def reduce_po_lat(l_csl_p, l_p_po, tol):
l_p_po = np.array(l_p_po, dtype='double')
l_csl_po = l_p_po.dot(l_csl_p)
lInt_csl_po, m1 = iman.int_approx(l_csl_po, tol)
inp_args = {}
inp_args['mat'] = lInt_csl_po
lllInt_csl_po = call_sage_math('./compute_LLL.py', inp_args)
lllInt_csl_po = Matrix(lllInt_csl_po)
Sz = lllInt_csl_po.shape
if Sz[0] == Sz[1]:
if lllInt_csl_po.det() < 0:
if Sz[0] == 3:
M4 = Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
lllInt_csl_po = lllInt_csl_po * M4
if Sz[0] == 2:
M4 = Matrix([[0, 1], [1, 0]])
lllInt_csl_po = lllInt_csl_po * M4
Tmat = ((Matrix(lInt_csl_po)).inv()) * (lllInt_csl_po)
else:
A1 = (np.array(lllInt_csl_po, dtype='int64'))
A2 = (np.array(lInt_csl_po, dtype='int64'))
# A1inv = np.linalg.pinv(A1)
A2inv = np.linalg.pinv(A2)
Tmat = Matrix(np.dot(A2inv, A1))
cond1 = iman.check_int_mat(Tmat, 1e-12)
if cond1:
Tmat1 = Matrix(
np.array(np.around(np.array(Tmat, dtype='double')), dtype='int64'))
return Tmat1
else:
raise Exception("Tmat is not an integer matrix.")
def _find_plane(p: tuple, v1: np.ndarray, v2: np.ndarray) -> Expr:
""" Helper method tp find plane that fits point p and two non-collinear vectors """
x, y, z = symbols('x, y, z')
matrix = Matrix([[x - p[0], y - p[1], z - p[2]], [v1[0], v1[1], v1[2]],
[v2[0], v2[1], v2[2]]])
plane = matrix.det()
assert plane != Zero, 'vector1 and vector2 must be non-collinear'
return plane
def cv9():
m_a = Matrix([[a, 1, 1, 1], [1, a, 1, 1], [1, 1, a, 1], [1, 1, 1, a]])
b = Matrix([1, 1, 1, 1])
det_a = m_a.det()
p_s = linsolve((m_a, b), [p_1, p_2, p_3, p_4])
print("\\mA={} \\\\".format(latex(m_a)))
print("det(A)={}={} \\\\".format(latex(det_a), latex(factor(det_a))))
print("p={}".format(latex(p_s)))
def _jacobian(self, x: Symbol, y: Symbol, p: tuple) -> float:
subs_dict = {self._x: p[0], self._y: p[1], self._z: p[2]}
matrix = Matrix([[
diff(self._f, x).subs(subs_dict),
diff(self._f, y).subs(subs_dict)
], [
diff(self._g, x).subs(subs_dict),
diff(self._g, y).subs(subs_dict)
]])
if self._logging:
print(
f'Jacobian d(f, g) / d({x}, {y}) at point {p} is |{matrix}| = {matrix.det()}'
)
return matrix.det()
def sylvester_det(p4, p2):
p4degree = len(p4) - 1
p2degree = len(p2) - 1
smatsize = p4degree + p2degree
nzeros4 = (smatsize - len(p4))
nzeroes2 = (smatsize - len(p2))
smatlist = []
for i in range(nzeros4 + 1):
smatlist.append([0]*i + p4 + [0]*(nzeros4 - i))
for i in range(nzeroes2 + 1):
smatlist.append([0]*i + p2 + [0]*(nzeroes2 - i))
smat = Matrix(smatlist)
det = smat.det(method='berkowitz')
det = sympy.expand(det)
return det
def handle(msg):
chat_id = msg["chat"]["id"]
if not "text" in msg:
bot.sendMessage(chat_id, "I only understand text at the moment :(")
return
text = msg["text"]
print "%s: %s" % (msg["from"]["first_name"], text)
if text == "/start":
bot.sendMessage(chat_id, start_message)
return
matrix = parseMatrix(text)
if matrix is None:
bot.sendMessage(chat_id,
"Can't parse matrix, please use a different format!")
return
if not check_dim(matrix):
bot.sendMessage(
chat_id, "Different row length detected, please check your input!")
return
if len(matrix) < 2 or len(matrix[0]) < 2:
bot.sendMessage(chat_id, "Matrix have to be at least 2x2!")
return
if len(matrix) != len(matrix[0]):
bot.sendMessage(chat_id,
"The two dimensions of the matrix must be equal!")
return
# calculate determinante
m = Matrix(matrix)
det = m.det()
if len(det.free_symbols) > 0:
bot.sendMessage(chat_id, 'determinante: %s' % det)
else:
bot.sendMessage(chat_id, 'determinante: %s = %s' % (det, det.evalf(4)))
def alexander_polynomial(braid, t=t, modulus=None, seifert=False):
""" Calculate Alexander Polynomial."""
if seifert:
M = Matrix(seifert_matrix(braid).tolist())
if modulus is None:
apoly = M - t*M.transpose()
else:
apoly = (M - t*M.transpose()) % modulus
apoly = apoly.det()
else:
deleted = False
h = generators(braid)
M = alexander_matrix(braid, t, modulus)
for i in range(len(braid)):
if h[i] == 0:
deleted = True
M.row_del(i)
M.col_del(i)
break
if not deleted:
M.row_del(-1)
M.col_del(-1)
apoly = M.det()
return _normalize_apoly(apoly)
def test_legacy_det():
# Minimal support for legacy keys for 'method' in det()
# Partially copied from test_determinant()
M = Matrix(((3, -2, 0, 5), (-2, 1, -2, 2), (0, -2, 5, 0), (5, 0, 3, 4)))
assert M.det(method="bareis") == -289
assert M.det(method="det_lu") == -289
assert M.det(method="det_LU") == -289
M = Matrix(((3, 2, 0, 0, 0), (0, 3, 2, 0, 0), (0, 0, 3, 2, 0),
(0, 0, 0, 3, 2), (2, 0, 0, 0, 3)))
assert M.det(method="bareis") == 275
assert M.det(method="det_lu") == 275
assert M.det(method="Bareis") == 275
M = Matrix(((1, 0, 1, 2, 12), (2, 0, 1, 1, 4), (2, 1, 1, -1, 3),
(3, 2, -1, 1, 8), (1, 1, 1, 0, 6)))
assert M.det(method="bareis") == -55
assert M.det(method="det_lu") == -55
assert M.det(method="BAREISS") == -55
M = Matrix(((3, 0, 0, 0), (-2, 1, 0, 0), (0, -2, 5, 0), (5, 0, 3, 4)))
assert M.det(method="bareiss") == 60
assert M.det(method="berkowitz") == 60
assert M.det(method="lu") == 60
M = Matrix(((1, 0, 0, 0), (5, 0, 0, 0), (9, 10, 11, 0), (13, 14, 15, 16)))
assert M.det(method="bareiss") == 0
assert M.det(method="berkowitz") == 0
assert M.det(method="lu") == 0
M = Matrix(((3, 2, 0, 0, 0), (0, 3, 2, 0, 0), (0, 0, 3, 2, 0),
(0, 0, 0, 3, 2), (0, 0, 0, 0, 3)))
assert M.det(method="bareiss") == 243
assert M.det(method="berkowitz") == 243
assert M.det(method="lu") == 243
M = Matrix(((-5, 2, 3, 4, 5), (1, -4, 3, 4, 5), (1, 2, -3, 4, 5),
(1, 2, 3, -2, 5), (1, 2, 3, 4, -1)))
assert M.det(method="bareis") == 11664
assert M.det(method="det_lu") == 11664
assert M.det(method="BERKOWITZ") == 11664
M = Matrix(((2, 7, -1, 3, 2), (0, 0, 1, 0, 1), (-2, 0, 7, 0, 2),
(-3, -2, 4, 5, 3), (1, 0, 0, 0, 1)))
assert M.det(method="bareis") == 123
assert M.det(method="det_lu") == 123
assert M.det(method="LU") == 123
def find_linear_recurrence(self,n,d=None,gfvar=None):
"""
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*(-21*x + 5)/((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 == 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 == 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))
tlr = taylor_to_var(r_star, rho, alpha_pi, alpha_z, sigma_R, tau_1, tau_2)
vcx = [a13, a23, a33, b13, b23, b33, b43]
eq = []
for i in range(7):
eq.append(vcx[i] - tlr[i])
inv_taylor = sympy.solve(
eq, [r_star, rho, alpha_pi, alpha_z, sigma_R, tau_1, tau_2])[0]
g_inv = sympy.lambdify(vcx, inv_taylor)
from sympy.matrices import Matrix
from scipy.stats import multivariate_normal
J = Matrix(7, 7, lambda i, j: sympy.diff(inv_taylor[i], vcx[j]))
Jdet = sympy.lambdify(vcx, J.det())
from collections import OrderedDict
class TaylorRulePrior(SimsZhaSVARPrior):
def __init__(self, taylor_args, *args, **kwargs):
# Initialize the baseclass
super(TaylorRulePrior, self).__init__(*args, **kwargs)
# r_loc = ordering[0]
# pi_loc = ordering[1]
# y_loc = ordering[2]
npara = self.n * (self.n +
class Context(object):
"""some context where variables exists"""
def __init__(self, size):
self.numberOfVariables = size
self.x = s.symbols('x:' + str(size))
self.fn = None
self.hessian = None
self.detHessian = None
def setFn(self, expression):
self.fn = expression
def getHessian(self):
expr = self.fn
hessianRows = []
for idx in range(0, self.numberOfVariables):
hessianRow = []
diffWrt = s.diff(expr, self.x[idx])
for idx2ndDeriv in range(0, self.numberOfVariables):
hessianRow.append(s.diff(diffWrt, self.x[idx2ndDeriv]))
hessianRows.append(hessianRow)
self.hessian = Matrix(hessianRows)
return self.hessian
# compute the determinate of the hessian at point [x1 .. xn]
def detHessianAt(self, subs):
if len(subs) != self.numberOfVariables:
raise Exception('airity mismatch: detHessianAt')
if self.hessian == None:
self.hessian = self.getHessian()
if self.detHessian == None:
self.detHessian = self.hessian.det()
detHessian = self.detHessian
for idx in range(0, self.numberOfVariables):
detHessian = detHessian.subs(self.x[idx], subs[idx])
return detHessian
def secondPartialTest(self, subs):
dValue = self.detHessianAt(subs).evalf()
if (dValue < 0):
return "Saddle Point at f(" + str(subs) + ")"
if (dValue == 0):
return "Inconclusive at f(" + str(subs) + ")"
# get the function at the top left of the hession
secondDerivAtSub = self.hessian[0, 0]
value = secondDerivAtSub
for idx in range(0, self.numberOfVariables):
value = value.subs(self.x[idx], subs[idx])
value = value.evalf()
if value > 0:
return "Relative Minimum at f(" + str(subs) + ")"
if value < 0:
return "Relative Maximum at f(" + str(subs) + ")"
class Context(object):
"""some context where variables exists"""
def __init__(self, size):
self.numberOfVariables = size
self.x = s.symbols('x:'+str(size))
self.fn = None
self.hessian = None
self.detHessian = None
def setFn(self, expression):
self.fn = expression
def getHessian(self):
expr = self.fn
hessianRows = []
for idx in range(0, self.numberOfVariables):
hessianRow = []
diffWrt = s.diff(expr, self.x[idx])
for idx2ndDeriv in range(0, self.numberOfVariables):
hessianRow.append(s.diff(diffWrt, self.x[idx2ndDeriv]))
hessianRows.append(hessianRow)
self.hessian = Matrix(hessianRows)
return self.hessian
# compute the determinate of the hessian at point [x1 .. xn]
def detHessianAt(self, subs):
if len(subs) != self.numberOfVariables:
raise Exception('airity mismatch: detHessianAt')
if self.hessian == None:
self.hessian = self.getHessian()
if self.detHessian == None:
self.detHessian = self.hessian.det()
detHessian = self.detHessian
for idx in range(0, self.numberOfVariables):
detHessian = detHessian.subs(self.x[idx], subs[idx]);
return detHessian
def secondPartialTest(self, subs):
dValue = self.detHessianAt(subs).evalf()
if(dValue < 0):
return "Saddle Point at f(" + str(subs) + ")"
if(dValue == 0):
return "Inconclusive at f(" + str(subs) + ")"
# get the function at the top left of the hession
secondDerivAtSub = self.hessian[0,0]
value = secondDerivAtSub
for idx in range(0, self.numberOfVariables):
value = value.subs(self.x[idx], subs[idx]);
value = value.evalf()
if value > 0:
return "Relative Minimum at f(" + str(subs) + ")"
if value < 0:
return "Relative Maximum at f(" + str(subs) + ")"
def test_det():
a = Matrix(2, 3, [1, 2, 3, 4, 5, 6])
raises(NonSquareMatrixError, lambda: a.det())
z = zeros_Determinant(2)
ey = eye_Determinant(2)
assert z.det() == 0
assert ey.det() == 1
x = Symbol('x')
a = Matrix(0, 0, [])
b = Matrix(1, 1, [5])
c = Matrix(2, 2, [1, 2, 3, 4])
d = Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 8])
e = Matrix(4, 4, [x, 1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 14])
from sympy.abc import i, j, k, l, m, n
f = Matrix(3, 3, [i, l, m, 0, j, n, 0, 0, k])
g = Matrix(3, 3, [i, 0, 0, l, j, 0, m, n, k])
h = Matrix(3, 3, [x**3, 0, 0, i, x**-1, 0, j, k, x**-2])
# the method keyword for `det` doesn't kick in until 4x4 matrices,
# so there is no need to test all methods on smaller ones
assert a.det() == 1
assert b.det() == 5
assert c.det() == -2
assert d.det() == 3
assert e.det() == 4 * x - 24
assert e.det(method="domain-ge") == 4 * x - 24
assert e.det(method='bareiss') == 4 * x - 24
assert e.det(method='berkowitz') == 4 * x - 24
assert f.det() == i * j * k
assert g.det() == i * j * k
assert h.det() == 1
raises(ValueError, lambda: e.det(iszerofunc="test"))
def test_determinant():
M = Matrix()
assert M.det() == 1
# Evaluating these directly because they are never reached via M.det()
assert M._eval_det_bareiss() == 1
assert M._eval_det_berkowitz() == 1
assert M._eval_det_lu() == 1
M = Matrix([[0]])
assert M.det() == 0
assert M._eval_det_bareiss() == 0
assert M._eval_det_berkowitz() == 0
assert M._eval_det_lu() == 0
M = Matrix([[5]])
assert M.det() == 5
assert M._eval_det_bareiss() == 5
assert M._eval_det_berkowitz() == 5
assert M._eval_det_lu() == 5
M = Matrix(((-3, 2), (8, -5)))
assert M.det(method="domain-ge") == -1
assert M.det(method="bareiss") == -1
assert M.det(method="berkowitz") == -1
assert M.det(method="lu") == -1
M = Matrix(((x, 1), (y, 2 * y)))
assert M.det(method="domain-ge") == 2 * x * y - y
assert M.det(method="bareiss") == 2 * x * y - y
assert M.det(method="berkowitz") == 2 * x * y - y
assert M.det(method="lu") == 2 * x * y - y
M = Matrix(((1, 1, 1), (1, 2, 3), (1, 3, 6)))
assert M.det(method="domain-ge") == 1
assert M.det(method="bareiss") == 1
assert M.det(method="berkowitz") == 1
assert M.det(method="lu") == 1
M = Matrix(((3, -2, 0, 5), (-2, 1, -2, 2), (0, -2, 5, 0), (5, 0, 3, 4)))
assert M.det(method="domain-ge") == -289
assert M.det(method="bareiss") == -289
assert M.det(method="berkowitz") == -289
assert M.det(method="lu") == -289
M = Matrix(((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16)))
assert M.det(method="domain-ge") == 0
assert M.det(method="bareiss") == 0
assert M.det(method="berkowitz") == 0
assert M.det(method="lu") == 0
M = Matrix(((3, 2, 0, 0, 0), (0, 3, 2, 0, 0), (0, 0, 3, 2, 0),
(0, 0, 0, 3, 2), (2, 0, 0, 0, 3)))
assert M.det(method="domain-ge") == 275
assert M.det(method="bareiss") == 275
assert M.det(method="berkowitz") == 275
assert M.det(method="lu") == 275
M = Matrix(((3, 0, 0, 0), (-2, 1, 0, 0), (0, -2, 5, 0), (5, 0, 3, 4)))
assert M.det(method="domain-ge") == 60
assert M.det(method="bareiss") == 60
assert M.det(method="berkowitz") == 60
assert M.det(method="lu") == 60
M = Matrix(((1, 0, 0, 0), (5, 0, 0, 0), (9, 10, 11, 0), (13, 14, 15, 16)))
assert M.det(method="domain-ge") == 0
assert M.det(method="bareiss") == 0
assert M.det(method="berkowitz") == 0
assert M.det(method="lu") == 0
M = Matrix(((3, 2, 0, 0, 0), (0, 3, 2, 0, 0), (0, 0, 3, 2, 0),
(0, 0, 0, 3, 2), (0, 0, 0, 0, 3)))
assert M.det(method="domain-ge") == 243
assert M.det(method="bareiss") == 243
assert M.det(method="berkowitz") == 243
assert M.det(method="lu") == 243
M = Matrix(((1, 0, 1, 2, 12), (2, 0, 1, 1, 4), (2, 1, 1, -1, 3),
(3, 2, -1, 1, 8), (1, 1, 1, 0, 6)))
assert M.det(method="domain-ge") == -55
assert M.det(method="bareiss") == -55
assert M.det(method="berkowitz") == -55
assert M.det(method="lu") == -55
M = Matrix(((-5, 2, 3, 4, 5), (1, -4, 3, 4, 5), (1, 2, -3, 4, 5),
(1, 2, 3, -2, 5), (1, 2, 3, 4, -1)))
assert M.det(method="domain-ge") == 11664
assert M.det(method="bareiss") == 11664
assert M.det(method="berkowitz") == 11664
assert M.det(method="lu") == 11664
M = Matrix(((2, 7, -1, 3, 2), (0, 0, 1, 0, 1), (-2, 0, 7, 0, 2),
(-3, -2, 4, 5, 3), (1, 0, 0, 0, 1)))
assert M.det(method="domain-ge") == 123
assert M.det(method="bareiss") == 123
assert M.det(method="berkowitz") == 123
assert M.det(method="lu") == 123
M = Matrix(((x, y, z), (1, 0, 0), (y, z, x)))
assert M.det(method="domain-ge") == z**2 - x * y
assert M.det(method="bareiss") == z**2 - x * y
assert M.det(method="berkowitz") == z**2 - x * y
assert M.det(method="lu") == z**2 - x * y
# issue 13835
a = symbols('a')
M = lambda n: Matrix([[i + a * j for i in range(n)] for j in range(n)])
assert M(5).det() == 0
assert M(6).det() == 0
assert M(7).det() == 0
def test_determinant():
for M in [Matrix(), Matrix([[1]])]:
assert (M.det() == M._eval_det_bareiss() == M._eval_det_berkowitz() ==
M._eval_det_lu() == 1)
M = Matrix(((-3, 2), (8, -5)))
assert M.det(method="bareiss") == -1
assert M.det(method="berkowitz") == -1
assert M.det(method="lu") == -1
M = Matrix(((x, 1), (y, 2 * y)))
assert M.det(method="bareiss") == 2 * x * y - y
assert M.det(method="berkowitz") == 2 * x * y - y
assert M.det(method="lu") == 2 * x * y - y
M = Matrix(((1, 1, 1), (1, 2, 3), (1, 3, 6)))
assert M.det(method="bareiss") == 1
assert M.det(method="berkowitz") == 1
assert M.det(method="lu") == 1
M = Matrix(((3, -2, 0, 5), (-2, 1, -2, 2), (0, -2, 5, 0), (5, 0, 3, 4)))
assert M.det(method="bareiss") == -289
assert M.det(method="berkowitz") == -289
assert M.det(method="lu") == -289
M = Matrix(((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16)))
assert M.det(method="bareiss") == 0
assert M.det(method="berkowitz") == 0
assert M.det(method="lu") == 0
M = Matrix(((3, 2, 0, 0, 0), (0, 3, 2, 0, 0), (0, 0, 3, 2, 0),
(0, 0, 0, 3, 2), (2, 0, 0, 0, 3)))
assert M.det(method="bareiss") == 275
assert M.det(method="berkowitz") == 275
assert M.det(method="lu") == 275
M = Matrix(((3, 0, 0, 0), (-2, 1, 0, 0), (0, -2, 5, 0), (5, 0, 3, 4)))
assert M.det(method="bareiss") == 60
assert M.det(method="berkowitz") == 60
assert M.det(method="lu") == 60
M = Matrix(((1, 0, 0, 0), (5, 0, 0, 0), (9, 10, 11, 0), (13, 14, 15, 16)))
assert M.det(method="bareiss") == 0
assert M.det(method="berkowitz") == 0
assert M.det(method="lu") == 0
M = Matrix(((3, 2, 0, 0, 0), (0, 3, 2, 0, 0), (0, 0, 3, 2, 0),
(0, 0, 0, 3, 2), (0, 0, 0, 0, 3)))
assert M.det(method="bareiss") == 243
assert M.det(method="berkowitz") == 243
assert M.det(method="lu") == 243
M = Matrix(((1, 0, 1, 2, 12), (2, 0, 1, 1, 4), (2, 1, 1, -1, 3),
(3, 2, -1, 1, 8), (1, 1, 1, 0, 6)))
assert M.det(method="bareiss") == -55
assert M.det(method="berkowitz") == -55
assert M.det(method="lu") == -55
M = Matrix(((-5, 2, 3, 4, 5), (1, -4, 3, 4, 5), (1, 2, -3, 4, 5),
(1, 2, 3, -2, 5), (1, 2, 3, 4, -1)))
assert M.det(method="bareiss") == 11664
assert M.det(method="berkowitz") == 11664
assert M.det(method="lu") == 11664
M = Matrix(((2, 7, -1, 3, 2), (0, 0, 1, 0, 1), (-2, 0, 7, 0, 2),
(-3, -2, 4, 5, 3), (1, 0, 0, 0, 1)))
assert M.det(method="bareiss") == 123
assert M.det(method="berkowitz") == 123
assert M.det(method="lu") == 123
M = Matrix(((x, y, z), (1, 0, 0), (y, z, x)))
assert M.det(method="bareiss") == z**2 - x * y
assert M.det(method="berkowitz") == z**2 - x * y
assert M.det(method="lu") == z**2 - x * y
# issue 13835
a = symbols('a')
M = lambda n: Matrix([[i + a * j for i in range(n)] for j in range(n)])
assert M(5).det() == 0
assert M(6).det() == 0
assert M(7).det() == 0
# Create the identity matrix
eye(3)
# Create an empty matrix
zeros(3,2)
# Create a matrix with ones
ones(4,2)
# Creates a square matrix with only the diagonal elements filled in
diag(2,2,2,2,2)
# To compute the determinant, use det
M = Matrix([[1, 0, 1], [2, -1, 3], [4, 3, 2]])
M.det()
# Conversions between radians to degrees
rtd = 180./np.pi # radians to degrees
dtr = np.pi/180. # degrees to radians
# Now we create the rotation matrices for elementary rotations about the X, Y, and Z axes, respectively.
# The about Z rotation matrix is derived on page 45
R_x = Matrix([[ 1, 0, 0],
[ 0, cos(q1), -sin(q1)],
[ 0, sin(q1), cos(q1)]])
# Rotation about X axis
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))
# In[59]: C * A # El **determinante** de una matriz podemos calcularlo mediante la función `det`: # In[60]: det(B) # O bien, mediante el propio método `det`: # In[61]: B.det() # La **matriz inversa** podemos calcularla utilizando el método `inv`: # In[62]: A.inv() # In[63]: A.inv() * A # La **transpuesta** de una matriz se puede obtener accediendo al atributo `T` # In[64]: