def test_diff_integrate():
M = Matrix([x, 1]).as_immutable()
assert M.integrate(x) == Matrix([x**2 / 2, x])
assert M.diff(x) == Matrix([1, 0])
assert M.limit(x, 1) == Matrix([1, 1])
assert zeros(2).as_immutable().integrate(x) == zeros(2)
def test_function_return_types():
# Lets ensure that decompositions of immutable matrices remain immutable
# I.e. do MatrixBase methods return the correct class?
X = ImmutableMatrix([[1, 2], [3, 4]])
Y = ImmutableMatrix([[1], [0]])
q, r = X.QRdecomposition()
assert (type(q), type(r)) == (ImmutableMatrix, ImmutableMatrix)
assert type(X.LUsolve(Y)) == ImmutableMatrix
assert type(X.QRsolve(Y)) == ImmutableMatrix
X = ImmutableMatrix([[1, 2], [2, 1]])
assert X.T == X
assert X.is_symmetric
assert type(X.cholesky()) == ImmutableMatrix
L, D = X.LDLdecomposition()
assert (type(L), type(D)) == (ImmutableMatrix, ImmutableMatrix)
assert X.is_diagonalizable()
assert X.berkowitz_det() == -3
assert X.norm(2) == 3
assert type(X.eigenvects()[0][2][0]) == ImmutableMatrix
assert type(zeros(3, 3).as_immutable().nullspace()[0]) == ImmutableMatrix
X = ImmutableMatrix([[1, 0], [2, 1]])
assert type(X.lower_triangular_solve(Y)) == ImmutableMatrix
assert type(X.T.upper_triangular_solve(Y)) == ImmutableMatrix
assert type(X.minorMatrix(0, 0)) == ImmutableMatrix
def test_function_return_types():
# Lets ensure that decompositions of immutable matrices remain immutable
# I.e. do MatrixBase methods return the correct class?
X = ImmutableMatrix([[1, 2], [3, 4]])
Y = ImmutableMatrix([[1], [0]])
q, r = X.QRdecomposition()
assert (type(q), type(r)) == (ImmutableMatrix, ImmutableMatrix)
assert type(X.LUsolve(Y)) == ImmutableMatrix
assert type(X.QRsolve(Y)) == ImmutableMatrix
X = ImmutableMatrix([[1, 2], [2, 1]])
assert X.T == X
assert X.is_symmetric
assert type(X.cholesky()) == ImmutableMatrix
L, D = X.LDLdecomposition()
assert (type(L), type(D)) == (ImmutableMatrix, ImmutableMatrix)
assert X.is_diagonalizable()
assert X.berkowitz_det() == -3
assert X.norm(2) == 3
assert type(X.eigenvects()[0][2][0]) == ImmutableMatrix
assert type(zeros(3, 3).as_immutable().nullspace()[0]) == ImmutableMatrix
X = ImmutableMatrix([[1, 0], [2, 1]])
assert type(X.lower_triangular_solve(Y)) == ImmutableMatrix
assert type(X.T.upper_triangular_solve(Y)) == ImmutableMatrix
assert type(X.minorMatrix(0, 0)) == ImmutableMatrix
def vandermonde(order, dim=1, syms='a b c d'):
"""Computes a Vandermonde matrix of given order and dimension.
Define syms to give beginning strings for temporary variables.
Returns the Matrix, the temporary variables, and the terms for the
polynomials.
"""
syms = syms.split()
n = len(syms)
if n < dim:
new_syms = []
for i in range(dim - n):
j, rem = divmod(i, n)
new_syms.append(syms[rem] + str(j))
syms.extend(new_syms)
terms = []
for i in range(order + 1):
terms.extend(comb_w_rep(dim, i))
rank = len(terms)
V = zeros(rank)
generators = [symbol_gen(syms[i]) for i in range(dim)]
all_syms = []
for i in range(rank):
row_syms = [next(g) for g in generators]
all_syms.append(row_syms)
for j, term in enumerate(terms):
v_entry = 1
for k in term:
v_entry *= row_syms[k]
V[i * rank + j] = v_entry
return V, all_syms, terms
def create_matrix(equations, coeffs):
A = zeros(len(equations))
i = 0
j = 0
for j in range(0, len(coeffs)):
c = coeffs[j]
for i in range(0, len(equations)):
e = equations[i]
d, _ = reduced(e, [c])
A[i, j] = d[0]
return A
def test_rotation_matrix():
N = CoordSysCartesian('N')
A = N.orient_new_axis('A', q1, N.k)
B = A.orient_new_axis('B', q2, A.i)
C = B.orient_new_axis('C', q3, B.j)
D = N.orient_new_axis('D', q4, N.j)
E = N.orient_new_space('E', q1, q2, q3, '123')
F = N.orient_new_quaternion('F', q1, q2, q3, q4)
G = N.orient_new_body('G', q1, q2, q3, '123')
assert N.rotation_matrix(C) == Matrix([
[- sin(q1) * sin(q2) * sin(q3) + cos(q1) * cos(q3), - sin(q1) *
cos(q2), sin(q1) * sin(q2) * cos(q3) + sin(q3) * cos(q1)],
[sin(q1) * cos(q3) + sin(q2) * sin(q3) * cos(q1),
cos(q1) * cos(q2), sin(q1) * sin(q3) - sin(q2) * cos(q1) *
cos(q3)], [- sin(q3) * cos(q2), sin(q2), cos(q2) * cos(q3)]])
test_mat = D.rotation_matrix(C) - Matrix(
[[cos(q1) * cos(q3) * cos(q4) - sin(q3) * (- sin(q4) * cos(q2) +
sin(q1) * sin(q2) * cos(q4)), - sin(q2) * sin(q4) - sin(q1) *
cos(q2) * cos(q4), sin(q3) * cos(q1) * cos(q4) + cos(q3) *
(- sin(q4) * cos(q2) + sin(q1) * sin(q2) * cos(q4))],
[sin(q1) * cos(q3) + sin(q2) * sin(q3) * cos(q1), cos(q1) *
cos(q2), sin(q1) * sin(q3) - sin(q2) * cos(q1) * cos(q3)],
[sin(q4) * cos(q1) * cos(q3) - sin(q3) * (cos(q2) * cos(q4) +
sin(q1) * sin(q2) *
sin(q4)), sin(q2) *
cos(q4) - sin(q1) * sin(q4) * cos(q2), sin(q3) *
sin(q4) * cos(q1) + cos(q3) * (cos(q2) * cos(q4) +
sin(q1) * sin(q2) * sin(q4))]])
assert test_mat.expand() == zeros(3, 3)
assert E.rotation_matrix(N) == Matrix(
[[cos(q2)*cos(q3), sin(q3)*cos(q2), -sin(q2)],
[sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1),
sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), sin(q1)*cos(q2)],
[sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3), -
sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1), cos(q1)*cos(q2)]])
assert F.rotation_matrix(N) == Matrix([[
q1**2 + q2**2 - q3**2 - q4**2,
2*q1*q4 + 2*q2*q3, -2*q1*q3 + 2*q2*q4], [ -2*q1*q4 + 2*q2*q3,
q1**2 - q2**2 + q3**2 - q4**2, 2*q1*q2 + 2*q3*q4],
[2*q1*q3 + 2*q2*q4,
-2*q1*q2 + 2*q3*q4,
q1**2 - q2**2 - q3**2 + q4**2]])
assert G.rotation_matrix(N) == Matrix([[
cos(q2)*cos(q3), sin(q1)*sin(q2)*cos(q3) + sin(q3)*cos(q1),
sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3)], [
-sin(q3)*cos(q2), -sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3),
sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1)], [
sin(q2), -sin(q1)*cos(q2), cos(q1)*cos(q2)]])
pytest.raises(TypeError, lambda: G.rotation_matrix(a))
def main():
t = ReferenceSimplex(2)
fe = Lagrange(2, 2)
u = 0
# compute u = sum_i u_i N_i
us = []
for i in range(0, fe.nbf()):
ui = Symbol("u_%d" % i)
us.append(ui)
u += ui * fe.N[i]
J = zeros(fe.nbf())
for i in range(0, fe.nbf()):
Fi = u * fe.N[i]
print(Fi)
for j in range(0, fe.nbf()):
uj = us[j]
integrands = diff(Fi, uj)
print(integrands)
J[j, i] = t.integrate(integrands)
pprint(J)
def test_sparse_zeros_sparse_eye():
assert SparseMatrix.eye(3) == eye(3, cls=SparseMatrix)
assert len(SparseMatrix.eye(3)._smat) == 3
assert SparseMatrix.zeros(3) == zeros(3, cls=SparseMatrix)
assert len(SparseMatrix.zeros(3)._smat) == 0
def test_eq():
A = SparseMatrix(((1, 2), (3, 4)))
assert A != 1
assert A != zeros(2, 1)