def test_covariance_deriv():
# now test full:
# do covariant derivatives
# check that it works, preseving everything
dvec = np.array([10, 3, 2, 3])
p = dvec.shape[0] - 1
alpha = randint(1, 10, (p, p + 1)) * .1
man = ComplexFlag(dvec, alpha=alpha)
n = man.n
d = man.d
Y = man.rand()
slp = np.random.randn(n * d)
aa = np.random.randn(n * d, n * d)
def omg_func(Y):
return (aa @ Y.reshape(-1) + slp).reshape(n, d)
xi = man.randvec(Y)
egrad = omg_func(Y)
ehess = (aa @ xi.reshape(-1)).reshape(n, d)
val1 = man.ehess2rhess(Y, egrad, ehess, xi)
def rgrad_func(W):
return man.proj_g_inv(W, omg_func(W))
if False:
d_xi_rgrad = num_deriv(man, Y, xi, rgrad_func)
rgrad = man.proj_g_inv(Y, egrad)
fourth = man.christoffel_form(Y, xi, rgrad)
val1c = man.proj(Y, d_xi_rgrad) + man.proj_g_inv(Y, fourth)
if False:
first = ehess
a = man.J(Y, man.g_inv(Y, egrad))
rgrad = man.proj_g_inv(Y, egrad)
second = -man.D_g(Y, xi, man.g_inv(Y, egrad))
aout = man.solve_J_g_inv_Jst(Y, a)
third = -man.proj(Y, man.D_g_inv_Jst(Y, xi, aout))
fourth = man.christoffel_form(Y, xi, rgrad)
val1a = man.proj_g_inv(Y, first + second + fourth) + third
d_xi_rgrad = num_deriv(man, Y, xi, rgrad_func)
rgrad = man.proj_g_inv(Y, egrad)
fourth = man.christoffel_form(Y, xi, rgrad)
val1b = man.proj(Y, d_xi_rgrad) + man.proj_g_inv(Y, fourth)
print(check_zero(val1 - val1b))
# nabla_v_xi, dxi, cxxi
# val2a, _, _ = calc_covar_numeric(man, Y, xi, omg_func)
val2, _, _ = calc_covar_numeric(man, Y, xi, rgrad_func)
# val2_p = project(prj, val2)
val2_p = man.proj(Y, val2)
# print(val1)
# print(val2_p)
print(check_zero(val1 - val2_p))
def test_christ_flat():
"""now test that christofel preserve metrics:
on the flat space
d_xi <v M v> = 2 <v M nabla_xi v>
v = proj(W) @ (aa W + b)
"""
dvec = np.array([10, 3, 2, 3])
p = dvec.shape[0] - 1
alpha = randint(1, 10, (p, p + 1)) * .1
man = ComplexFlag(dvec, alpha=alpha)
Y = man.rand()
n = man.n
d = man.d
xi = man.randvec(Y)
aa = crandn(n * d, n * d)
bb = crandn(n * d)
def v_func_flat(Y):
return (aa @ Y.reshape(-1) + bb).reshape(n, d)
vv = v_func_flat(Y)
dlt = 1e-7
Ynew = Y + dlt * xi
vnew = v_func_flat(Ynew)
val = man.inner(Y, vv, vv)
valnew = man.inner(Ynew, vnew, vnew)
d1 = (valnew - val) / dlt
dv = (vnew - vv) / dlt
nabla_xi_v = dv + man.g_inv(Y, man.christoffel_form(Y, xi, vv))
d2 = man.inner(Y, vv, nabla_xi_v)
print(d1)
print(2 * d2)
def test_all_projections():
dvec = np.array([10, 3, 2, 3])
p = dvec.shape[0] - 1
alpha = randint(1, 10, (p, p + 1)) * .1
man = ComplexFlag(dvec, alpha=alpha)
Y = man.rand()
U = man._rand_ambient()
Upr = man.proj(Y, U)
test_inner(man, Y)
test_J(man, Y)
# now check metric, Jst etc
# check Jst: vectorize the operator J then compare Jst with jmat.T.conjugate()
jmat = make_j_mat(man, Y)
test_Jst(man, Y, jmat)
ginv_mat = make_g_inv_mat(man, Y)
# test g_inv_Jst
for ii in range(10):
a = man._rand_range_J()
avec = man._vec_range_J(a)
jtout = man._unvec(ginv_mat @ jmat.T @ avec)
jtout2 = man.g_inv_Jst(Y, a)
diff = check_zero(jtout - jtout2)
print(diff)
# test projection
test_projection(man, Y)
for i in range(20):
Uran = man._rand_ambient()
Upr = man.proj(Y, man.g_inv(Y, Uran))
Upr2 = man.proj_g_inv(Y, Uran)
print(check_zero(Upr - Upr2))
for ii in range(10):
a = man._rand_range_J()
xi = man._rand_ambient()
jtout2 = man.Jst(Y, a)
dlt = 1e-7
Ynew = Y + dlt * xi
jtout2a = man.Jst(Ynew, a)
d1 = (jtout2a - jtout2) / dlt
d2 = man.D_Jst(Y, xi, a)
print(check_zero(d2 - d1))
for ii in range(10):
Y = man.rand()
eta = man._rand_ambient()
xi = man.randvec(Y)
a1 = man.J(Y, eta)
dlt = 1e-7
Ynew = Y + dlt * xi
a2 = man.J(Ynew, eta)
d1 = (man._vec_range_J(a2) - man._vec_range_J(a1)) / dlt
d2 = man._vec_range_J(man.D_J(Y, xi, eta))
print(check_zero(d2 - d1))
for ii in range(10):
a = man._rand_range_J()
xi = man._rand_ambient()
jtout2 = man.g_inv_Jst(Y, a)
dlt = 1e-7
Ynew = Y + dlt * xi
jtout2a = man.g_inv_Jst(Ynew, a)
d1 = (jtout2a - jtout2) / dlt
d2 = man.D_g_inv_Jst(Y, xi, a)
print(check_zero(d2 - d1))
for ii in range(10):
arand = man._rand_range_J()
a2 = man.solve_J_g_inv_Jst(Y, arand)
a1 = man.J(Y, man.g_inv_Jst(Y, a2))
print(check_zero(man._vec_range_J(a1) - man._vec_range_J(arand)))
# derives
for ii in range(10):
Y1 = man.rand()
xi = man.randvec(Y1)
omg1 = man._rand_ambient()
omg2 = man._rand_ambient()
dlt = 1e-7
Y2 = Y1 + dlt * xi
p1 = man.inner(Y1, omg1, omg2)
p2 = man.inner(Y2, omg1, omg2)
der1 = (p2 - p1) / dlt
der2 = man.base_inner_ambient(man.D_g(Y1, xi, omg2), omg1)
print(check_zero(der1 - der2))
# cross term for christofel
for i in range(10):
Y1 = man.rand()
xi = man.randvec(Y1)
omg1 = man._rand_ambient()
omg2 = man._rand_ambient()
dr1 = man.D_g(Y1, xi, omg1)
x12 = man.contract_D_g(Y1, omg1, omg2)
p1 = trace(dr1 @ omg2.T.conjugate()).real
p2 = trace(x12 @ xi.T.conjugate()).real
print(p1, p2, p1 - p2)
# now test christofel:
# two things: symmetric on vector fields
# and christofel relation
# in the case metric
for i in range(10):
Y1 = man.rand()
xi = man.randvec(Y1)
eta1 = man.randvec(Y1)
eta2 = man.randvec(Y1)
p1 = man.proj_g_inv(Y1, man.christoffel_form(Y1, xi, eta1))
p2 = man.proj_g_inv(Y1, man.christoffel_form(Y1, eta1, xi))
print(check_zero(p1 - p2))
v1 = man.base_inner_ambient(man.christoffel_form(Y1, eta1, eta2), xi)
v2 = man.base_inner_ambient(man.D_g(Y1, eta1, eta2), xi)
v3 = man.base_inner_ambient(man.D_g(Y1, eta2, eta1), xi)
v4 = man.base_inner_ambient(man.D_g(Y1, xi, eta1), eta2)
print(v1, 0.5 * (v2 + v3 - v4), v1 - 0.5 * (v2 + v3 - v4))
"""