 * I / (J1 * J2)], [2, 0, 0]]
)
assert linsolve(Augmatrix, A, B) == FiniteSet((0, I / (J1 * J2)))
# Issue #10121 - Assignment of free variables
a, b, c, d, e = symbols("a, b, c, d, e")
Augmatrix = Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]])
assert linsolve(Augmatrix, a, b, c, d, e) == FiniteSet((a, 0, c, 0, e))
def _contains(self, other):
from sympy.matrices import Matrix
from sympy.solvers.solveset import solveset, linsolve
from sympy.utilities.iterables import is_sequence, iterable, cartes
L = self.lamda
if is_sequence(other):
if not is_sequence(L.expr):
return S.false
if len(L.expr) != len(other):
raise ValueError(filldedent('''
Dimensions of other and output of Lambda are different.'''))
elif iterable(other):
raise ValueError(filldedent('''
`other` should be an ordered object like a Tuple.'''))
solns = None
if self._is_multivariate():
if not is_sequence(L.expr):
# exprs -> (numer, denom) and check again
# XXX this is a bad idea -- make the user
# remap self to desired form
return other.as_numer_denom() in self.func(
Lambda(L.variables, L.expr.as_numer_denom()), self.base_set)
eqs = [expr - val for val, expr in zip(other, L.expr)]
variables = L.variables
free = set(variables)
if all(i.is_number for i in list(Matrix(eqs).jacobian(variables))):
solns = list(linsolve([e - val for e, val in
zip(L.expr, other)], variables))
else:
syms = [e.free_symbols & free for e in eqs]
solns = {}
for i, (e, s, v) in enumerate(zip(eqs, syms, other)):
if not s:
if e != v:
return S.false
solns[vars[i]] = [v]
continue
elif len(s) == 1:
sy = s.pop()
sol = solveset(e, sy)
if sol is S.EmptySet:
return S.false
elif isinstance(sol, FiniteSet):
solns[sy] = list(sol)
else:
raise NotImplementedError
else:
raise NotImplementedError
solns = cartes(*[solns[s] for s in variables])
else:
x = L.variables[0]
if isinstance(L.expr, Expr):
# scalar -> scalar mapping
solnsSet = solveset(L.expr - other, x)
if solnsSet.is_FiniteSet:
solns = list(solnsSet)
else:
msgset = solnsSet
else:
# scalar -> vector
for e, o in zip(L.expr, other):
solns = solveset(e - o, x)
if solns is S.EmptySet:
return S.false
for soln in solns:
try:
if soln in self.base_set:
break # check next pair
except TypeError:
if self.base_set.contains(soln.evalf()):
break
else:
return S.false # never broke so there was no True
return S.true
if solns is None:
raise NotImplementedError(filldedent('''
Determining whether %s contains %s has not
been implemented.''' % (msgset, other)))
for soln in solns:
try:
if soln in self.base_set:
return S.true
except TypeError:
return self.base_set.contains(soln.evalf())
return S.false