def test_issue_17247_expression_blowup_32():
M = Matrix([[x + 1, 1 - x, 0, 0], [1 - x, x + 1, 0, x + 1],
[0, 1 - x, x + 1, 0], [0, 0, 0, x + 1]])
assert M.LUsolve(ones(4, 1)) == Matrix([[(x + 1) / (4 * x)],
[(x - 1) / (4 * x)],
[(x + 1) / (4 * x)],
[1 / (x + 1)]])
def solve(A, b):
"""
linear equation system Ax = b -> x
"""
A = Matrix(A)
b = Matrix(b)
x = A.LUsolve(b)
return x
def test_LUsolve():
A = Matrix([[2, 3, 5], [3, 6, 2], [8, 3, 6]])
x = Matrix(3, 1, [3, 7, 5])
b = A * x
soln = A.LUsolve(b)
assert soln == x
A = Matrix([[0, -1, 2], [5, 10, 7], [8, 3, 4]])
x = Matrix(3, 1, [-1, 2, 5])
b = A * x
soln = A.LUsolve(b)
assert soln == x
A = Matrix([[2, 1], [1, 0], [1, 0]]) # issue 14548
b = Matrix([3, 1, 1])
assert A.LUsolve(b) == Matrix([1, 1])
b = Matrix([3, 1, 2]) # inconsistent
raises(ValueError, lambda: A.LUsolve(b))
A = Matrix([[0, -1, 2], [5, 10, 7], [8, 3, 4], [2, 3, 5], [3, 6, 2],
[8, 3, 6]])
x = Matrix([2, 1, -4])
b = A * x
soln = A.LUsolve(b)
assert soln == x
A = Matrix([[0, -1, 2], [5, 10, 7]]) # underdetermined
x = Matrix([-1, 2, 0])
b = A * x
raises(NotImplementedError, lambda: A.LUsolve(b))
A = Matrix(4, 4, lambda i, j: 1 / (i + j + 1) if i != 3 else 0)
b = Matrix.zeros(4, 1)
raises(NonInvertibleMatrixError, lambda: A.LUsolve(b))
def evalMat(Mlist, blist):
M = Matrix(Mlist)
b = Matrix(blist)
c = M.LUsolve(b)
print
print " ================= Case: ", b.transpose()
print
print c
return c
def analytical_coefs_sympy():
import sympy
from sympy.matrices import Matrix
x0, x1, y0, y1, dy0, dy1, x = sympy.symbols('x0 x1 y0 y1 dy0 dy1 x')
#a1,a2,a3,a4 = sympy.symbols('a0 a1 a2 a3')
# ===== setup
bas1 = 1 + 0 * x
bas2 = x**2
bas3 = x**4
bas4 = x**6
# ===== Main
dbas1 = sympy.diff(bas1, x)
dbas2 = sympy.diff(bas2, x)
dbas3 = sympy.diff(bas3, x)
dbas4 = sympy.diff(bas4, x)
#print ( " dbas1: ", dbas1); print ( " dbas2: ", dbas2); print ( " dbas3: ", dbas3); print ( " dbas4: ", dbas4)
A = Matrix([[
bas1.subs(x, x0),
bas2.subs(x, x0),
bas3.subs(x, x0),
bas4.subs(x, x0)
], [
bas1.subs(x, x1),
bas2.subs(x, x1),
bas3.subs(x, x1),
bas4.subs(x, x1)
],
[
dbas1.subs(x, x0),
dbas2.subs(x, x0),
dbas3.subs(x, x0),
dbas4.subs(x, x0)
],
[
dbas1.subs(x, x1),
dbas2.subs(x, x1),
dbas3.subs(x, x1),
dbas4.subs(x, x1)
]])
system = Matrix(4, 1, [y0, y1, dy0, dy1])
print("Solving matrix ...")
coefs = A.LUsolve(system)
print("Symplyfying ...")
for i, coef in enumerate(coefs):
coef = coef.factor()
coef = coef.collect([x0, x1])
#print ( " coef %i: " %i, coef )
coef_subs = sympy.cse(coef)
print(" coef %i: " % i, coef_subs)
def find_linear_recurrence(self, n, d=None, gfvar=None):
r"""
Finds the shortest linear recurrence that satisfies the first n
terms of sequence of order `\leq` ``n/2`` if possible.
If ``d`` is specified, find shortest linear recurrence of order
`\leq` min(d, n/2) if possible.
Returns list of coefficients ``[b(1), b(2), ...]`` corresponding to the
recurrence relation ``x(n) = b(1)*x(n-1) + b(2)*x(n-2) + ...``
Returns ``[]`` if no recurrence is found.
If gfvar is specified, also returns ordinary generating function as a
function of gfvar.
Examples
========
>>> from sympy import sequence, sqrt, oo, lucas
>>> from sympy.abc import n, x, y
>>> sequence(n**2).find_linear_recurrence(10, 2)
[]
>>> sequence(n**2).find_linear_recurrence(10)
[3, -3, 1]
>>> sequence(2**n).find_linear_recurrence(10)
[2]
>>> sequence(23*n**4+91*n**2).find_linear_recurrence(10)
[5, -10, 10, -5, 1]
>>> sequence(sqrt(5)*(((1 + sqrt(5))/2)**n - (-(1 + sqrt(5))/2)**(-n))/5).find_linear_recurrence(10)
[1, 1]
>>> sequence(x+y*(-2)**(-n), (n, 0, oo)).find_linear_recurrence(30)
[1/2, 1/2]
>>> sequence(3*5**n + 12).find_linear_recurrence(20,gfvar=x)
([6, -5], 3*(5 - 21*x)/((x - 1)*(5*x - 1)))
>>> sequence(lucas(n)).find_linear_recurrence(15,gfvar=x)
([1, 1], (x - 2)/(x**2 + x - 1))
"""
from sympy.matrices import Matrix
x = [simplify(expand(t)) for t in self[:n]]
lx = len(x)
if d is None:
r = lx // 2
else:
r = min(d, lx // 2)
coeffs = []
for l in range(1, r + 1):
l2 = 2 * l
mlist = []
for k in range(l):
mlist.append(x[k:k + l])
m = Matrix(mlist)
if m.det() != 0:
y = simplify(m.LUsolve(Matrix(x[l:l2])))
if lx == l2:
coeffs = flatten(y[::-1])
break
mlist = []
for k in range(l, lx - l):
mlist.append(x[k:k + l])
m = Matrix(mlist)
if m * y == Matrix(x[l2:]):
coeffs = flatten(y[::-1])
break
if gfvar is None:
return coeffs
else:
l = len(coeffs)
if l == 0:
return [], None
else:
n, d = x[l - 1] * gfvar**(l - 1), 1 - coeffs[l - 1] * gfvar**l
for i in range(l - 1):
n += x[i] * gfvar**i
for j in range(l - i - 1):
n -= coeffs[i] * x[j] * gfvar**(i + j + 1)
d -= coeffs[i] * gfvar**(i + 1)
return coeffs, simplify(factor(n) / factor(d))
print
print " ==== Solution of 6th order polygon coeficients ==== "
p0, p1, v0, v1, a0, a1, t = symbols('p0 p1 v0 v1 a0 a1 t')
" 1 t t2 t3 t4 t5 "
A = Matrix([
[1, 0, 0, 0, 0, 0], # p0
[1, 1, 1, 1, 1, 1], # p1
[0, 1, 0, 0, 0, 0], # v0
[0, 1, 2, 3, 4, 5], # v1
[0, 0, 2, 0, 0, 0], # a0
[0, 0, 2, 6, 12, 20]
]) # a1
b = Matrix(6, 1, [p0, p1, v0, v1, a0, a1])
CP = A.LUsolve(b)
print "Coefficients "
print CP
print
print " ==== Make A4 A5 zero ==== "
A = Matrix([[1.5, -1.0], [-0.5, 0.5]])
b = Matrix(2, 1, [
-(15 * p0 - 15 * p1 + 8 * v0 + 7 * v1),
-(-6 * p0 + 6 * p1 - 3 * v0 - 3 * v1)
])
c = A.LUsolve(b)
print " Coefficients: "
print c
