def test_interpolation_2d():
'''
Test function
f(x,y) = x^2 + y^2
f(x,y) = exp(-(x^2+y^2))
'''
Nx = 32
Ny = 32
Mx = 100
My = 200
ix = np.arange(Nx + 1)
x = np.cos(ix * np.pi / Nx)
wx = barycentric_weights_cgl(Nx)
iy = np.arange(Ny + 1)
y = np.cos(iy * np.pi / Ny)
wy = barycentric_weights_cgl(Ny)
X, Y = np.meshgrid(x, y)
#f = X**2 + Y**2
f = np.exp(-(X**2 + Y**2))
plt.plot(X[Ny / 2, :], f[Ny / 2., :], '.')
kx = np.arange(Mx + 1)
u = (2. / Mx) * kx - 1.
ky = np.arange(My + 1)
v = (2. / My) * ky - 1.
U, V = np.meshgrid(u, v)
#f0 = U**2 + V**2
f0 = np.exp(-(U**2 + V**2))
f1 = np.zeros_like(f)
f2 = np.zeros_like(f)
print 'cheb interpolation'
t = time()
f1 = cheb_interpolation_2d(u, v, f)
print 'cheb time: ', time() - t
print 'general interpolation'
t = time()
f2 = interpolation_2d(u, v, f, x, y, wx, wy)
print 'general time: ', time() - t
print f2 - f1
print 'Is general and cheb interpolation the same? ',
print almost_equal(f1, f2)
print 'Interpolation error: ', np.linalg.norm(f1 - f0)
plt.plot(U[My / 2, :], f1[My / 2., :])
plt.show()
def test_clencurt_weights():
N = 257
M = 500
t = time()
for i in xrange(M):
w1 = clencurt_weights(N)
print 'direct sum time: ', time() - t
t = time()
for i in xrange(M):
w2 = clencurt_weights_fft(N)
print 'fft time: ', time() - t
print w1 - w2
print almost_equal(w1, w2)
def testThualComplex(self):
L = np.array([[1, 1, 1, 1, 1, 1], [3 + 1j, 2, 1, 0, 0, 0],
[0, 6, 5, 4 - 1j, 0, 0], [0, 0, 7, 9, 8, 0],
[0, 0, 0, 3, 1, 0], [0, 0, 0, 0, 2, 7]])
f = np.array([0, 3, 5, 8, 7, 4])
u0 = np.linalg.solve(L, f)
p = np.array([3 + 1j, 6, 7, 3, 2])
q = np.array([2, 5, 9, 1, 7])
r = np.array([1, 4 - 1j, 8, 0, 0])
c = np.array([1, 1, 1, 1, 1, 1])
u = solve_tridiag_complex_thual(p, q, r, c, f)
# it will fail if tol < 40
self.assertTrue(almost_equal(u0.real, u.real, 40))
self.assertTrue(almost_equal(u0.imag, u.imag, 40))
def test_interpolation_1d():
'''
Test function
f(x) = |x| + x/2 - x^2
'''
N = 64
M = 1000
ii = np.arange(N + 1)
xx = np.cos(ii * np.pi / N)
w = barycentric_weights_cgl(N)
f = np.abs(xx) + .5 * xx - xx**2
plt.plot(xx, f, '.')
kk = np.arange(M + 1)
yy = (2. / M) * kk - 1.
f1 = np.zeros_like(yy)
f2 = np.zeros_like(yy)
f3 = np.zeros_like(yy)
f4 = np.zeros_like(yy)
print 'Interpolation by point-by-point'
t = time()
for j in xrange(M + 1):
f1[j] = cheb_interpolation_point(yy[j], f)
f2[j] = interpolation_point(yy[j], f, xx, w)
print 'point-by-point time: ', time() - t
print f2 - f1
print almost_equal(f1, f2)
plt.plot(yy, f1)
print 'Interpolation by matrix'
t = time()
f3 = cheb_interpolation_1d(yy, f)
f4 = interpolation_1d(yy, f, xx, w)
print 'matrix time: ', time() - t
print f3 - f1
print f4 - f1
print almost_equal(f3, f1), almost_equal(f4, f1)
plt.plot(yy, f3, 'r')
plt.show()
def test_barycentric_weights():
N = 16
ii = np.arange(N + 1)
print 'Chebyshev Gauss-Lobatto points'
xx = np.cos(ii * np.pi / N)
w = barycentric_weights(xx)
w1 = barycentric_weights_cgl(N)
print w / np.max(w)
print w1
print w / np.max(w) - w1
print almost_equal(w / np.max(w), w1)
print 'Chebyshev Gauss points'
xx = np.cos((2 * ii + 1) * np.pi / (2 * N + 2))
w = barycentric_weights(xx)
w1 = barycentric_weights_cg(N)
print w / np.max(w)
print w1
print w / np.max(w) - w1
print almost_equal(w / np.max(w), w1)
def testThual(self):
L = np.array([[1, 1, 1, 1, 1, 1], [3, 2, 1, 0, 0,
0], [0, 6, 5, 4, 0, 0],
[0, 0, 7, 9, 8, 0], [0, 0, 0, 3, 1, 0],
[0, 0, 0, 0, 2, 7]])
f = np.array([0, 3, 5, 8, 7, 4])
u0 = np.linalg.solve(L, f)
p = np.array([3, 6, 7, 3, 2])
q = np.array([2, 5, 9, 1, 7])
r = np.array([1, 4, 8, 0, 0])
c = np.array([1, 1, 1, 1, 1, 1])
u = solve_tridiag_thual(p, q, r, c, f)
self.assertTrue(almost_equal(u0, u))