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)
"""
alpha = randint(1, 10, 2) * .1
beta = randint(1, 10, 2)[0] * .1
n = 20
d = 3
man = ComplexPositiveSemidefinite(n, d, alpha=alpha, beta=beta)
S = man.rand()
xi = man.randvec(S)
xi = man.randvec(S)
aa = crandn(n * d, n * d)
bb = crandn(n * d)
cc = crandn(d * d, d * d)
dd = hsym(crandn(d, d))
def v_func_flat(S):
# a function from the manifold
# to ambient
csp = hsym((cc @ S.P.reshape(-1)).reshape(d, d))
return psd_ambient((aa @ S.Y.reshape(-1) + bb).reshape(n, d), csp + dd)
vv = v_func_flat(S)
dlt = 1e-7
Snew = psd_point(S.Y + dlt * xi.tY, S.P + dlt * xi.tP)
vnew = v_func_flat(Snew)
val = man.inner_product_amb(S, vv)
valnew = man.inner_product_amb(Snew, vnew)
d1 = (valnew - val) / dlt
dv = (vnew - vv).scalar_mul(1 / dlt)
nabla_xi_v = dv + man.g_inv(S, man.christoffel_form(S, xi, vv))
nabla_xi_va = dv + man.g_inv(
S,
super(ComplexPositiveSemidefinite, man).christoffel_form(S, xi, vv))
print(check_zero(man._vec(nabla_xi_v) - man._vec(nabla_xi_va)))
d2 = man.inner(S, vv, nabla_xi_v)
print(d1)
print(2 * d2)
def test_covariance_deriv():
# now test full:
# do covariant derivatives
# check that it works, preseving everything
n, d = (5, 3)
alpha = randint(1, 10, 2) * .1
beta = randint(1, 10, 2)[0] * .01
man = ComplexPositiveSemidefinite(n, d, alpha=alpha, beta=beta)
S = man.rand()
aa = crandn(n * d, n * d)
cc = crandn(d * d, d * d)
icpt = man._rand_ambient()
def omg_func(S):
csp = hsym((cc @ S.P.reshape(-1)).reshape(d, d))
return psd_ambient(
(aa @ S.Y.reshape(-1) + icpt.tY.reshape(-1)).reshape(n, d),
csp + icpt.tP)
xi = man.randvec(S)
egrad = omg_func(S)
ecsp = hsym((cc @ xi.tP.reshape(-1)).reshape(d, d))
ehess = psd_ambient((aa @ xi.tY.reshape(-1)).reshape(n, d), ecsp)
val1 = man.ehess2rhess(S, egrad, ehess, xi)
def rgrad_func(W):
return man.proj_g_inv(W, omg_func(W))
if False:
first = ehess
a = man.J_g_inv(S, egrad)
rgrad = man.proj_g_inv(S, egrad)
second = man.D_g(S, xi, man.g_inv(S, egrad)).scalar_mul(-1)
aout = man.solve_J_g_inv_Jst(S, a)
third = man.proj(S, man.D_g_inv_Jst(S, xi, aout)).scalar_mul(-1)
fourth = man.christoffel_form(S, xi, rgrad)
val1a1 = man.proj_g_inv(S, first + second + fourth) + third
print(check_zero(man._vec(val1 - val1a1)))
elif True:
d_xi_rgrad = num_deriv_amb(man, S, xi, rgrad_func)
rgrad = man.proj_g_inv(S, egrad)
fourth = man.christoffel_form(S, xi, rgrad)
val1a = man.proj(S, d_xi_rgrad) + man.proj_g_inv(S, fourth)
print(check_zero(man._vec(val1 - val1a)))
# nabla_v_xi, dxi, cxxi
val2a, _, _ = calc_covar_numeric(man, S, xi, omg_func)
val2, _, _ = calc_covar_numeric(man, S, xi, rgrad_func)
# val2_p = project(prj, val2)
val2_p = man.proj(S, val2)
# print(val1)
# print(val2_p)
print(check_zero(man._vec(val1) - man._vec(val2_p)))
if True:
H = xi
valrangeA_ = ehess + man.g(S, man.D_proj(
S, H, man.g_inv(S, egrad))) - man.D_g(
S, H, man.g_inv(S, egrad)) +\
man.christoffel_form(S, H, man.proj_g_inv(S, egrad))
valrangeB = man.proj_g_inv(S, valrangeA_)
valrange = man.ehess2rhess_alt(S, egrad, ehess, xi)
print(check_zero(man._vec(valrange) - man._vec(val2_p)))
print(check_zero(man._vec(valrange) - man._vec(val1)))
print(check_zero(man._vec(valrange) - man._vec(valrangeB)))
def test_all_projections():
alpha = randint(1, 10, 2) * .1
beta = randint(1, 10, 1)[0] * .02
n = 5
d = 3
man = ComplexPositiveSemidefinite(n, d, alpha=alpha, beta=beta)
print(man)
S = man.rand()
test_inner(man, S)
test_J(man, S)
# now check metric, Jst etc
# check Jst: vectorize the operator J then compare Jst with jmat.T
jmat = make_j_mat(man, S)
test_Jst(man, S, jmat)
ginv_mat = make_g_inv_mat(man, S)
ee = man._rand_ambient()
g1 = man._unvec(ginv_mat @ man._vec(ee))
print(check_zero(man.g_inv(S, ee).tP - g1.tP))
print(check_zero(man.g_inv(S, ee).tY - g1.tY))
# test g_inv_Jst
for ii in range(10):
a = man._rand_range_J()
avec = man._vec_range_J(a)
jtout = ginv_mat @ jmat.T @ avec
jtout2 = man._vec(man.g_inv_Jst(S, a))
diff = check_zero(jtout - jtout2)
print(diff)
# test projection
test_projection(man, S)
# now diff projection
for i in range(1):
e = man._rand_ambient()
S1 = man.rand()
xi = man.randvec(S1)
dlt = 1e-7
S2 = psd_point(S1.Y + dlt * xi.tY, S1.P + dlt * xi.tP)
# S = psd_point(S1.Y, S1.P)
d1P = (man.proj(S2, e).tP - man.proj(S1, e).tP) / dlt
d1Y = (man.proj(S2, e).tY - man.proj(S1, e).tY) / dlt
d2 = man.D_proj(S1, xi, e)
print(check_zero(d1P - d2.tP) + check_zero(d1Y - d2.tY))
for i in range(10):
a = man._rand_range_J()
eta = man._rand_ambient()
print(man.base_inner_ambient(eta, man.Jst(S, a)))
print((trace((eta.tP.T.conjugate() - eta.tP) @ a['P'] +
(man.alpha[1] * eta.tY.T.conjugate() @ S.Y + man.beta *
(S.Pinv @ eta.tP.T.conjugate() -
eta.tP.T.conjugate() @ S.Pinv)) @ a['YP'])).real)
print((trace(2 * eta.tP.T.conjugate() @ a['P']) + trace(
(man.alpha[1] * eta.tY.T.conjugate() @ S.Y + man.beta *
(S.Pinv @ eta.tP.T.conjugate() - eta.tP.T.conjugate() @ S.Pinv))
@ a['YP'])).real)
print((trace(
eta.tP.T.conjugate() @ (2 * a['P'] + man.beta *
(a['YP'] @ S.Pinv - S.Pinv @ a['YP']))) +
trace(eta.tY.T.conjugate() @ (man.alpha[1] * S.Y @ a['YP']))
).real)
print(man.base_inner_E_J(man.J(S, eta), a))
print((
trace((eta.tP - eta.tP.T.conjugate()).T.conjugate() @ a['P'] +
(man.alpha[1] * S.Y.T.conjugate() @ eta.tY + man.beta *
(eta.tP @ S.Pinv - S.Pinv @ eta.tP)).T.conjugate() @ a['YP']
)).real)
for i in range(10):
a = man._rand_range_J()
beta = man.beta
alf = man.alpha
anew1 = man.J(S, man.g_inv_Jst(S, a))
anew = {}
saYP = a['YP'] + a['YP'].T.conjugate()
anew['P'] = 4 / beta * S.P @ a['P'] @ S.P + S.P @ saYP - saYP @ S.P
anew['YP'] = alf[1] * a['YP'] + beta * (
((2 / man.beta) * S.P @ a['P'] @ S.P + S.P @ a['YP'] -
a['YP'] @ S.P) @ S.Pinv - S.Pinv @ (
(2 / man.beta) * S.P @ a['P'] @ S.P + S.P @ a['YP'] -
a['YP'] @ S.P))
anew['YP'] = alf[1] * a['YP'] + (
(2 * S.P @ a['P'] + beta * S.P @ a['YP'] @ S.Pinv - beta * a['YP'])
- (2 * a['P'] @ S.P + beta * a['YP'] -
beta * S.Pinv @ a['YP'] @ S.P))
anew['YP'] = (alf[1] - 2 * beta) * a['YP'] + (
(2 * S.P @ a['P'] + beta * S.P @ a['YP'] @ S.Pinv) -
(2 * a['P'] @ S.P - beta * S.Pinv @ a['YP'] @ S.P))
anew['YP'] = (alf[1] - 2 * beta) * a['YP'] + (
(2 * S.P @ a['P'] - 2 * a['P'] @ S.P +
beta * S.P @ a['YP'] @ S.Pinv + beta * S.Pinv @ a['YP'] @ S.P))
print(check_zero(man._vec_range_J(anew1) - man._vec_range_J(anew)))
for i in range(10):
a = man._rand_range_J()
b1 = man.J(S, man.g_inv_Jst(S, a))
b2 = man.J_g_inv_Jst(S, a)
print(check_zero(man._vec_range_J(b1) - man._vec_range_J(b2)))
a1 = man.solve_J_g_inv_Jst(S, b1)
print(check_zero(man._vec_range_J(a) - man._vec_range_J(a1)))
for ii in range(10):
E = man._rand_ambient()
a2 = man.J_g_inv(S, E)
a1 = man.J(S, man.g_inv(S, E))
print(check_zero(man._vec_range_J(a1) - man._vec_range_J(a2)))
for i in range(20):
Uran = man._rand_ambient()
Upr = man.proj(S, man.g_inv(S, Uran))
Upr2 = man.proj_g_inv(S, Uran)
print(check_zero(man._vec(Upr) - man._vec(Upr2)))
for ii in range(10):
a = man._rand_range_J()
xi = man.randvec(S)
jtout2 = man.Jst(S, a)
dlt = 1e-7
Snew = psd_point(S.Y + dlt * xi.tY, S.P + dlt * xi.tP)
jtout2a = man.Jst(Snew, a)
d1 = (jtout2a - jtout2).scalar_mul(1 / dlt)
d2 = man.D_Jst(S, xi, a)
print(check_zero(man._vec(d2) - man._vec(d1)))
for ii in range(10):
S1 = man.rand()
eta = man._rand_ambient()
xi = man.randvec(S1)
a1 = man.J(S1, eta)
dlt = 1e-8
Snew = psd_point(S1.Y + dlt * xi.tY, S1.P + dlt * xi.tP)
a2 = man.J(Snew, eta)
d1 = {
'P': (a2['P'] - a1['P']) / dlt,
'YP': (a2['YP'] - a1['YP']) / dlt
}
d2 = man.D_J(S1, xi, eta)
print(check_zero(man._vec_range_J(d2) - man._vec_range_J(d1)))
# derives metrics
for ii in range(10):
S1 = man.rand()
xi = man.randvec(S1)
omg1 = man._rand_ambient()
omg2 = man._rand_ambient()
dlt = 1e-7
S2 = psd_point(S1.Y + dlt * xi.tY, S1.P + dlt * xi.tP)
p1 = man.inner(S1, omg1, omg2)
p2 = man.inner(S2, omg1, omg2)
der1 = (p2 - p1) / dlt
der2 = man.base_inner_ambient(man.D_g(S1, xi, omg2), omg1)
print(der1 - der2)
if np.abs(der1 - der2) > 1e-4:
print(der1, der2)
break
# cross term for christofel
for i in range(10):
S1 = man.rand()
xi = man.randvec(S1)
eta1 = man.randvec(S1)
eta2 = man.randvec(S1)
dr1 = man.D_g(S1, xi, eta1)
x12 = man.contract_D_g(S1, eta1, eta2)
p1 = man.base_inner_ambient(dr1, eta2)
p2 = man.base_inner_ambient(x12, xi)
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):
S1 = man.rand()
xi = man.randvec(S1)
eta1 = man.randvec(S1)
eta2 = man.randvec(S1)
p1 = man.proj_g_inv(S1, man.christoffel_form(S1, xi, eta1))
p2 = man.proj_g_inv(S1, man.christoffel_form(S1, eta1, xi))
print(check_zero(man._vec(p1) - man._vec(p2)))
v1 = man.base_inner_ambient(man.christoffel_form(S1, eta1, eta2), xi)
v2 = man.base_inner_ambient(man.D_g(S1, eta1, eta2), xi)
v3 = man.base_inner_ambient(man.D_g(S1, eta2, eta1), xi)
v4 = man.base_inner_ambient(man.D_g(S1, xi, eta1), eta2)
print(v1, 0.5 * (v2 + v3 - v4), v1 - 0.5 * (v2 + v3 - v4))