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_chris_vectorfields():
# now test that it works on embedded metrics
# we test that D_xi (eta g eta) = 2(eta g nabla_xi eta)
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
slp = crandn(n * d)
Y0 = man.rand()
slpxi = crandn(n * d)
aa = crandn(n * d, n * d)
aaxi = crandn(n * d, n * d)
def v_func(Y):
return man.proj(Y, (aa @ (Y - Y0).reshape(-1) + slp).reshape(n, d))
YY = Y0.copy()
xi = man.proj(YY, slpxi.reshape(n, d))
nabla_xi_v, dv, cxv = calc_covar_numeric(man, YY, xi, v_func)
def xi_func(Y):
return man.proj(Y, (aaxi @ (Y - Y0).reshape(-1) + slpxi).reshape(n, d))
vv = v_func(YY)
nabla_v_xi, dxi, cxxi = calc_covar_numeric(man, YY, vv, xi_func)
diff = nabla_xi_v - nabla_v_xi
# print(diff)
# now do Lie bracket:
dlt = 1e-7
YnewXi = YY + dlt * xi
Ynewvv = YY + dlt * vv
vnewxi = v_func(YnewXi)
xnewv = xi_func(Ynewvv)
dxiv = (vnewxi - vv) / dlt
dvxi = (xnewv - xi) / dlt
diff2 = man.proj(YY, dxiv - dvxi)
print(check_zero(diff - diff2))
def test_rhess_02():
np.random.seed(0)
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()
UU = {}
p = alpha.shape[0]
VV = {}
gidx = man._g_idx
for rr in range(p):
UU[rr] = make_sym_pos(n)
VV[rr] = crandn(n, dvec[rr + 1])
def f(Y):
ss = 0
for rr in range(p):
br, er = gidx[rr + 1]
wr = Y[:, br:er]
ss += trace(UU[rr] @ wr @ wr.T.conjugate()).real
return ss
def df(W):
ret = np.zeros_like(W)
for rr in range(p):
br, er = gidx[rr + 1]
wr = W[:, br:er]
ret[:, br:er] += 2 * UU[rr] @ wr
return ret
def ehess_form(W, xi, eta):
ss = 0
for rr in range(p):
br, er = gidx[rr + 1]
ss += 2 * trace(
UU[rr] @ xi[:, br:er] @ eta[:, br:er].T.conjugate()).real
return ss
def ehess_vec(W, xi):
ret = np.zeros_like(W)
for rr in range(p):
br, er = gidx[rr + 1]
ret[:, br:er] += 2 * UU[rr] @ xi[:, br:er]
return ret
xxi = crandn(n, d)
dlt = 1e-8
Ynew = Y + dlt * xxi
d1 = (f(Ynew) - f(Y)) / dlt
d2 = df(Y)
print(d1 - trace(d2 @ xxi.T.conjugate()).real)
eeta = crandn(n, d)
d1 = trace((df(Ynew) - df(Y)) @ eeta.T.conjugate()).real / dlt
ehess_val = ehess_form(Y, xxi, eeta)
# ehess_val2 = ehess_form(Y, eeta, xxi)
dv2 = ehess_vec(Y, xxi)
print(trace(dv2 @ eeta.T.conjugate()).real)
print(d1, ehess_val, d1 - ehess_val)
# now check the formula: ehess = xi (eta_func(f)) - <D_xi eta, df(Y)>
# promote eta to a vector field.
m1 = crandn(n, n)
m2 = crandn(d, d)
def eta_field(Yin):
return m1 @ (Yin - Y) @ m2 + eeta
# xietaf: should go to ehess(xi, eta) + df(Y) @ etafield)
xietaf = trace(
df(Ynew) @ eta_field(Ynew).T.conjugate() -
df(Y) @ eta_field(Y).T.conjugate()).real / dlt
# appy eta_func to f: should go to tr(m1 @ xxi @ m2 @ df(Y).T.conjugate())
Dxietaf = trace(
(eta_field(Ynew) - eta_field(Y)) @ df(Y).T.conjugate()).real / dlt
# this is ehess. should be same as d1 or ehess_val
print(xietaf - Dxietaf)
print(xietaf - Dxietaf - ehess_val)
# now check: rhess. Need to make sure xi, eta in the tangent space.
# first compare this with numerical differentiation
xi1 = man.proj(Y, xxi)
eta1 = man.proj(Y, eeta)
egvec = df(Y)
ehvec = ehess_vec(Y, xi1)
rhessvec = man.ehess2rhess(Y, egvec, ehvec, xi1)
# check it numerically:
def rgrad_func(Y):
return man.proj_g_inv(Y, df(Y))
val2, _, _ = calc_covar_numeric(man, Y, xi1, rgrad_func)
val2_p = man.proj(Y, val2)
# print(rhessvec)
# print(val2_p)
print(check_zero(rhessvec - val2_p))
rhessval = man.inner(Y, rhessvec, eta1)
print(man.inner(Y, val2, eta1))
print(rhessval)
# check symmetric:
ehvec_e = ehess_vec(Y, eta1)
ehess_valp = ehess_form(Y, xi1, eta1)
rhessvec_e = man.ehess2rhess(Y, egvec, ehvec_e, eta1)
rhessval_e = man.inner(Y, rhessvec_e, xi1)
rhessval_e1 = man.rhess02(Y, xi1, eta1, egvec, ehess_valp)
# rhessval_e2 = man.rhess02_alt(Y, xi1, eta1, egvec,
# trace([email protected]()).real)
# print(rhessval_e, rhessval_e1, rhessval_e2)
print(rhessval_e, rhessval_e1, rhessval_e - rhessval_e1)
print('rhessval_e %f ' % rhessval_e)
# the above computed inner_prod(Nabla_xi Pi * df, eta)
# in the following check. Extend eta1 to eta_proj
# (Pi Nabla_hat Pi g_inv df, g eta)
# = D_xi (Pi g_inv df, g eta) - (Pi g_inv df g Pi Nabla_hat eta)
def eta_proj(Y):
return man.proj(Y, eta_field(Y))
print(check_zero(eta1 - eta_proj(Y)))
e1 = man.inner(Y, man.proj_g_inv(Y, df(Y)), eta_proj(Y))
e1a = trace(df(Y) @ eta_proj(Y).T.conjugate()).real
print(e1, e1a, e1 - e1a)
Ynew = Y + xi1 * dlt
e2 = man.inner(Ynew, man.proj_g_inv(Ynew, df(Ynew)), eta_proj(Ynew))
e2a = trace(df(Ynew) @ eta_proj(Ynew).T.conjugate()).real
print(e2, e2a, e2 - e2a)
first = (e2 - e1) / dlt
first1 = trace(
df(Ynew) @ eta_proj(Ynew).T.conjugate() -
df(Y) @ eta_proj(Y).T.conjugate()).real / dlt
print(first - first1)
val3, _, _ = calc_covar_numeric(man, Y, xi1, eta_proj)
second = man.inner(Y, man.proj_g_inv(Y, df(Y)), man.proj(Y, val3))
second2 = man.inner(Y, man.proj_g_inv(Y, df(Y)), val3)
print(second, second2, second - second2)
print('same as rhess_val %f' % (first - second))
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))
"""