def createP_sigma(lx, ly, band):
C = table.C_Sigma(ly / (2 / mp.sqrt(3)), lx / (mp.sqrt(3) / 2), band)
#Values tabulated on a rectangular grid, flipped and compensated for.
C = [
C[0], C[1], C[2], C[3] / (3**(1 / 2)), C[4] / (3**(1 / 2)),
C[5] / (3**(1 / 2))
]
#For sqrt(3) normalisation, see paper for reference.
f = lambda kx, ky, kz: C[0] * fi_As(lx, ly, kx, ky, kz) + C[1] * fi_Ax(
lx, ly, kx, ky, kz) + C[2] * fi_Ay(lx, ly, kx, ky, kz) + C[3] * fi_Bs(
lx, ly, kx, ky, kz) + C[4] * fi_Bx(lx, ly, kx, ky, kz) + C[
5] * fi_By(lx, ly, kx, ky, kz)
nrm = 1 / mp.sqrt(
mp.conj(C[0]) * C[0] + mp.conj(C[1]) * C[1] + mp.conj(C[2]) * C[2] +
3 *
(mp.conj(C[3]) * C[3] + mp.conj(C[4]) * C[4] + mp.conj(C[5]) * C[5]) +
2 * mp.re((mp.conj(C[0]) * C[3] * pss + mp.conj(C[0]) * C[4] * psx +
mp.conj(C[0]) * C[5] * psy + mp.conj(C[1]) * C[3] * psx +
mp.conj(C[1]) * C[4] * pxx + mp.conj(C[1]) * C[5] * pxy +
mp.conj(C[2]) * C[3] * psy + mp.conj(C[2]) * C[4] * pxy +
mp.conj(C[2]) * C[5] * pyy) *
(mp.exp(1j * mp.fdot([lx, ly, 0], R_1)) +
mp.exp(1j * mp.fdot([lx, ly, 0], R_2)) +
mp.exp(1j * mp.fdot([lx, ly, 0], R_3)))))
return lambda kx, ky, kz: f(kx, ky, kz) * nrm
def cumulants(self, gamma):
"""
Compute
.. math::
\Lambda(\gamma) = \log\left(\int_{\mathbb{R}^n}
e^{\gamma \|z\|^2_2} F(dz)\right)
as well as its first two derivatives with respect to $\gamma$,
where $F$ is the empirical distribution of `self.sample`.
"""
norm_squared = self.sample
M = norm_squared.mean()
M0 = float(np.mean([mp.exp(gamma * (ns - M)) for ns in norm_squared]))
M0 *= mp.exp(gamma * M)
M1 = np.mean([
np.exp(float(gamma * ns + np.log(ns) - mp.log(M0)))
for ns in norm_squared
])
M2 = np.mean([
np.exp(float(gamma * ns + 2 * np.log(ns) - mp.log(M0)))
for ns in norm_squared
])
return M0, M1, (M2 - M1**2)
def test_gauss_quadrature_dynamic(verbose = False):
n = 5
A = mp.randmatrix(2 * n, 1)
def F(x):
r = 0
for i in xrange(len(A) - 1, -1, -1):
r = r * x + A[i]
return r
def run(qtype, FW, R, alpha = 0, beta = 0):
X, W = mp.gauss_quadrature(n, qtype, alpha = alpha, beta = beta)
a = 0
for i in xrange(len(X)):
a += W[i] * F(X[i])
b = mp.quad(lambda x: FW(x) * F(x), R)
c = mp.fabs(a - b)
if verbose:
print(qtype, c, a, b)
assert c < 1e-5
run("legendre", lambda x: 1, [-1, 1])
run("legendre01", lambda x: 1, [0, 1])
run("hermite", lambda x: mp.exp(-x*x), [-mp.inf, mp.inf])
run("laguerre", lambda x: mp.exp(-x), [0, mp.inf])
run("glaguerre", lambda x: mp.sqrt(x)*mp.exp(-x), [0, mp.inf], alpha = 1 / mp.mpf(2))
run("chebyshev1", lambda x: 1/mp.sqrt(1-x*x), [-1, 1])
run("chebyshev2", lambda x: mp.sqrt(1-x*x), [-1, 1])
run("jacobi", lambda x: (1-x)**(1/mp.mpf(3)) * (1+x)**(1/mp.mpf(5)), [-1, 1], alpha = 1 / mp.mpf(3), beta = 1 / mp.mpf(5) )
def FreeFermions(eigvec, subsystem, FermiVector): r=range(FermiVector) Cij=mp.matrix([[mp.fsum([eigvec[i,k]*eigvec[j,k] for k in r]) for i in subsystem] for j in subsystem]) C_eigval=mp.eigsy(Cij, eigvals_only=True) EH_eigval=mp.matrix([mp.log(mp.fdiv(mp.fsub(mp.mpf(1.0),x),x)) for x in C_eigval]) S=mp.re(mp.fsum([mp.log(mp.mpf(1.0)+mp.exp(-x))+mp.fdiv(x,mp.exp(x)+mp.mpf(1.0)) for x in EH_eigval])) return(S)
def abeff_lemmabound(N, y, t, cond):
sigma1 = 0.5 * (1 + y)
sum1, sum2, sum3, sum5 = [0.0 for _ in range(4)]
b1 = 1
a1 = mp.power(N, -0.4)
xN = 4 * mp.pi() * N * N - mp.pi() * t / 4.0
xNp1 = 4 * mp.pi() * (N + 1) * (N + 1) - mp.pi() * t / 4.0
delta = mp.pi() * y / (2 * (xN - 6 - (14 + 2 * y) / mp.pi())) + 2 * y * (
7 + y) * mp.log(abs(1 + y + 1j * xNp1) / (4 * mp.pi)) / (xN * xN)
expdelta = mp.exp(delta)
for n in range(2, 30 * N + 1):
nf = float(n)
denom = mp.power(nf, sigma1 + (t / 4.0) * mp.log(N * N))
#print([cond[n][i] for i in range(1,9)])
common1 = mp.exp((t / 4.0) * mp.power(mp.log(nf), 2))
common2 = common1 * mp.power(nf / N, y)
common3 = expdelta * (mp.exp(t * y * mp.log(n) / (2 * (xN - 6))) - 1)
bn, bn2, bn3, bn5 = [common1 * cond[n][2 * i - 1] for i in range(1, 5)]
an, an2, an3, an5 = [common2 * cond[n][2 * i] for i in range(1, 5)]
en, en2, en3, en5 = an * common3, an2 * common3, an3 * common3, an5 * common3
sum1 += (en + max(
(1 - a1) * abs(bn + an) / (1 + a1), abs(bn - an))) / denom
sum2 += (en2 + max(
(1 - a1) * abs(bn2 + an2) / (1 + a1), abs(bn2 - an2))) / denom
sum3 += (en3 + max(
(1 - a1) * abs(bn3 + an3) / (1 + a1), abs(bn3 - an3))) / denom
sum5 += (en5 + max(
(1 - a1) * abs(bn5 + an5) / (1 + a1), abs(bn5 - an5))) / denom
return [N, expdelta] + [1 - a1 - j for j in [sum1, sum2, sum3, sum5]]
def abeff_trianglebound(N, y, t, cond):
sigma1 = 0.5 * (1 + y)
sum1, sum2, sum3, sum5 = [0.0 for _ in range(4)]
b1 = 1
a1 = mp.power(N, -0.4)
xN = 4 * mp.pi() * N * N - mp.pi() * t / 4.0
xNp1 = 4 * mp.pi() * (N + 1) * (N + 1) - mp.pi() * t / 4.0
delta = mp.pi() * y / (2 * (xN - 6 - (14 + 2 * y) / mp.pi())) + 2 * y * (
7 + y) * mp.log(abs(1 + y + 1j * xNp1) / (4 * mp.pi)) / (xN * xN)
expdelta = mp.exp(delta)
for n in range(1, 30 * N + 1):
nf = float(n)
denom = mp.power(nf, sigma1 + (t / 4.0) * mp.log(N * N))
common1 = mp.exp((t / 4.0) * mp.power(mp.log(nf), 2))
common2 = common1 * mp.power(nf / N, y) * expdelta * mp.exp(
t * y * mp.log(n) / (2 * (xN - 6)))
bn, bn2, bn3, bn5 = [
common1 * abs(cond[n][2 * i - 1]) for i in range(1, 5)
]
an, an2, an3, an5 = [
common2 * abs(cond[n][2 * i]) for i in range(1, 5)
]
sum1 += (bn + an) / denom
sum2 += (bn2 + an2) / denom
sum3 += (bn3 + an3) / denom
sum5 += (bn5 + an5) / denom
return [N, expdelta] + [2 - j for j in [sum1, sum2, sum3, sum5]]
def system(t, y):
# Variables
i, a, m, z, s = y
# Auxiliary equations
g = i + a + m
tot = g + s
star_elem = m + α * a
τS = star_elem**(1 / 3.0) / (K * g**(1 / 3.0) * tot**(2 / 3.0))
ψ = star_elem / τS
ηion = ηi_lim * (1 - mp.exp(-a / Σion))
ηdiss = ηd_lim * (1 - mp.exp(-m / Σdiss))
Z = z / g
τR = C2 * (1 + T1 * ψ / (Twarm * g)) / (g * tot)
τC = C4 * (1 + T1 * ψ / (Tcold * g)) * Zsun / (g * tot * (Z + Zeff))
recombination = i / τR
cloud_formation = a / τC
# ODE system
return [
-recombination + (ηion + R) * ψ,
-cloud_formation + recombination +
(ηdiss - ηion - α * a / star_elem) * ψ,
cloud_formation - (ηdiss + m / star_elem) * ψ,
(Zsn * R - Z) * ψ,
(1 - R) * ψ,
]
def test_levin_2():
# [2] A. Sidi - "Pratical Extrapolation Methods" p.373
mp.dps = 17
z = mp.mpf(10)
eps = mp.mpf(mp.eps)
with mp.extraprec(2 * mp.prec):
L = mp.levin(method="sidi", variant="t")
n = 0
while 1:
s = (-1)**n * mp.fac(n) * z**(-n)
v, e = L.step(s)
n += 1
if e < eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
eps = mp.exp(0.9 * mp.log(eps))
exact = mp.quad(lambda x: mp.exp(-x) / (1 + x / z), [0, mp.inf])
# there is also a symbolic expression for the integral:
# exact = z * mp.exp(z) * mp.expint(1,z)
err = abs(v - exact)
assert err < eps
w = mp.nsum(lambda n: (-1)**n * mp.fac(n) * z**(-n), [0, mp.inf],
method="sidi",
levin_variant="t")
assert err < eps
def test_levin_3():
mp.dps = 17
z = mp.mpf(2)
eps = mp.mpf(mp.eps)
with mp.extraprec(
7 * mp.prec
): # we need copious amount of precision to sum this highly divergent series
L = mp.levin(method="levin", variant="t")
n, s = 0, 0
while 1:
s += (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4**n))
n += 1
v, e = L.step_psum(s)
if e < eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
eps = mp.exp(0.8 * mp.log(eps))
exact = mp.quad(lambda x: mp.exp(-x * x / 2 - z * x**4),
[0, mp.inf]) * 2 / mp.sqrt(2 * mp.pi)
# there is also a symbolic expression for the integral:
# exact = mp.exp(mp.one / (32 * z)) * mp.besselk(mp.one / 4, mp.one / (32 * z)) / (4 * mp.sqrt(z * mp.pi))
err = abs(v - exact)
assert err < eps
w = mp.nsum(lambda n: (-z)**n * mp.fac(4 * n) /
(mp.fac(n) * mp.fac(2 * n) * (4**n)), [0, mp.inf],
method="levin",
levin_variant="t",
workprec=8 * mp.prec,
steps=[2] + [1 for x in xrange(1000)])
err = abs(v - w)
assert err < eps
def system(t, y):
# Variables
i, a, m, z, s = y
# Auxiliary equations
g = i + a + m
tot = g + s
star_elem = m + α * a
τS = star_elem**(1 / 3.0) / (K * g**(1 / 3.0) * tot**(2 / 3.0))
ψ = star_elem / τS
Ii = ΣI * mp.exp(-t / τ) / (τ * (1.0 - mp.exp(-T / τ)))
outflow = ω * ψ / g
Oi = outflow * i
Oa = outflow * a
Om = outflow * m
Oz = outflow * z
ηion = ηi_lim * (1 - mp.exp(-a / Σion))
ηdiss = ηd_lim * (1 - mp.exp(-m / Σdiss))
Z = z / g
τR = C2 * (1 + T1 * ψ / (Twarm * g)) / (g * tot)
τC = C4 * (1 + T1 * ψ / (Tcold * g)) * Zsun / (g * tot * (Z + Zeff))
recombination = i / τR
cloud_formation = a / τC
# ODE system
return [
Ii - Oi - recombination + (ηion + R) * ψ,
-Oa - cloud_formation + recombination +
(ηdiss - ηion - α * a / star_elem) * ψ,
-Om + cloud_formation - (ηdiss + m / star_elem) * ψ,
-Oz + (Zsn * R - Z) * ψ,
(1 - R) * ψ,
]
def xmultibound(x):
N = int(mp.sqrt(0.25 * x / mp.pi()))
sum1_1, sum1_2, sum12_1, sum12_2, sum123_1, sum123_2, sum1235_1, sum1235_2 = [
0.0 for i in range(8)
]
factor2 = 1 - 1 / mp.power(2.0, 0.7 + 0.1 * mp.log(N * N / 2.0))
factor3 = 1 - 1 / mp.power(3.0, 0.7 + 0.1 * mp.log(N * N / 3.0))
factor5 = 1 - 1 / mp.power(5.0, 0.7 + 0.1 * mp.log(N * N / 5.0))
factorN = 1 / mp.power(N, 0.4)
pow2, pow3, pow5, pow6, pow10, pow15, pow30 = [
mp.power(j, 0.4) for j in [2, 3, 5, 6, 10, 15, 30]
]
expo2, expo3, expo5 = 0.2 * mp.log(2), 0.2 * mp.log(3), 0.2 * mp.log(5)
expo6, expo10, expo15, expo30 = expo2 + expo3, expo2 + expo5, expo3 + expo5, expo2 + expo3 + expo5
exp23, exp25, exp35 = mp.exp(0.2 * mp.log(2) * mp.log(3)), mp.exp(
0.2 * mp.log(2) * mp.log(5)), mp.exp(0.2 * mp.log(3) * mp.log(5))
exp235 = exp23 * exp25 * exp35
for n in range(1, 30 * N + 1):
L1 = deltaN(n, N)
L2 = deltaN(n, 2 * N) * divdelta(n, 2)
if L2 > 0: L2 /= mp.power((n / 2.0), expo2)
L3 = deltaN(n, 3 * N) * divdelta(n, 3)
if L3 > 0: L3 /= mp.power((n / 3.0), expo3)
L5 = deltaN(n, 5 * N) * divdelta(n, 5)
if L5 > 0: L5 /= mp.power((n / 5.0), expo5)
L6 = deltaN(n, 6 * N) * divdelta(n, 6)
if L6 > 0: L6 /= (mp.power((n / 6.0), expo6) * exp23)
L10 = deltaN(n, 10 * N) * divdelta(n, 10)
if L10 > 0: L10 /= (mp.power((n / 10.0), expo10) * exp25)
L15 = deltaN(n, 15 * N) * divdelta(n, 15)
if L15 > 0: L15 /= (mp.power((n / 15.0), expo15) * exp35)
L30 = deltaN(n, 30 * N) * divdelta(n, 30)
if L30 > 0: L30 /= (mp.power((n / 30.0), expo30) * exp235)
R1 = L1
R2 = L2 / pow2
R3 = L3 / pow3
R5 = L5 / pow5
R6 = L6 / pow6
R10 = L10 / pow10
R15 = L15 / pow15
R30 = L30 / pow30
n = float(n)
denom1 = mp.power(n, 0.7 + 0.1 * mp.log(N * N / n))
denom2 = mp.power(n, 0.3 + 0.1 * mp.log(N * N / n))
sum1_1 += abs(L1) / denom1
sum1_2 += abs(R1) / denom2
sum12_1 += abs(L1 - L2) / denom1
sum12_2 += abs(R1 - R2) / denom2
sum123_1 += abs(L1 - L2 - L3 + L6) / denom1
sum123_2 += abs(R1 - R2 - R3 + R6) / denom2
sum1235_1 += abs(L1 - L2 - L3 - L5 + L6 + L10 + L15 - L30) / denom1
sum1235_2 += abs(R1 - R2 - R3 - R5 + R6 + R10 + R15 - R30) / denom2
finalsum1 = sum1_1 - 1 + sum1_2 * factorN
finalsum12 = (sum12_1 - 1 + sum12_2 * factorN) / factor2
finalsum123 = (sum123_1 - 1 + sum123_2 * factorN) / (factor2 * factor3)
finalsum1235 = (sum1235_1 - 1 + sum1235_2 * factorN) / (factor2 * factor3 *
factor5)
return [finalsum1, finalsum12, finalsum123, finalsum1235]
def fi_Bx(lx, ly, kx, ky, kz):
return (mp.exp(1j * mp.fdot(
[lx + kx / ct.hbar, ly + ky / ct.hbar, kz / ct.hbar], R_1)) +
mp.exp(1j * mp.fdot(
[lx + kx / ct.hbar, ly + ky / ct.hbar, kz / ct.hbar], R_2)) +
mp.exp(1j * mp.fdot(
[lx + kx / ct.hbar, ly + ky / ct.hbar, kz / ct.hbar], R_3))
) * w2px(kx, ky, kz)
def phi_decay(u,n_max=100):
running_sum=0
u=mp.mpc(u)
for n in range(1,n_max+1):
term1=2*PI_sq*mp.power(n,4)*mp.exp(9*u) - 3*PI*mp.power(n,2)*mp.exp(5*u)
term2=mp.exp(-1*PI*mp.power(n,2)*mp.exp(4*u))
running_sum += term1*term2
#print n,term1, term2, running_sum
return running_sum
def XGEON_integrand_nBA(y, n, RA, RB, rh, l, pm1, Om, lam, sig, deltaphi):
bA = mp.sqrt(RA**2 - rh**2) / l
bB = mp.sqrt(RB**2 - rh**2) / l
K = 1
alp2 = bA**2 * bB**2 / 2 / (bA**2 + bB**2) / sig**2
bet2 = (bA + bB) * bA * bB / (bA**2 + bB**2)
E = (bB - bA) / fp.sqrt(2) / fp.sqrt(bB**2 + bA**2) * (
(bB + bA) * y / 2 / sig + fp.j * sig * Om)
return K*mp.exp(-alp2*y**2)*mp.exp(-fp.j*bet2*Om*y) *fp.erfc(E) \
* XGEON_denoms_n(y,n,RA,RB,rh,l,pm1,Om,lam,sig,deltaphi)
def FreeFermions(subsystem, C):
C = mp.matrix([[C[x, y] for x in subsystem] for y in subsystem])
C_eigval = mp.eigh(C, eigvals_only=True)
EH_eigval = mp.matrix(
[mp.log(mp.fdiv(mp.fsub(mp.mpf(1.0), x), x)) for x in C_eigval])
S = mp.re(
mp.fsum([
mp.log(mp.mpf(1.0) + mp.exp(-x)) + mp.fdiv(x,
mp.exp(x) + mp.mpf(1.0))
for x in EH_eigval
]))
return (S)
def FreeFermions(subsystem,
C_t): #implements free fermion technique by peschel
C = mp.matrix([[C_t[x, y] for x in subsystem] for y in subsystem])
C_eigval = mp.eigh(C, eigvals_only=True)
EH_eigval = mp.matrix(
[mp.log(mp.fdiv(mp.fsub(mp.mpf(1.0), x), x)) for x in C_eigval])
S = mp.re(
mp.fsum([
mp.log(mp.mpf(1.0) + mp.exp(-x)) + mp.fdiv(x,
mp.exp(x) + mp.mpf(1.0))
for x in EH_eigval
]))
return (S)
def f02(y, n, R, rh, l, pm1, Om, lam, sig):
K = lam**2 * sig / 2 / fp.sqrt(2 * fp.pi)
a = (R**2 - rh**2) * l**2 / 4 / sig**2 / rh**2
b = fp.sqrt(R**2 - rh**2) * Om * l / rh
Zp = mp.mpf((R**2 + rh**2) / (R**2 - rh**2))
if Zp - mp.cosh(y) > 0:
return fp.mpf(K * mp.exp(-a * y**2) * mp.cos(b * y) /
mp.sqrt(Zp - mp.cosh(y)))
elif Zp - mp.cosh(y) < 0:
return fp.mpf(-K * mp.exp(-a * y**2) * mp.sin(b * y) /
mp.sqrt(mp.cosh(y) - Zp))
else:
return 0
def calc_lmsr_marginal_price(token_count, token_index, net_outcome_tokens_sold,
funding):
mp.dps = 100
mp.pretty = True
b = mpf(funding) / mp.log(len(net_outcome_tokens_sold))
result = b * mp.log(
sum(
mp.exp(share_count / b + token_count / b)
for share_count in net_outcome_tokens_sold) - sum(
mp.exp(share_count / b)
for index, share_count in enumerate(net_outcome_tokens_sold) if
index != token_index)) - net_outcome_tokens_sold[token_index]
return result
def fn1(y, n, R, rh, l, pm1, Om, lam, sig):
K = lam**2 * sig / 2 / fp.sqrt(2 * fp.pi)
a = (R**2 - rh**2) * l**2 / 4 / sig**2 / rh**2
b = fp.sqrt(R**2 - rh**2) * Om * l / rh
Zm = mp.mpf(rh**2 / (R**2 - rh**2) *
(R**2 / rh**2 * fp.cosh(2 * fp.pi * rh / l * n) - 1))
if Zm == mp.cosh(y):
return 0
elif Zm - fp.cosh(y) > 0:
return fp.mpf(K * mp.exp(-a * y**2) * mp.cos(b * y) /
mp.sqrt(Zm - mp.cosh(y)))
else:
return fp.mpf(-K * mp.exp(-a * y**2) * mp.sin(b * y) /
mp.sqrt(mp.cosh(y) - Zm))
def get_rate_constants(self):
"""This function calculates all rate constants based on
the DFT energies for each species.
No coverage dependence is included.
"""
# Gas phase entropies
kbT = kb * self.T # in eV
dE = self.get_rxn_energies()
dS = self.get_rxn_entropies()
Ea = self.get_Eacts()
STS = self.get_TS_entropies()
ZPEs = self.get_ZPEs_rxn()
TSZPEs = self.get_ZPEs_act()
dE += ZPEs
Ea += TSZPEs
nlevels = self.nlevels
if self.alkali_promotion:
for level in range(nlevels):
dE[level] += 0.02 # N2 inc by 0.01
Ea[level] -= 0.15 # Barrier lowered by 0.15
dE[level + 2] += 0.24 - 0.01 # NH inc by 0.24
dE[level + 3] += 0.27 - 0.24 # NH2 inc by 0.27
dE[level + 4] += 0 - 0.27 # NH3 inc by 0.54, but not adsorbed
self.dE = dE
self.dS = dS
self.Ea = Ea
self.STS = STS
self.ZPEs = ZPEs
self.TSZPEs = TSZPEs
# Calculate equilibrium and rate constants
K = np.zeros(len(dE)) # equilibrium constants
kf = np.zeros(len(dE)) # forward rate constants
kr = np.zeros(len(dE)) # reverse rate constants
for i in range(len(dE)):
dG = dE[i] - self.T * dS[i]
K[i] = mp.exp(-dG / kbT)
Ea[i] = max([0, dE[i], Ea[i]]) # Enforce Ea > 0, and Ea > dE
kf[i] = kbT / h * mp.exp(STS[i] / kb) * mp.exp(-Ea[i] / kbT)
kr[i] = kf[i] / K[i] # enforce thermodynamic consistency
self.kf = kf
self.kr = kr
return (kf, kr)
def test_svd_test_case():
# a test case from Golub and Reinsch
# (see wilkinson/reinsch: handbook for auto. comp., vol ii-linear algebra, 134-151(1971).)
eps = mp.exp(0.8 * mp.log(mp.eps))
a = [[22, 10, 2, 3, 7],
[14, 7, 10, 0, 8],
[-1, 13, -1, -11, 3],
[-3, -2, 13, -2, 4],
[ 9, 8, 1, -2, 4],
[ 9, 1, -7, 5, -1],
[ 2, -6, 6, 5, 1],
[ 4, 5, 0, -2, 2]]
a = mp.matrix(a)
b = mp.matrix([mp.sqrt(1248), 20, mp.sqrt(384), 0, 0])
S = mp.svd_r(a, compute_uv = False)
S -= b
assert mp.mnorm(S) < eps
S = mp.svd_c(a, compute_uv = False)
S -= b
assert mp.mnorm(S) < eps
def Ht_AFE_A(z, t):
"""
This is the much more accurate approx functional eqn posted by Terry at
https://terrytao.wordpress.com/2018/02/02/polymath15-second-thread-generalising-the-riemann-siegel-approximate-functional-equation/#comment-492182
:param z: point at which H_t is computed
:param t: the "time" parameter
:return: the A part in Ht
"""
z, t = mp.mpc(z), mp.mpc(t)
s = (1 + 1j * z.real - z.imag) / 2
tau = mp.sqrt(s.imag / (2 * mp.pi()))
N = int(tau)
A_pre = (1/16) * s * (s-1) \
* mp.power(mp.pi(), -1*s/2) * mp.gamma(s/2)
A_sum = 0.0
for n in range(1, N + 1):
if t.real > 0:
A_sum += mp.exp(
(t / 16) * mp.power(mp.log(
(s + 4) / (2 * mp.pi() * n * n)), 2)) / mp.power(n, s)
else:
A_sum += 1 / mp.power(n, s)
return A_pre * A_sum
def run_hessenberg(A, verbose=0):
if verbose > 1:
print("original matrix (hessenberg):\n", A)
n = A.rows
Q, H = mp.hessenberg(A)
if verbose > 1:
print("Q:\n", Q)
print("H:\n", H)
B = Q * H * Q.transpose_conj()
eps = mp.exp(0.8 * mp.log(mp.eps))
err0 = 0
for x in xrange(n):
for y in xrange(n):
err0 += abs(A[y, x] - B[y, x])
err0 /= n * n
err1 = 0
for x in xrange(n):
for y in xrange(x + 2, n):
err1 += abs(H[y, x])
if verbose > 0:
print("difference (H):", err0, err1)
if verbose > 1:
print("B:\n", B)
assert err0 < eps
assert err1 == 0
def plot_C_pi():
C = []
for m in [0, 1]:
Cm = []
for i in range(0, len(lmx)):
Crow = []
for j in range(0, len(lmy)):
if m == 0:
Crow.append(1)
if m == 1:
Crow.append(
float(
mp.norm(-mp.exp(1j * (mp.atan(
mp.im(eq5.fl(lmx[i][j], lmy[i][j], 0)) /
mp.re(eq5.fl(lmx[i][j], lmx[i][j], 0))))))))
Cm.append(Crow)
C.append(Cm)
for m in range(0, len(C)):
plt.figure()
plt.title("C matrix for band pi, element number " + str(m))
plt.contourf(lmx, lmy, C[m], cmap=plt.get_cmap("coolwarm"))
plt.colorbar()
plt.xlabel(r'$l_x$ $[m^{-1}]$')
plt.ylabel(r'$l_y$ $[m^{-1}]$')
def run_eig(A, verbose=0):
if verbose > 1:
print("original matrix (eig):\n", A)
n = A.rows
E, EL, ER = mp.eig(A, left=True, right=True)
if verbose > 1:
print("E:\n", E)
print("EL:\n", EL)
print("ER:\n", ER)
eps = mp.exp(0.8 * mp.log(mp.eps))
err0 = 0
for i in xrange(n):
B = A * ER[:, i] - E[i] * ER[:, i]
err0 = max(err0, mp.mnorm(B))
B = EL[i, :] * A - EL[i, :] * E[i]
err0 = max(err0, mp.mnorm(B))
err0 /= n * n
if verbose > 0:
print("difference (E):", err0)
assert err0 < eps
def run_hessenberg(A, verbose = 0):
if verbose > 1:
print("original matrix (hessenberg):\n", A)
n = A.rows
Q, H = mp.hessenberg(A)
if verbose > 1:
print("Q:\n",Q)
print("H:\n",H)
B = Q * H * Q.transpose_conj()
eps = mp.exp(0.8 * mp.log(mp.eps))
err0 = 0
for x in xrange(n):
for y in xrange(n):
err0 += abs(A[y,x] - B[y,x])
err0 /= n * n
err1 = 0
for x in xrange(n):
for y in xrange(x + 2, n):
err1 += abs(H[y,x])
if verbose > 0:
print("difference (H):", err0, err1)
if verbose > 1:
print("B:\n", B)
assert err0 < eps
assert err1 == 0
def run_eig(A, verbose = 0):
if verbose > 1:
print("original matrix (eig):\n", A)
n = A.rows
E, EL, ER = mp.eig(A, left = True, right = True)
if verbose > 1:
print("E:\n", E)
print("EL:\n", EL)
print("ER:\n", ER)
eps = mp.exp(0.8 * mp.log(mp.eps))
err0 = 0
for i in xrange(n):
B = A * ER[:,i] - E[i] * ER[:,i]
err0 = max(err0, mp.mnorm(B))
B = EL[i,:] * A - EL[i,:] * E[i]
err0 = max(err0, mp.mnorm(B))
err0 /= n * n
if verbose > 0:
print("difference (E):", err0)
assert err0 < eps
def run_eigsy(A, verbose = False):
if verbose:
print("original matrix:\n", str(A))
D, Q = mp.eigsy(A)
B = Q * mp.diag(D) * Q.transpose()
C = A - B
E = Q * Q.transpose() - mp.eye(A.rows)
if verbose:
print("eigenvalues:\n", D)
print("eigenvectors:\n", Q)
NC = mp.mnorm(C)
NE = mp.mnorm(E)
if verbose:
print("difference:", NC, "\n", C, "\n")
print("difference:", NE, "\n", E, "\n")
eps = mp.exp( 0.8 * mp.log(mp.eps))
assert NC < eps
assert NE < eps
return NC
def w(sigma, t, T0dash):
wterm1 = 1 + (sigma**2) / (T0dash**2)
wterm2 = 1 + ((1 - sigma)**2) / (T0dash**2)
wterm3 = (sigma - 1) * mp.log(wterm1) / 4.0 + nonnegative(
(T0dash / 2.0) * mp.atan(sigma / T0dash) -
sigma / 2.0) + 1 / (12.0 * (T0dash - 0.33))
return mp.sqrt(wterm1) * mp.sqrt(wterm2) * mp.exp(wterm3)
def run_eigsy(A, verbose=False):
if verbose:
print("original matrix:\n", str(A))
D, Q = mp.eigsy(A)
B = Q * mp.diag(D) * Q.transpose()
C = A - B
E = Q * Q.transpose() - mp.eye(A.rows)
if verbose:
print("eigenvalues:\n", D)
print("eigenvectors:\n", Q)
NC = mp.mnorm(C)
NE = mp.mnorm(E)
if verbose:
print("difference:", NC, "\n", C, "\n")
print("difference:", NE, "\n", E, "\n")
eps = mp.exp(0.8 * mp.log(mp.eps))
assert NC < eps
assert NE < eps
return NC
def Ht_AFE_B(z, t):
"""
This is the much more accurate approx functional eqn posted by Terry at
https://terrytao.wordpress.com/2018/02/02/polymath15-second-thread-generalising-the-riemann-siegel-approximate-functional-equation/#comment-492182
:param z: point at which H_t is computed
:param t: the "time" parameter
:return: the B part in Ht
"""
z, t = mp.mpc(z), mp.mpc(t)
s = (1 + 1j * z.real - z.imag) / 2
tau = mp.sqrt(s.imag / (2 * mp.pi()))
M = int(tau)
B_pre = (1 / 16.0) * s * (s - 1) * mp.power(mp.pi(), 0.5 *
(s - 1)) * mp.gamma(0.5 *
(1 - s))
B_sum = 0.0
for m in range(1, M + 1):
if t.real > 0:
B_sum += mp.exp(
(t / 16.0) * mp.power(mp.log(
(5 - s) / (2 * mp.pi() * m * m)), 2)) / mp.power(m, 1 - s)
else:
B_sum += 1 / mp.power(m, 1 - s)
return B_pre * B_sum
def log_likelihood(self, x, S=10):
# define the posterior q(z|x) / encode x into q(z|x)
qz = self.posterior(x)
# define the prior p(z)
pz = self.prior(batch_size=x.size(0))
# sample S samples from the posterior per data point x
z = qz.rsample([S]) # [S, batchsize, latentdim]
# define the observation model p(x|z) = B(x | g(z))
px = self.observation_model(z)
# Calculating Monte Carlo Estimate of log likelihood
sum_log_lik = px.log_prob(x).sum(-1) + pz.log_prob(z).sum(
-1) - qz.log_prob(z).sum(-1)
log_lik = torch.zeros(x.shape[0])
for i in range(x.shape[0]):
tmp = mp.log(
sum([mp.exp(t)
for t in sum_log_lik[:, i].detach().numpy()]) / S)
log_lik[i] = float(tmp)
ave_log_lik = log_lik.mean()
n_in_ave = x.shape[0]
return {
"log_like": log_lik,
"average_log_like": ave_log_lik,
"n": n_in_ave
}
def abtoy_generalbound(N, numfactors=1):
pset = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31]
pset = pset[:numfactors]
pprod = reduce(mul, pset)
ppset = powerset(pset)[1:]
L_sum, R_sum = 0.0, 0.0
factorN = 1 / mp.power(N, 0.4)
for n in range(1, pprod * N + 1):
lcond = deltaN(n, N)
rcond = deltaN(n, N)
for comb in ppset:
combprod = reduce(mul, comb)
if len(comb) > 1:
subcomb = findsubsets(comb, 2)
subcombprods = [mp.log(i[0]) * mp.log(i[1]) for i in subcomb]
sumexpcombprod = mp.exp(0.2 * sum(subcombprods))
denom2 = mp.power(n / float(combprod),
0.2 * mp.log(combprod)) * sumexpcombprod
else:
denom2 = mp.power(n / float(combprod), 0.2 * mp.log(combprod))
lterm = ((-1)**len(comb)) * deltaN(n, combprod * N) * divdelta(
n, combprod) / denom2
rterm = lterm / mp.power(combprod, 0.4)
lcond += lterm
rcond += rterm
L_sum += abs(lcond) / mp.power(n, 0.7 + 0.1 * mp.log(N * N / n))
R_sum += abs(rcond) / mp.power(n, 0.3 + 0.1 * mp.log(N * N / n))
L_sum = L_sum - 1
R_sum = R_sum * factorN
return [N, pset, L_sum, R_sum, 1 - L_sum - R_sum]
def cumulants(self, gamma):
"""
Compute
.. math::
\Lambda(\gamma) = \log\left(\int_{\mathbb{R}^n}
e^{\gamma \|z\|^2_2} F(dz)\right)
as well as its first two derivatives with respect to $\gamma$,
where $F$ is the empirical distribution of `self.sample`.
"""
norm_squared = self.sample
M = norm_squared.mean()
M0 = float(np.mean([mp.exp(gamma*(ns-M)) for ns in norm_squared]))
M0 *= mp.exp(gamma*M)
M1 = np.mean([np.exp(float(gamma*ns+np.log(ns)-mp.log(M0))) for ns in norm_squared])
M2 = np.mean([np.exp(float(gamma*ns+2*np.log(ns)-mp.log(M0))) for ns in norm_squared])
return M0, M1, (M2-M1**2)
def test_levin_2():
# [2] A. Sidi - "Pratical Extrapolation Methods" p.373
mp.dps = 17
z=mp.mpf(10)
eps = mp.mpf(mp.eps)
with mp.extraprec(2 * mp.prec):
L = mp.levin(method = "sidi", variant = "t")
n = 0
while 1:
s = (-1)**n * mp.fac(n) * z ** (-n)
v, e = L.step(s)
n += 1
if e < eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
eps = mp.exp(0.9 * mp.log(eps))
exact = mp.quad(lambda x: mp.exp(-x)/(1+x/z),[0,mp.inf])
# there is also a symbolic expression for the integral:
# exact = z * mp.exp(z) * mp.expint(1,z)
err = abs(v - exact)
assert err < eps
w = mp.nsum(lambda n: (-1) ** n * mp.fac(n) * z ** (-n), [0, mp.inf], method = "sidi", levin_variant = "t")
assert err < eps
def test_levin_3():
mp.dps = 17
z=mp.mpf(2)
eps = mp.mpf(mp.eps)
with mp.extraprec(7*mp.prec): # we need copious amount of precision to sum this highly divergent series
L = mp.levin(method = "levin", variant = "t")
n, s = 0, 0
while 1:
s += (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4 ** n))
n += 1
v, e = L.step_psum(s)
if e < eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
eps = mp.exp(0.8 * mp.log(eps))
exact = mp.quad(lambda x: mp.exp( -x * x / 2 - z * x ** 4), [0,mp.inf]) * 2 / mp.sqrt(2 * mp.pi)
# there is also a symbolic expression for the integral:
# exact = mp.exp(mp.one / (32 * z)) * mp.besselk(mp.one / 4, mp.one / (32 * z)) / (4 * mp.sqrt(z * mp.pi))
err = abs(v - exact)
assert err < eps
w = mp.nsum(lambda n: (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4 ** n)), [0, mp.inf], method = "levin", levin_variant = "t", workprec = 8*mp.prec, steps = [2] + [1 for x in xrange(1000)])
err = abs(v - w)
assert err < eps
def test_cohen_alt_0():
mp.dps = 17
AC = mp.cohen_alt()
S, s, n = [], 0, 1
while 1:
s += -((-1) ** n) * mp.one / (n * n)
n += 1
S.append(s)
v, e = AC.update_psum(S)
if e < mp.eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
eps = mp.exp(0.9 * mp.log(mp.eps))
err = abs(v - mp.pi ** 2 / 12)
assert err < eps
def test_cohen_alt_1():
mp.dps = 17
A = []
AC = mp.cohen_alt()
n = 1
while 1:
A.append( mp.loggamma(1 + mp.one / (2 * n - 1)))
A.append(-mp.loggamma(1 + mp.one / (2 * n)))
n += 1
v, e = AC.update(A)
if e < mp.eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
v = mp.exp(v)
err = abs(v - 1.06215090557106)
assert err < 1e-12
def entropy(NMZ, NM, NZ, NZW, M, K, J, alpha, phi):
'''
compute perplexity as a function of entropy of the model
'''
AK = K * alpha
N = 0
ent = 0
for m, d in enumerate(DTM):
#print "m:", m
#print "d", d
theta = NMZ[m, :] / (M + AK)
#print theta
ent -= mp.log(np.inner(dryrun[:,m],theta))
#print "ent:", ent
N += M
return mp.exp(ent/N)
def run_schur(A, verbose = 0):
if verbose > 1:
print("original matrix (schur):\n", A)
n = A.rows
Q, R = mp.schur(A)
if verbose > 1:
print("Q:\n", Q)
print("R:\n", R)
B = Q * R * Q.transpose_conj()
C = Q * Q.transpose_conj()
eps = mp.exp(0.8 * mp.log(mp.eps))
err0 = 0
for x in xrange(n):
for y in xrange(n):
err0 += abs(A[y,x] - B[y,x])
err0 /= n * n
err1 = 0
for x in xrange(n):
for y in xrange(n):
if x == y:
C[y,x] -= 1
err1 += abs(C[y,x])
err1 /= n * n
err2 = 0
for x in xrange(n):
for y in xrange(x + 1, n):
err2 += abs(R[y,x])
if verbose > 0:
print("difference (S):", err0, err1, err2)
if verbose > 1:
print("B:\n", B)
assert err0 < eps
assert err1 < eps
assert err2 == 0
def test_levin_nsum():
mp.dps = 17
with mp.extraprec(mp.prec):
z = mp.mpf(10) ** (-10)
a = mp.nsum(lambda n: n**(-(1+z)), [1, mp.inf], method = "l") - 1 / z
assert abs(a - mp.euler) < 1e-10
eps = mp.exp(0.8 * mp.log(mp.eps))
a = mp.nsum(lambda n: (-1)**(n-1) / n, [1, mp.inf], method = "sidi")
assert abs(a - mp.log(2)) < eps
z = 2 + 1j
f = lambda n: mp.rf(2 / mp.mpf(3), n) * mp.rf(4 / mp.mpf(3), n) * z**n / (mp.rf(1 / mp.mpf(3), n) * mp.fac(n))
v = mp.nsum(f, [0, mp.inf], method = "levin", steps = [10 for x in xrange(1000)])
exact = mp.hyp2f1(2 / mp.mpf(3), 4 / mp.mpf(3), 1 / mp.mpf(3), z)
assert abs(exact - v) < eps
def test_levin_1():
mp.dps = 17
eps = mp.mpf(mp.eps)
with mp.extraprec(2 * mp.prec):
L = mp.levin(method = "levin", variant = "v")
A, n = [], 1
while 1:
s = mp.mpf(n) ** (2 + 3j)
n += 1
A.append(s)
v, e = L.update(A)
if e < eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
eps = mp.exp(0.9 * mp.log(eps))
err = abs(v - mp.zeta(-2-3j))
assert err < eps
w = mp.nsum(lambda n: n ** (2 + 3j), [1, mp.inf], method = "levin", levin_variant = "v")
err = abs(v - w)
assert err < eps
def test_levin_0():
mp.dps = 17
eps = mp.mpf(mp.eps)
with mp.extraprec(2 * mp.prec):
L = mp.levin(method = "levin", variant = "u")
S, s, n = [], 0, 1
while 1:
s += mp.one / (n * n)
n += 1
S.append(s)
v, e = L.update_psum(S)
if e < eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
eps = mp.exp(0.9 * mp.log(eps))
err = abs(v - mp.pi ** 2 / 6)
assert err < eps
w = mp.nsum(lambda n: 1/(n * n), [1, mp.inf], method = "levin", levin_variant = "u")
err = abs(v - w)
assert err < eps
def run_svd_r(A, full_matrices = False, verbose = True):
m, n = A.rows, A.cols
eps = mp.exp(0.8 * mp.log(mp.eps))
if verbose:
print("original matrix:\n", str(A))
print("full", full_matrices)
U, S0, V = mp.svd_r(A, full_matrices = full_matrices)
S = mp.zeros(U.cols, V.rows)
for j in xrange(min(m, n)):
S[j,j] = S0[j]
if verbose:
print("U:\n", str(U))
print("S:\n", str(S0))
print("V:\n", str(V))
C = U * S * V - A
err = mp.mnorm(C)
if verbose:
print("C\n", str(C), "\n", err)
assert err < eps
D = V * V.transpose() - mp.eye(V.rows)
err = mp.mnorm(D)
if verbose:
print("D:\n", str(D), "\n", err)
assert err < eps
E = U.transpose() * U - mp.eye(U.cols)
err = mp.mnorm(E)
if verbose:
print("E:\n", str(E), "\n", err)
assert err < eps
from mpmath import mp
mp.dps = 50
N = long(sys.argv[1]) if len(sys.argv) > 1 else 1000000
inicio = -1*mp.pi
fim = 1*mp.pi
dx = (fim-inicio)/(N-1)
Idx = mp.mpf(1/dx)
with open('RealExponencial.h', 'w') as f:
f.write("\n".join([
'using namespace std;',
'',
'#define R_EXP_N {N}',
'#define R_EXP_inicio {inicio}',
'#define R_EXP_fim {fim}',
'',
'',
]).format(**locals()))
f.write('const double RealExponencial[] = {')
for i in range(N):
value = mp.exp(i*dx + inicio)
v = "%.50e" % value
f.write( v + ' ,\n')
f.write('\n};')
def make_f(m):
return lambda x: (0.5 * (1 + mp.erf( (x - E_w)/mp.sqrt(2*V_w/m) )))**power * 1/(mp.sqrt(2 * mp.pi * V_c/m)) * mp.exp( -(x-E_c)**2/(2*V_c/m) )
h = mp.mpf(h.value)
k_B = mp.mpf(k_B.value)
for f in files:
t = mp.mpf(rrlmod.str2val(f.split('_')[3]))
data = np.loadtxt(f, dtype=str)
bn = data[:,-1]
freq = crrls.n2f(map(float, data[:,0]), line, n_max=float(data[-1,0])+1)
dn = fc.set_dn(line)
beta = np.empty(len(bn), dtype=mp.mpf)
betabn = np.empty(len(bn), dtype=mp.mpf)
for i in xrange(len(bn)):
if i < len(bn)-dn:
#bnn = np.divide(bn[i+dn,-1], bn[i,-1])
nu = mp.mpf(freq[i])
bnn = mp.mpf(bn[i+dn]) / mp.mpf(bn[i])
e = mp.mpf(-h*nu*1e6/(k_B*t))
exp = mp.exp(e)
beta[i] = (mp.mpf('1') - bnn*exp)/(mp.mpf('1') - exp)
if beta[i]*mp.mpf(bn[i]) != 'None':
betabn[i] = beta[i]*mp.mpf(bn[i])
else:
betabn[i] = -9999
np.savetxt('{0}/bbn2_{1}/{2}bn_beta'.format(cwd, line, f), np.column_stack((data[:-30,0], betabn[:-30])), fmt=('%s', '%s'))
os.chdir(cwd)
def _mp_fn(x):
"""
Actual function that gets evaluated. The caller just vectorizes.
"""
#return mp.mpf(2)*mp.j1(x)/x
return mp.exp(mp.loggamma(x))
B = int((((Vmax_float**2)*rho*e)/(pn - pc_float)))
A = int(Rmax_float)**int(B)
print Rmax_float
print B
wind_speed = []
radius_roci = []
print range(1,int(roci_float))
for i in range(1,int(roci_float),1):
Vg = (mp.sqrt((((A*B*(pn - int(pc_float)))*mp.exp(int(-A)/(int(i)**int(B))))/(((int(rho)*int(i)**int(B) + ((int(i)**2*int(f)**2)/4))))))) - (((int(i)* int(f))/2))
wind_speed.append(float(Vg))
radius_roci.append(i)
print Vg
df_windspeed = pd.DataFrame(wind_speed)
df_radius = pd.DataFrame(radius_roci)
plot_dataframe = pd.DataFrame({'Radius (km)' : radius_roci, 'Windspeed' : wind_speed})
#print plot_dataframe
#print datetime
#print forecast_time
plt.figure()