mat[i, j] = (1-(-1.)**(i+j-1))*i*j/(i+j-1)
return mat
def derivative_matrix(deg):
'''Matrix of \int_{-1}^{1}(P_i, P_j`) of monomials of degree up to deg.'''
n = deg + 1
mat = np.zeros((n, n))
for i in range(n):
for j in range(1, n):
mat[i, j] = (1 - (-1.)**(i + j))*j/(i + j)
return mat
# -----------------------------------------------------------------------------
if __name__ == '__main__':
from tests import test_M, test_A, test_C
from sympy.plotting import plot
from sympy import Symbol
deg = 5
basis_fs = basis_functions(deg=deg)
x = Symbol('x')
ps = plot(basis_fs[0], (x, -1, 1), show=False)
[ps.append(plot(f, (x, -1, 1), show=False)[0]) for f in basis_fs[1:]]
ps.show()
assert test_M(basis_fs, mass_matrix(deg))
assert test_A(basis_fs, stiffness_matrix(deg))
assert test_C(basis_fs, derivative_matrix(deg))
return common.stiffness_matrix(deg, basis_functions, unroll)
def derivative_matrix(deg, unroll=True):
'''
Matrix of \int_{-1}^{1}(L_i, L_j`) from Chebyshev polynomials of first kind.
'''
return common.derivative_matrix(deg, basis_functions, unroll)
# -----------------------------------------------------------------------------
if __name__ == '__main__':
from tests import test_M, test_A, test_C
from sympy.plotting import plot
from sympy import simplify
deg = 5
basis_fs = basis_functions(deg=deg)
x = Symbol('x')
# Need to check the definition
reference = [1, x, 2*x**2-1, 4*x**3-3*x, 8*x**4-8*x**2+1]
assert not any(map(simplify, (f-f_ for f, f_ in zip(basis_fs, reference))))
ps = plot(basis_fs[0], (x, -1, 1), show=False)
[ps.append(plot(f, (x, -1, 1), show=False)[0]) for f in basis_fs[1:]]
# ps.show()
assert test_M(basis_fs, mass_matrix(deg, True))
assert test_A(basis_fs, stiffness_matrix(deg, True))
assert test_C(basis_fs, derivative_matrix(deg, True))