def d_theta(angle,k_v,R_v,alpha_v,An):
'''Off-axis level, :math:`D_{\\theta}`, for piston in a sphere
Parameters
----------
angle : mpmath.mpf
Azimuth/elevation angle in radians
k_v : mpmath.mpf
Wavenumber
alpha: mpmath.mpf
Piston equivalent aperture in radians.
An : mpmath.matrix
An is a required pre-calculated matrix which results from the
satisfying the conditions of the model - specific to each parameter
set.
Returns
-------
abs(rel_level()) : mpmath.mpf
The off-axis value at angle :math:`\theta`
'''
num = 4
N_v = An.rows
denom = (k_v**2)*(R_v**2)*mpmath.sin(alpha_v)**2
part1 = num/denom
jn_matrix = mpmath.matrix([I**f for f in range(N_v)])
legendre_matrix = mpmath.matrix([legendre(n_v, mpmath.cos(angle)) for n_v in range(N_v)])
Anjn = HP(An,jn_matrix).doit()
part2_matrix = HP(Anjn,legendre_matrix).doit()
part2 = sum(part2_matrix)
rel_level = lambdify([], -part1*part2, 'mpmath')
return abs(rel_level())
def jac_Diode_In_mpmath(x, b, c, y):
"""
[Реестровая]
непосредственная проверка невозможна
:param x:
:param b:
:param c:
:param y:
:return:
"""
global FT
dfdbMPM = lambda y, x, b, c: mpm.matrix([[
mpm.exp((-b[2] * y[0] + x[0]) / (FT * b[1])) - 1, -b[0] *
(-b[2] * y[0] + x[0]) * mpm.exp((-b[2] * y[0] + x[0]) / (FT * b[1])) /
(FT * b[1]**2), -b[0] * y[0] * mpm.exp((-b[2] * y[0] + x[0]) /
(FT * b[1])) / (FT * b[1])
]])
dfdyMPM = lambda y, x, b, c: mpm.matrix([[
-1 - b[0] * b[2] * mpm.exp((-b[2] * y[0] + x[0]) / (FT * b[1])) /
(FT * b[1])
]])
jacf = dfdyMPM(y, x, b, c)**-1 * dfdbMPM(y, x, b, c)
return jacf
def SL(pts, prec):
título("Sistema Linear", "=")
# Definir Precisão
mm.mp.dps = prec
nv = len(pts[0])
pol = []
# Criar Matriz Polinômio
matA = []
matY = []
for i in range(len(pts[0])):
matA.append([])
for g in range(nv):
matA[i].append(pts[0][i]**g)
matY.append(pts[1][i])
# Resolver Matriz
matA = mm.matrix(matA)
print("Matriz A")
print(str(matA))
matY = mm.matrix(matY)
print("\nMatriz Y")
print(str(matY))
matInvA = matA**-1
print("\nMatriz Inversa A")
print(str(matInvA))
matX = matInvA * matY
print("\nMatriz X")
print(str(matX))
# Transformar Resposta em String Polinômio
sPol = pol2str(matX)
print("\nPolinômio: ",sPol)
return sPol
def Newton(pts, prec):
título("Newton", "=")
# Definir Precisão
mm.mp.dps = prec
nv = len(pts[0])
matX = pts[0]
matY = [pts[1]]
for i in range(1, nv):
matY.append([])
for j in range(nv - i):
matY[i].append((matY[i - 1][j + 1] - matY[i - 1][j]) /
(matX[j + i] - matX[j]))
print("\nTabela\n\n", mm.matrix(matY))
# matP = [matY[1][0]]
# for i in range(nv-1):
# mult = 1
# for n in range(i):
# mult *= -matX[n]
# matP.append(mult)
matX = pts[0]
matY = pts[1]
matP = mm.matrix(list(reversed(Polynomial(lagrange(matX, matY)).coef)))
# Transformar Resposta em String Polinômio
sPol = pol2str(matP)
print("\nPolinômio: ", sPol)
return sPol
def cons(o1, p11):
b = matrix(o1)
# print('b = ',o)
c = matrix(p11)
# print('c = ', c)
a = b * c
# print('a = ', a)
return a
def test_poincare_reflect0():
z = mpm.matrix([[0., 0., 0.5]])
x = mpm.matrix([[0.1, -0.2, 0.1]])
assert mpm.norm(poincare_reflect0(z, z)) < TOL
y = poincare_reflect0(z, x)
assert mpm.norm(poincare_reflect0(z, y) - x) < TOL
def test_poincare_reflect():
x = mpm.matrix([[0.3, -0.3, 0.0]])
a = mpm.matrix([[0.5, 0., 0.0]])
y = poincare_reflect(a, x)
R = mpm.norm(a)
r1 = mpm.norm(x - a)
r2 = mpm.norm(y - a)
assert R**2 - r1 * r2 < TOL
def zero():
#this function resets the initial position, velocity, and time variables; called when restarting
global time, s0, v0
time = 0
s0 = matrix([[orbrad['earth']],
[0]])
v0 = matrix([[0],
[-30000]])
def test_poincare_dist():
x = mpm.matrix([[0., 0., -0.4]])
y = mpm.matrix([[0.0, 0.0, 0.1]])
o = mpm.zeros(1, 3)
assert poincare_dist(o, y) - poincare_dist0(y) < TOL
assert poincare_dist(o, x) - poincare_dist0(x) < TOL
assert poincare_dist(o, x) + poincare_dist(o, y) - poincare_dist(x,
y) < TOL
def CF_bounded_Pareto(u, P):
"""
Use: CF_bounded_Pareto(u, P)
This function returns the characteristic function of an FPP with exponential
pulses and bounded Pareto amplitudes [3] as a complex (u.size, 1)-matrix.
The parameters are:
gamma, the shape parameter of the distribution,
alpha, the scale of the Pareto distribution,
L, the lower bound of the Pareto distribution,
H, the upper bound of the Pareto distribution.
Input:
u: The variable of the characteristic function. .......... 1D np array
P=[gamma, alpha, L, H]: The parameters of the distribution. ... list
Output:
res: the characteristic function. ........ complex (u.size,1) np array
References:
[1] A. Theodorsen and O. E. Garcia, PPCF 60 (2018) 034006
https://doi.org/10.1088/1361-6587/aa9f9c
[2] O. E. Garcia and A. Theodorsen POP 25 (2018) 014506
https://doi.org/10.1063/1.5020555
[3] https://en.wikipedia.org/wiki/Pareto_distribution
"""
import numpy as np
import mpmath as mm
#res = np.zeros([u.size, 1], dtype=complex)
g_m = mm.mpf(P[0])
a_m = mm.mpf(P[1])
L_m = mm.mpf(P[2])
H_m = mm.mpf(P[3])
u_m = mm.matrix(-1.j * u)
C = mm.matrix(u.size, 1)
def tmp(x, a):
return -mm.log(x) - mm.gammainc(0, x) + x**a * mm.gammainc(-a, x)
const_0 = -g_m * (a_m**(-1) + mm.euler)
const_1 = g_m / (H_m**a_m - L_m**a_m)
for i in range(u.size):
if u_m[i] == 0:
lnC = 0
else:
lnCtmp = (H_m**a_m * tmp(L_m * u_m[i], a_m) -
L_m**a_m * tmp(H_m * u_m[i], a_m))
lnC = const_0 + const_1 * lnCtmp
C[i] = mm.exp(lnC)
return np.array(C.tolist(), dtype=np.cfloat)
def runTest(self):
data = {}
data[1.] = mpmath.matrix([[1., 1.], [1., 1.]])
data[2.] = mpmath.matrix([[2., 2.], [2., 2.]])
rk.use_mpmath_types()
cal = rk.get_asym_calc(cu.rydbergs, [0, 0])
dmat = rk.get_dmat_from_discrete(rk.Smat, data, cal, "dum2")
self.assertEqual(nw.mode, nw.mode_mpmath)
self.assertEqual(nw.dps, nw.dps_default_mpmath)
for mat in dmat.values():
nw.shape(mat) # Will get an exception if types are wrong.
def poly_fixed(p,
a,
b,
x_exp,
bits,
y_exp=None,
dtype=numpy.int64,
epsilon=EPSILON):
c, shr, exp = poly_fixed_multi(p.transpose(), mpmath.matrix([a]),
mpmath.matrix([b]), x_exp, bits, y_exp,
dtype, epsilon)
return c[0, :], shr, exp
def absposition(planet):
#this function is used to get the position of all planets other than the earth
#it is assumed that they are perfectly circular in orbit
if(planet == 'sun'):
#the sun is always at the origin
pos = matrix([[0],
[0]])
else:
angle = angularvelocity(planet)*time
pos = matrix([[orbrad[planet]*cos(angle)],
[orbrad[planet]*sin(angle)]])
return pos
def main():
# Initialise initial conditions
mpm.dps = 32 # Quadruple precision calculations
#T = 2. * mpm.mpf(1.657935803684451)
#x0 = mpm.matrix([1.006057199440406, 0.0, 0.0, 0.022036094262319])
#x0 = mpm.matrix([1.009433860747752, 0.0, 0.0, 0.003760276142823])
x0 = mpm.matrix([
1.0060556387918027, 2.4281528205782993e-06, -8.1318569756237198e-06,
0.022040604156795093
]) # Velocities have been inverted (+ -> -) to spoof reverse time
# T = mpm.mpf(2 * 3.061660616555952)
x_des = mpm.matrix([
1.0039053449563018, 0.0024706864707973696, -0.012281740632575283,
0.024133313701580495
])
# Call the Taylor integrator
f = mpm.odefun(PCR3BP, 0.0, x0, tol=1e-30, degree=100, verbose=True)
#for i in range(4):
# print(f(T)[i] - f(0.0)[i])
print(mpm.pi)
min_dist = 10.
with open('output.dat', 'w+') as fiyul:
for time in mpm.linspace(0., 2 * mpm.pi, 1000):
temp = f(time)
distance = mpm.sqrt((temp[0] - x_des[0])**mpm.mpf(2.0) +
(temp[1] - x_des[1])**mpm.mpf(2.0))
print("Distance between desired point: %s" % (distance))
if distance < min_dist:
min_dist = distance
fiyul.write("%s, %s, %s, %s\n" %
(temp[0], temp[1], temp[2], temp[3]))
print("Minimum distance was %s" % min_dist)
return None
def multivariate_normalization(data, variables_size):
mpmath.mp.dps = 20
for j in range(0, len(data[0])):
means = []
ts_s = []
for u in range(0, variables_size):
means.append(numpy.mean(data[u][j]))
ts_s.append(data[u][j])
covariance_matrix = mpmath.matrix(numpy.cov(ts_s))
w, v = mpmath.eig(covariance_matrix)
diagonal = mpmath.diag(w)
try:
result = mpmath.sqrtm(diagonal)
except ZeroDivisionError as error:
print("j: ", j)
print("ts:", ts_s)
print("not invertible sqrtm")
print("covariance_matrix:", covariance_matrix)
sys.exit()
B = v * result
try:
inverse_B = B**-1
except ZeroDivisionError as error:
# Not invertible. Skip this one.
# Non invertable cases Uwave
print("j: ", j)
print("ts:", ts_s)
print("not invertible")
print("covariance_matrix:", covariance_matrix)
sys.exit()
except Exception as exception:
# Not invertible. Skip this one.
print("j: ", j)
print("ts:", ts_s)
print("not invertible")
print("covariance_matrix:", covariance_matrix)
sys.exit()
for i in range(0, len(data[0][j])):
atributes_together = []
for u in range(0, variables_size):
atributes_together.append(data[u][j][i])
atributes_together = mpmath.matrix(atributes_together)
result = atributes_together - mpmath.matrix(means)
result = inverse_B * result
for u in range(0, variables_size):
if type(result[u]) is mpmath.mpc:
data[u][j][i] = result[u].real
else:
data[u][j][i] = result[u]
return data
def g(theta):
# calculate output of alpha-max-beta-min for each theta-value
y = utils.poly.eval_multi(p_alpha_cos_beta_sin,
mpmath.matrix(theta))
# optionally apply some iterations of Newton-Raphson square root
# this means the alpha, beta will be optimized to return the best
# error AFTER the Newton-Raphson has been applied at run-time
for i in range(NEWTON_ITERS):
y = mpmath.matrix(
[.5 * (y[j] + 1. / y[j]) for j in range(y.rows)])
return y - 1.
def _semicolon(a, b):
if not isinstance(a, mpmath.matrix):
a = mpmath.matrix([[a]])
if not isinstance(b, mpmath.matrix):
b = mpmath.matrix([[b]])
assert a.cols == b.cols, 'matrix cols not equal'
r = mpmath.matrix(a.rows + b.rows, a.cols)
for j in range(a.cols):
for i in range(a.rows):
r[(i, j)] = a[(i, j)]
for i in range(b.rows):
r[(i + a.rows, j)] = b[(i, j)]
return r
def tranformRectangle(self, orignalRectangle, uVector, vVector):
LeftButton = mpmath.matrix([orignalRectangle[0], orignalRectangle[1]])
LeftButton = self.tranformCoordinate(LeftButton, uVector, vVector)
RightUp = mpmath.matrix([orignalRectangle[2], orignalRectangle[3]])
RightUp = self.tranformCoordinate(RightUp, uVector, vVector)
Left = min(LeftButton[0], RightUp[0])
Button = min(LeftButton[1], RightUp[1])
Right = max(LeftButton[0], RightUp[0])
Up = max(LeftButton[1], RightUp[1])
return [Left, Button, Right, Up]
def d_theta(angle, k_v, R_v, alpha_v, An):
num = 4
N_v = An.rows
denom = (k_v**2) * (R_v**2) * mpmath.sin(alpha_v)**2
part1 = num / denom
jn_matrix = mpmath.matrix([I**f for f in range(N_v)])
legendre_matrix = mpmath.matrix(
[legendre(n_v, mpmath.cos(angle)) for n_v in range(N_v)])
Anjn = HP(An, jn_matrix).doit()
part2_matrix = HP(Anjn, legendre_matrix).doit()
part2 = sum(part2_matrix)
rel_level = lambdify([], -part1 * part2, 'mpmath')
return abs(rel_level())
def traverseTheHierarchy(self, startingStructureName=None, delegateFunction = None,
transformPath = [], rotateAngle = 0, transFlags = (0,0,0), coordinates = (0,0)):
#since this is a recursive function, must deal with the default
#parameters explicitly
if startingStructureName == None:
startingStructureName = self.rootStructureName
#set up the rotation matrix
if(rotateAngle == None or rotateAngle == ""):
rotateAngle = 0
else:
rotateAngle = math.radians(float(rotateAngle))
mRotate = mpmath.matrix([[math.cos(rotateAngle),-math.sin(rotateAngle),0.0],
[math.sin(rotateAngle),math.cos(rotateAngle),0.0],[0.0,0.0,1.0],])
#set up the translation matrix
translateX = float(coordinates[0])
translateY = float(coordinates[1])
mTranslate = mpmath.matrix([[1.0,0.0,translateX],[0.0,1.0,translateY],[0.0,0.0,1.0]])
#set up the scale matrix (handles mirror X)
scaleX = 1.0
if(transFlags[0]):
scaleY = -1.0
else:
scaleY = 1.0
mScale = mpmath.matrix([[scaleX,0.0,0.0],[0.0,scaleY,0.0],[0.0,0.0,1.0]])
#we need to keep track of all transforms in the hierarchy
#when we add an element to the xy tree, we apply all transforms from the bottom up
transformPath += [(mRotate,mScale,mTranslate)]
if delegateFunction != None:
delegateFunction(startingStructureName, transformPath)
#starting with a particular structure, we will recursively traverse the tree
#********might have to set the recursion level deeper for big layouts!
if(len(self.structures[startingStructureName].srefs)>0): #does this structure reference any others?
#if so, go through each and call this function again
#if not, return back to the caller (caller can be this function)
for sref in self.structures[startingStructureName].srefs:
#here, we are going to modify the sref coordinates based on the parent objects rotation
# if (sref.sName.count("via") == 0):
self.traverseTheHierarchy(startingStructureName = sref.sName,
delegateFunction = delegateFunction,
transformPath = transformPath,
rotateAngle = sref.rotateAngle,
transFlags = sref.transFlags,
coordinates = sref.coordinates)
# else:
# print "WARNING: via encountered, ignoring:", sref.sName
#MUST HANDLE AREFs HERE AS WELL
#when we return, drop the last transform from the transformPath
del transformPath[-1]
return
def sarkar_embedding_3D(tree, root, **kwargs):
eps = kwargs.get("eps",0.1)
weighted = kwargs.get("weighted", True)
tau = kwargs.get("tau")
max_deg = max(tree.degree)[1]
if tau is None:
tau = (1+eps)/eps * mpm.log(2*max_deg/ mpm.pi)
prc = kwargs.get("precision")
if prc is None:
prc = _embedding_precision(tree,root,eps)
mpm.mp.dps = prc
n = tree.order()
emb = mpm.zeros(n,3)
place = []
# place the children of root
fib = fib_2D_code(tree.degree[root])
for i, v in enumerate(tree[root]):
r = mpm.tanh(tau*tree[root][v].get("weight",1.))
v_emb = r * mpm.matrix([[fib[i,0],fib[i,1],fib[i,2]]])
emb[v,:]=v_emb
place.append((root,v))
while place:
u, v = place.pop() # u is the parent of v
u_emb, v_emb = emb[u,:], emb[v,:]
# reflect and rotate so that embedding(v) is at (0,0,0)
# and embedding(u) is in the direction of (0,0,1)
u_emb = poincare_reflect0(v_emb, u_emb, precision=prc)
R = rotate_3D_mp(mpm.matrix([[0.,0.,1.]]), u_emb)
#u_emb = (R.T * u_emb).T
# place children of v
fib = fib_2D_code(tree.degree[v])
i=0
for w in tree[v]:
if w == u: # i=0 is for u (parent of v)
continue
i+=1
r = mpm.tanh(tau*tree[w][v].get("weight",1.))
w_emb = r * mpm.matrix([[fib[i,0],fib[i,1],fib[i,2]]])
#undo reflection and rotation
w_emb = (R * w_emb.T).T
w_emb = poincare_reflect0(v_emb, w_emb, precision=prc)
emb[w,:] = w_emb
place.append((v,w))
return emb
def evolution(self, pathways, dt, ddt, stationary=False):
# print(self.size)
# eng = matlab.engine.start_matlab()
rate_matrix = mp.matrix(self.size, self.size)
species_list = list(self.species.items())
population_array = mp.matrix([0 for i in range(self.size)])
# TODO: parallel optimization
self.timestamp += dt
time_array = np.arange(0, dt, ddt) + self.timestamp + ddt
for i in range(self.size):
diag_i = 0
population_array[i] = species_list[i][1]
for j in range(self.size):
if j == i: continue
rate_temp = pathways.get_rate(species_list[i][0],
species_list[j][0])
diag_i += rate_temp
rate_matrix[i, j] = rate_temp
rate_matrix[i, i] = -diag_i
# rate_matrix = matlab.double(rate_matrix)
# population_array = matlab.double(population_array)
if stationary:
intermediate_population_arrays = \
preprocessing.normalize([[boltzmann_factor(species[0].get_G()) for species in species_list]
for t in time_array], norm='l1', axis=1)
# intermediate_population_arrays = Propagate_stationary(rate_matrix, population_array, dt, ddt=ddt)
else:
'''
k_fastest = np.max(rate_matrix)
for i in range(self.size): # Make it a REAL sparse matrix
for j in range(self.size):
if rate_matrix[i][j] < k_fastest * rate_cutoff:
rate_matrix[i][j] = 0
rate_matrix[i][i] = -np.sum(rate_matrix[i])
'''
# Master Equation
intermediate_population_arrays = Propagate_sp(rate_matrix,
population_array,
dt,
ddt=ddt)
# TODO: Modify here!
population_array = intermediate_population_arrays[-1]
# Remapping
for i in range(self.size):
self.update_population(species_list[i][0], population_array[i])
return species_list, intermediate_population_arrays, time_array
def __init__(self, side, extended, distance):
self.distance = mp.mpf(distance)
rightPointX = mp.mpf('1.41949302') if extended else mp.sqrt(2)
self.side = side # side length of square
self.d = side / mp.cos(mp.pi / 12) # length of left edge
sin60 = mp.sqrt(3) / 2
self.top = matrix([self.d / 2, sin60 * self.d])
self.right = matrix([side * rightPointX, 0])
self.nleft = matrix([[-sin60, 0.5]])
self.ndown = matrix([[0, -1]])
v = self.right - self.top
norm = mp.norm(v)
self.nright = matrix([[-v[1] / norm, v[0] / norm]])
self.offset = self.nright * self.right
def solve_lu_mp(A, b):
"""
Solve a linear system using mpmath
:param ndarray A: The linear system
:param ndarray b: The right hand side
:return: ndarray
"""
A_mp = mpmath.matrix([list(row) for row in A])
b_mp = mpmath.matrix([list(row) for row in b])
x_mp = mpmath.lu_solve(A_mp, b_mp)
x = numpy.array(x_mp.tolist(), 'f8')
return x
def reduce_to_fpp(self, z):
# TODO: need to check what happens here when periods are infinite.
from mpmath import floor, matrix, lu_solve
T1, T2 = self.periods
R1, R2 = T1.real, T2.real
I1, I2 = T1.imag, T2.imag
A = matrix([[R1, R2], [I1, I2]])
b = matrix([z.real, z.imag])
x = lu_solve(A, b)
N = int(floor(x[0]))
M = int(floor(x[1]))
alpha = x[0] - N
beta = x[1] - M
assert (alpha >= 0 and beta >= 0)
return alpha, beta, N, M
def reduce_to_fpp(self,z): # TODO: need to check what happens here when periods are infinite. from mpmath import floor, matrix, lu_solve T1, T2 = self.__periods R1, R2 = T1.real, T2.real I1, I2 = T1.imag, T2.imag A = matrix([[R1,R2],[I1,I2]]) b = matrix([z.real,z.imag]) x = lu_solve(A,b) N = int(floor(x[0])) M = int(floor(x[1])) alpha = x[0] - N beta = x[1] - M assert(alpha >= 0 and beta >= 0) return alpha,beta,N,M
def iv_matrix_mid_as_mp(M):
"""
entrywise midpoint of interval matrix,
as (non-interval) mpmath matrix.
If the input is a mpmath matrix, it is returned unchanged.
If the input is a 2-dimensional numpy.ndarray, it is converted to mpmath.
"""
if isinstance(M, numpy.ndarray):
return mp.matrix(M)
Y = mp.matrix(M.rows, M.cols)
for i in range(M.rows):
for j in range(M.cols):
Y[i, j] = mp.mpi(M[i, j]).mid
return Y
def polish(self, init_shapes, flag_initial_error=True):
"""Use Newton's method to compute precise shapes from rough ones."""
precision = self._precision
manifold = self.manifold
#working_prec = precision + 32
working_prec = precision + 64
mpmath.mp.prec = working_prec
target_epsilon = mpmath.mpmathify(2.0)**-precision
det_epsilon = mpmath.mpmathify(2.0)**(32 - precision)
#shapes = [mpmath.mpmathify(z) for z in init_shapes]
shapes = mpmath.matrix(init_shapes)
init_equations = manifold.gluing_equations('rect')
target = mpmath.mpmathify(self.target_holonomy)
error = self._gluing_equation_error(init_equations, shapes, target)
if flag_initial_error and error > 0.000001:
raise GoodShapesNotFound(
'Initial solution not very good: error=%s' % error)
# Now begin the actual computation
eqns = enough_gluing_equations(manifold)
assert eqns[-1] == manifold.gluing_equations('rect')[-1]
for i in range(100):
errors = self._gluing_equation_errors(eqns, shapes, target)
if infinity_norm(errors) < target_epsilon:
break
derivative = [[
int(eqn[0][i]) / z - int(eqn[1][i]) / (1 - z)
for i, z in enumerate(shapes)
] for eqn in eqns]
derivative[-1] = [target * x for x in derivative[-1]]
derivative = mpmath.matrix(derivative)
#det = derivative.matdet().abs()
det = abs(mpmath.det(derivative))
if min(det, 1 / det) < det_epsilon:
raise GoodShapesNotFound(
'Gluing system is too singular (|det| = %s).' % det)
delta = mpmath.lu_solve(derivative, errors)
shapes = shapes - delta
# Check to make sure things worked out ok.
error = self._gluing_equation_error(init_equations, shapes, target)
#total_change = infinity_norm(init_shapes - shapes)
if error > 1000 * target_epsilon:
raise GoodShapesNotFound('Failed to find solution')
#if flag_initial_error and total_change > pari(0.0000001):
# raise GoodShapesNotFound('Moved too far finding a good solution')
self.shapelist = [Number(z, precision=precision) for z in shapes]
def makeMeasAccToPlan_lognorm(func, expplan:list, b:list, c:dict, Ve=[], n=1, outfilename="", listOfOutvars=None):
"""
MPMATH+CAPABLE
:param func: векторная функция
:param expplan: план эксперимента (список значений вектора x)
:param b: вектор b
:param c: вектор c
:param Ve: ковариационная матрица (np.array)
:param n: объём выборки y
:param outfilename: имя выходного файла, куда писать план
:param listOfOutvars: список выносимых переменных
:return: список экспериментальных данных в формате списка словарей 'x':..., 'y':...
"""
res = list()
for i in range(len(expplan)):
y=makeMeasOneDot_lognorm(func, expplan[i], b, c, Ve)
if y is None: #если функция вернула чушь, то в measdata её не записывать!
continue
res.append({'x':mpm.matrix(expplan[i]), 'y':y})
#res.append({'x':expplan[i], 'y':y})
return res
def dfdb (y,x,b,c):
global FT,XO,XT,XF
ft=FT
v=x[0]
i=y[0]
iss=IS=b[0]
n=N=b[1]
ikf=IKF=b[2]
isr=ISR=b[3]
nr=NR=b[4]
vj=VJ=b[5]
m=M=b[6]
rs=RS=b[7]
#ещё раз: ВСЕ константы должны быть в перспективе либо глобальными переменными, как FT, либо составляющими вектора c, но он кривой, потому лучше уж глобальными.
return mpm.matrix ([[iss*mpm.sqrt(ikf/(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO)))*(-mpm.exp((-i*rs + v)/(ft*n)) + XO)*(mpm.exp((-i*rs + v)/(ft*n)) - XO)/(XT*(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO))) + mpm.sqrt(ikf/(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO)))*(mpm.exp((-i*rs + v)/(ft*n)) - XO),
iss**XT*mpm.sqrt(ikf/(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO)))*(-i*rs + v)*(mpm.exp((-i*rs + v)/(ft*n)) - XO)*mpm.exp((-i*rs + v)/(ft*n))/(XT*ft*n**XT*(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO))) - iss*mpm.sqrt(ikf/(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO)))*(-i*rs + v)*mpm.exp((-i*rs + v)/(ft*n))/(ft*n**XT),
iss*mpm.sqrt(ikf/(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO)))*(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO))*(-ikf/(XT*(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO))**XT) + XO/(XT*(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO))))*(mpm.exp((-i*rs + v)/(ft*n)) - XO)/ikf,
((XO - (-i*rs + v)/vj)**XT + XF)**(m/XT)*(mpm.exp((-i*rs + v)/(ft*nr)) - XO),
-isr*(-i*rs + v)*((XO - (-i*rs + v)/vj)**XT + XF)**(m/XT)*mpm.exp((-i*rs + v)/(ft*nr))/(ft*nr**XT),
isr*m*(XO - (-i*rs + v)/vj)*(-i*rs + v)*((XO - (-i*rs + v)/vj)**XT + XF)**(m/XT)*(mpm.exp((-i*rs + v)/(ft*nr)) - XO)/(vj**XT*((XO - (-i*rs + v)/vj)**XT + XF)),
isr*((XO - (-i*rs + v)/vj)**XT + XF)**(m/XT)*(mpm.exp((-i*rs + v)/(ft*nr)) - XO)*mpm.log((XO - (-i*rs + v)/vj)**XT + XF)/XT,
i*iss**XT*mpm.sqrt(ikf/(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO)))*(mpm.exp((-i*rs + v)/(ft*n)) - XO)*mpm.exp((-i*rs + v)/(ft*n))/(XT*ft*n*(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO))) - i*iss*mpm.sqrt(ikf/(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO)))*mpm.exp((-i*rs + v)/(ft*n))/(ft*n)+ \
i*isr*m*(XO - (-i*rs + v)/vj)*((XO - (-i*rs + v)/vj)**XT + XF)**(m/XT)*(mpm.exp((-i*rs + v)/(ft*nr)) - XO)/(vj*((XO - (-i*rs + v)/vj)**XT + XF)) - i*isr*((XO - (-i*rs + v)/vj)**XT + XF)**(m/XT)*mpm.exp((-i*rs + v)/(ft*nr))/(ft*nr)
]])
def test_gth_solve_high_prec():
n = 1e-100
with mp.workdps(100):
P = mp.matrix([[-3*(mp.exp(n)-1), 3*(mp.exp(n)-1)],
[mp.exp(n)-1 , -(mp.exp(n)-1) ]])
run_gth_solve(P, verbose=VERBOSE)
def test_stoch_eig_high_prec():
n = 1e-100
with mp.workdps(100):
P = mp.matrix([[1-3*(mp.exp(n)-1), 3*(mp.exp(n)-1)],
[mp.exp(n)-1 , 1-(mp.exp(n)-1)]])
run_stoch_eig(P, verbose=VERBOSE)
def countNMpmath(A, b, bstart, bend):
N = mpm.matrix(len(b), len(b))
for j in range(len(b)):
partone = mpm.mpf(2) * A[j, 0] / (b[j] - bstart[j])**mpm.mpf(3)
parttwo = mpm.mpf(2) * A[j, 1] / (bend[j] - b[j])**mpm.mpf(3)
N[j, j] += parttwo + partone #так как матрица нулевая
return N
def vawe_func_plot(self, Energy):
x_min = min(self.structure_width_vector)
#x_max = max(self.structure_width_vector)
x = np.linspace(x_min, self.x_max, 150)
U = np.zeros_like(x)
for ind in range(len(x)):
U[ind] = self.U(x[ind])
v_F_vector = mp.matrix(len(Energy), len(x))
for energy in range(len(Energy)):
for ind in range(len(x)):
v_F_vector[energy,
ind] = self.vawe_function(x[ind], Energy[energy])
plt.plot(x / 10**(-7), U, color='k', linestyle='--', linewidth=1)
for energy in range(len(Energy)):
plt.plot(x / 10**(-7), v_F_vector[energy, :], linewidth=2)
plt.ylabel(r'$\Psi_{k}(x), U(x)$')
plt.xlabel(r'$x,10^{-9} м$')
plt.show()
def test_msvd():
output("""\
msvd_:{[b;x]
$[3<>count usv:.qml.msvd x;::;
not (.qml.mdim[u:usv 0]~2#d 0) and (.qml.mdim[v:usv 2]~2#d 1) and
.qml.mdim[s:usv 1]~d:.qml.mdim x;::;
not mortho[u] and mortho[v] and
all[0<=f:.qml.mdiag s] and mzero s _'til d 0;::;
not mzero x-.qml.mm[u] .qml.mm[s] flip v;::;
b;1b;(p*/:m#/:u;m#m#/:s;
(p:(f>.qml.eps*f[0]*max d)*1-2*0>m#u 0)*/:(m:min d)#/:v)]};""")
for A in subjects:
U, S, V = map(mp.matrix, mp.svd(mp.matrix(A)))
m = min(A.rows, A.cols)
p = mp.diag([mp.sign(s * u) for s, u in zip(mp.chop(S), U[0, :])])
U *= p
S = mp.diag(S)
V = V.T * p
test("msvd_[0b", A, (U, S, V))
reps(250)
for Aq in large_subjects:
test("msvd_[1b", qstr(Aq), qstr("1b"))
reps(10000)
def countNMpmath (A, b, bstart, bend):
N=mpm.matrix(len(b),len(b))
for j in range (len(b)):
partone=mpm.mpf(2)*A[j,0]/(b[j]-bstart[j])**mpm.mpf(3)
parttwo=mpm.mpf(2)*A[j,1]/(bend[j]-b[j])**mpm.mpf(3)
N[j,j]+=parttwo+partone #так как матрица нулевая
return N
def solver_Diode_In_mpmath (x,b,c=None):
"""
[Реестровая] +
:param x:
:param b:
:param c:
:return:
"""
global numnone
global FT
#первая часть решения: находим корень с небольшой точностью через numpy
dfdy=lambda y,x,b,c=None: np.array ([[ -1 - b[0]*b[2]*math.exp((-b[2]*y[0] + x[0])/(FT*b[1]))/(FT*b[1])]])
func = lambda y,x,b,c: [np.float(x) for x in func_Diode_In_mpmath(y,x,b,c)]
solvinit=[0.5]
try:
#solx=optimize.root(func, solvinit, args=(x,b,c), jac=dfdy, method='lm').x
solx=optimize.root(func, solvinit, args=(x,b,c), method='lm').x
#http://stackoverflow.com/questions/2566412/find-nearest-value-in-numpy-array
#http://codereview.stackexchange.com/questions/28207/is-this-the-fastest-way-to-find-the-closest-point-to-a-list-of-points-using-nump
except BaseException as e:
numnone+=1
print ("solver: ERROR: "+e.__str__())
return [None]
#вторая часть решения: находим корень с увеличенной точностью через mpmath
funcMPM = lambda y: func_Diode_In_mpmath ([y],x,b,c)
dfdyMPM=lambda y: mpm.matrix ([[ -1 - b[0]*b[2]*mpm.exp((-b[2]*[y][0] + x[0])/(FT*b[1]))/(FT*b[1])]])
solvinitMPM=solx[0]
try:
precsolx=mpm.calculus.optimization.findroot(mpm.mp, f=funcMPM, x0=solvinitMPM, solver=MDNewton, multidimensional=True, J=dfdyMPM, verify=False)
except BaseException as e:
numnone+=1
print ("solver MPM: ERROR: "+e.__str__())
return [None]
return precsolx
def dfdy (y,x,b,c):
global FT,XO,XT,XF
ft=FT
v=x[0]
i=y[0]
iss=IS=b[0]
n=N=b[1]
ikf=IKF=b[2]
isr=ISR=b[3]
nr=NR=b[4]
vj=VJ=b[5]
m=M=b[6]
rs=RS=b[7]
#fh = iss**mpm.mpf(2)*rs*mpm.sqrt(ikf/(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - mpm.mpf(1))))*(mpm.exp((-i*rs + v)/(ft*n)) - mpm.mpf(1))*mpm.exp((-i*rs + v)/(ft*n))/(mpm.mpf(2)*ft*n*(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - mpm.mpf(1)))) - iss*rs*mpm.sqrt(ikf/(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - mpm.mpf(1))))*mpm.exp((-i*rs + v)/(ft*n))/(ft*n)
#sh = isr*m*rs*(mpm.mpf(1) - (-i*rs + v)/vj)*((mpm.mpf(1) - (-i*rs + v)/vj)**mpm.mpf(2) + mpm.mpf('0.005'))**(m/mpm.mpf(2))*(mpm.exp((-i*rs + v)/(ft*nr)) - mpm.mpf(1))/(vj*((mpm.mpf(1) - (-i*rs + v)/vj)**mpm.mpf(2) + mpm.mpf('0.005'))) - isr*rs*((mpm.mpf(1) - (-i*rs + v)/vj)**mpm.mpf(2) + mpm.mpf('0.005'))**(m/mpm.mpf(2))*mpm.exp((-i*rs + v)/(ft*nr))/(ft*nr)
fh = iss**XT*rs*mpm.sqrt(ikf/(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO)))*(mpm.exp((-i*rs + v)/(ft*n)) - XO)*mpm.exp((-i*rs + v)/(ft*n))/(XT*ft*n*(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO))) - iss*rs*mpm.sqrt(ikf/(ikf + iss*(mpm.exp((-i*rs + v)/(ft*n)) - XO)))*mpm.exp((-i*rs + v)/(ft*n))/(ft*n)
sh = isr*m*rs*(XO - (-i*rs + v)/vj)*((XO - (-i*rs + v)/vj)**XT + XF)**(m/XT)*(mpm.exp((-i*rs + v)/(ft*nr)) - XO)/(vj*((XO - (-i*rs + v)/vj)**XT + XF)) - isr*rs*((XO - (-i*rs + v)/vj)**XT + XF)**(m/XT)*mpm.exp((-i*rs + v)/(ft*nr))/(ft*nr)
return mpm.matrix ([[fh+sh]])
def numerical_coeffs(Mn, n, dict_ss):
"""Numerical coefficients of polynomial of order (n - 3)! obtianed from determinant of elimination theory matrix."""
Mn = Mn.tolist()
Mn = [[_zs_sub(_ss_sub(str(entry))) for entry in line] for line in Mn]
zs = punctures(n)
values = [
mpmath.e**(2 * mpmath.pi * 1j * j / (math.factorial(n - 3) + 1))
for j in range(math.factorial(n - 3) + 1)
]
A = [[
value**exponent for exponent in range(math.factorial(n - 3) + 1)[::-1]
] for value in values]
b = []
for i, value in enumerate(values):
dict_zs = {
str(zs[-2]): value,
str(zs[-3]): 1
} # noqa --- used in eval function
nMn = mpmath.matrix([[eval(entry, None) for entry in line]
for line in Mn])
b += [mpmath.det(nMn)]
return mpmath.lu_solve(A, b).T.tolist()[0]
def test_gth_solve_iv():
P = mp.iv.matrix([[-0.1, 0.075, 0.025],
[0.15, -0.2 , 0.05 ],
[0.25, 0.25 , -0.5 ]])
x_expected = mp.matrix([[0.625, 0.3125, 0.0625]])
x_iv = mp.iv.gth_solve(P)
for value, interval in zip(x_expected, x_iv):
assert value in interval
def test_stoch_eig_iv():
P = mp.iv.matrix([[0.9 , 0.075, 0.025],
[0.15, 0.8 , 0.05 ],
[0.25, 0.25 , 0.5 ]])
x_expected = mp.matrix([[0.625, 0.3125, 0.0625]])
x_iv = mp.iv.stoch_eig(P)
for value, interval in zip(x_expected, x_iv):
assert value in interval
def QSdiagonalise(mat):
if QSMODE == MODE_NORM:
w, v = np.linalg.eig(mat)
P = np.transpose(np.matrix(v, dtype=np.complex128))
return np.dot(P, np.dot(mat, np.linalg.inv(P)))
else:
w, v = mpmath.eig(mat)
P = mpmath.matrix(v).T
return P * mat * P**-1
def QSatanElements(mat):
if QSMODE == MODE_NORM:
return np.arctan(mat)
else:
at = mpmath.matrix(mat.rows, mat.cols)
for i in range(mat.rows):
for j in range(mat.cols):
at[i,j] = mpmath.atan(mat[i,j])
return at
def ufrac_to_ucart(A, cell, uvals):
U11, U22, U33, U23, U13, U12 = uvals
U21 = U12
U32 = U23
U31 = U13
Uij = mpm.matrix([[U11, U12, U13], [U21, U22, U23], [U31, U32, U33]])
a, b, c, alpha, beta, gamma = cell
V = vol_unitcell(*cell)
# calculate reciprocal lattice vectors:
astar = (b * c * sin(radians(alpha))) / V
bstar = (c * a * sin(radians(beta))) / V
cstar = (a * b * sin(radians(gamma))) / V
# matrix with the reciprocal lattice vectors:
N = mpm.matrix([[astar, 0, 0],
[0, bstar, 0],
[0, 0, cstar]])
# Finally transform Uij values from fractional to cartesian axis system:
Ucart = A * N * Uij * N.T * A.T
return Ucart
def jac_Diode_In_mpmath (x,b,c,y):
"""
[Реестровая]
непосредственная проверка невозможна
:param x:
:param b:
:param c:
:param y:
:return:
"""
global FT
dfdbMPM=lambda y,x,b,c: mpm.matrix( [ [mpm.exp((-b[2]*y[0] + x[0])/(FT*b[1])) - 1,
-b[0]*(-b[2]*y[0] + x[0])*mpm.exp((-b[2]*y[0] + x[0])/(FT*b[1]))/(FT*b[1]**2),
-b[0]*y[0]*mpm.exp((-b[2]*y[0] + x[0])/(FT*b[1]))/(FT*b[1])] ])
dfdyMPM=lambda y,x,b,c: mpm.matrix ([[ -1 - b[0]*b[2]*mpm.exp((-b[2]*y[0] + x[0])/(FT*b[1]))/(FT*b[1])]])
jacf=dfdyMPM(y,x,b,c)**-1 * dfdbMPM(y,x,b,c)
return jacf
def test_nstr():
m = matrix([[0.75, 0.190940654, -0.0299195971],
[0.190940654, 0.65625, 0.205663228],
[-0.0299195971, 0.205663228, 0.64453125e-20]])
assert nstr(m, 4, min_fixed=-inf) == \
'''[ 0.75 0.1909 -0.02992]
[ 0.1909 0.6563 0.2057]
[-0.02992 0.2057 0.000000000000000000006445]'''
assert nstr(m, 4) == \
'''[ 0.75 0.1909 -0.02992]
def _readCoefficientsForFit(self, fit, ci, typeString):
fileName = self.resultFileHandler.getCoeffFileName(fit, ci, typeString)
if QSMODE == MODE_NORM:
return np.asmatrix(np.loadtxt(fileName, dtype=np.complex128, delimiter=","))
else:
f = open( fileName, "r" )
s1 = f.read()
l1 = s1.split("\n")
l2 = [self._splitmpRows(s) for s in l1]
l3 = [map(lambda s:s.replace("[","").replace("]","").replace("[","").replace(")",""),l) for l in l2]
return mpmath.matrix(l3)
def orthogonal_matrix(self):
"""
Converts von fractional to cartesian.
Invert the matrix to do the opposite.
"""
import mpmath as mpm
Am = mpm.matrix([[self.a, self.b * cos(self.gamma), self.c * cos(self.beta)],
[0, self.b * sin(self.gamma),
(self.c * (cos(self.alpha) - cos(self.beta) * cos(self.gamma)) / sin(self.gamma))],
[0, 0, self.V / (self.a * self.b * sin(self.gamma))]])
return Am
def diodeResistorIMPLICITJac(x, b, c, y):
FT = 0.02586419 # подогнанное по pspice
dfdb = lambda y, x, b, c: mpm.matrix(
[
[
math.exp((-b[2] * y[0] + x[0]) / (FT * b[1])) - 1,
-b[0] * (-b[2] * y[0] + x[0]) * math.exp((-b[2] * y[0] + x[0]) / (FT * b[1])) / (FT * b[1] ** 2),
-b[0] * y[0] * math.exp((-b[2] * y[0] + x[0]) / (FT * b[1])) / (FT * b[1]),
]
]
)
dfdy = lambda y, x, b, c: mpm.matrix(
[-1 - b[0] * b[2] * math.exp((-b[2] * y[0] + x[0]) / (FT * b[1])) / (FT * b[1])]
)
# возвращает структурную матрицу
# jacf=lambda x,b,c,y: jjacf(x,b,c,y,dfdb,dfdy)
jacf = (dfdy(y, x, b, c) ** -1) * dfdb(y, x, b, c)
return jacf
def test_mev():
output("""\
reim:{$[0>type x;1 0*x;2=count x;x;'`]};
mc:{((x[0]*y 0)-x[1]*y 1;(x[0]*y 1)+x[1]*y 0)};
mmc:{((.qml.mm[x 0]y 0)-.qml.mm[x 1]y 1;(.qml.mm[x 0]y 1)+.qml.mm[x 1]y 0)};
mev_:{[b;x]
if[2<>count wv:.qml.mev x;'`length];
if[not all over prec>=abs
mmc[flip vc;flip(flip')(reim'')flip x]-
flip(w:reim'[wv 0])mc'vc:(flip')(reim'')(v:wv 1);'`check];
/ Normalize sign; LAPACK already normalized to real
v*:1-2*0>{x a?max a:abs x}each vc[;0];
(?'[prec>=abs w[;1];w[;0];w];?'[b;v;0n])};""")
for A in eigenvalue_subjects:
if A.rows <= 3:
V = []
for w, n, r in A.eigenvects():
w = sp.simplify(sp.expand_complex(w))
if len(r) == 1:
r = r[0]
r = sp.simplify(sp.expand_complex(r))
r = r.normalized() / sp.sign(max(r, key=abs))
r = sp.simplify(sp.expand_complex(r))
else:
r = None
V.extend([(w, r)]*n)
V.sort(key=lambda (x, _): (-abs(x), -sp.im(x)))
else:
Am = mp.matrix(A)
# extra precision for complex pairs to be equal in sort
with mp.extradps(mp.mp.dps):
W, R = mp.eig(Am)
V = []
for w, r in zip(W, (R.column(i) for i in range(R.cols))):
w = mp.chop(w)
with mp.extradps(mp.mp.dps):
_, S, _ = mp.svd(Am - w*mp.eye(A.rows))
if sum(x == 0 for x in mp.chop(S)) == 1:
# nullity 1, so normalized eigenvector is unique
r /= mp.norm(r) * mp.sign(max(r, key=abs))
r = mp.chop(r)
else:
r = None
V.append((w, r))
V.sort(key=lambda (x, _): (-abs(x), -x.imag))
W, R = zip(*V)
test("mev_[%sb" % "".join("0" if r is None else "1" for r in R), A,
(W, [r if r is None else list(r) for r in R]), complex_pair=True)
def solve(cls, left_mat, right_mat):
"""Solve ``Ax = b`` for ``x``.
``A`` is given by ``left_mat`` and ``b`` by ``right_mat``.
This method seeks to mirror
:meth:`mpmath.matrices.linalg.LinearAlgebraMethods.lu_solve`,
which uses
:meth:`mpmath.matrices.linalg.LinearAlgebraMethods.LU_decomp`,
:meth:`mpmath.matrices.linalg.LinearAlgebraMethods.L_solve` and
:meth:`mpmath.matrices.linalg.LinearAlgebraMethods.U_solve`. Due
to limitations of :mod:`mpmath` we use modified helpers to
accomplish the upper- and lower-triangular solves. We also cache
the LU-factorization for future uses.
It's worth pointing out that :func:`numpy.linalg.solve` works in
exactly this fashion. From the `C source`_ there is a
``lapack_func`` that gets defined and is eventually used
`in Python`_ as ``gufunc``. Notice that the ``lapack_func`` is
``dgesv`` for doubles. Checking the `LAPACK docs`_ verifies
the ``dgesv`` does an LU and then two triangular solves.
.. _C source: https://github.com/numpy/numpy/blob/\
4123a2d55c353e71f665712b4de578f933bd4135/numpy/\
linalg/umath_linalg.c.src#L1579
.. _in Python: https://github.com/numpy/numpy/blob/\
4123a2d55c353e71f665712b4de578f933bd4135/\
numpy/linalg/linalg.py#L384
.. _LAPACK docs: http://www.netlib.org/lapack/explore-html/d7/d3b/\
group__double_g_esolve.html\
#ga5ee879032a8365897c3ba91e3dc8d512
"""
left_mat_id = id(left_mat)
if left_mat_id not in cls._solve_lu_cache:
# left_mat is a NumPy matrix, so must first be
# converted to a mpmath matrix.
lu_parts, pivots = mpmath.mp.LU_decomp(
mpmath.matrix(left_mat.tolist()))
# Convert back to a NumPy matrix.
lu_parts = np.array(lu_parts.tolist())
cls._solve_lu_cache[left_mat_id] = lu_parts, pivots
lu_parts, pivots = cls._solve_lu_cache[left_mat_id]
solution = _forward_substitution(lu_parts, right_mat, pivots)
solution = _back_substitution(lu_parts, solution)
return solution
def makeAinitMpmath (bstart, bend, Skbinit, binit, isBinitGood=True):
"""
расчёт ведётся по правилу "binit есть хорошее предположение в рамках области"
:param bstart:
:param bend:
:param Skbinit:
:param binit:
:return:
"""
A=mpm.matrix( len(binit), 2 )
for i in range (len(binit)):
if isBinitGood:
A[i,0] = mpm.mpf('0.001')*(binit[i]-bstart[i])*Skbinit
A[i,1] = mpm.mpf('0.001')*(bend[i]-binit[i])*Skbinit
else:
A[i,0] = A[i,1] = mpm.mpf('0.001')*(bend[i]-bstart[i])*Skbinit #универсально
return A
def __init__(self, screen, placement = (0, 0), scene_dimensions = None, bg_color = (100, 100, 100)):
self._world = None
#the target screen which the final image of the screen is blitted onto
self.screen = screen
#the placement of the final image of the scene on the screen defaults to the top left corner
#these are screen coordinates
self.screen_placement = placement
self.scr_t = mp.matrix([[1, 0, placement[X]],
[0, 1, placement[Y]],
[0, 0, 1]])
#Scene is a surface which everything in the world is blitted onto
if not scene_dimensions:
#the scene's width and height defaults to the width and height of the screen
scene_dimensions = (screen.get_width(), screen.get_height())
self.scene = pygame.Surface(scene_dimensions)
self.scene_rect = self.scene.get_rect()
#the placement of the viewport in terms of global coordinates
self.scene_placement = (0, 0)
self.scene_scale = 1
self.bg_color = bg_color
def global_to_map(self, global_coordinates):
"""
Maps global coordinates to map coordinates. Mainly used to tell which
voxel the mouse pointer is on. Implementation based on article:
http://www.alcove-games.com/advanced-tutorials/isometric-tile-picking/
"""
(gx, gy) = global_coordinates
#depth info comes from the currently activated layer
mz = self._active_layer
#turn the global coordinates into a homogenous vector
g_coord = mp.matrix([[gx],
[gy + mz * self._voxel_dimensions[DEPTH]],
[1]])
#Apply the isometric unprojection matrix from common_util
g_coord = UNPROJECT * g_coord
#divide and round the coordinates to get integer indexes.
mx = int(mp.nint(g_coord[X]) / self._side_length)
my = int(mp.nint(g_coord[Y]) / self._side_length)
return (mx, my, mz)
def apply_dephasing(number_of_spins, alpha, B,fm_str = 'AFM',kT = 0.0):
'''
takes number of spins, alpha, and the magnetic field and returns the
hamiltonian along with its ground state and the dephased ground state
choose fm_str = 'FM' for ferromagnetic interaction, and 'AFM' for anti-ferromagnetic
'''
if fm_str == 'FM':
calc = ising_calculator_FM(number_of_spins, alpha, B)
elif fm_str == 'AFM':
calc = ising_calculator_AFM(number_of_spins, alpha, B)
H = calc.get_H()
if kT == 0.0:
energy,groundstate = H.groundstate()
dm_groundstate = ket2dm(groundstate)
else:
Hdata = H.data.todense()
arr_mp = mp.matrix(-Hdata /kT)
exp_mp = mp.expm(arr_mp)
trace = np.array(exp_mp.tolist()).trace()
normalized_mp = exp_mp / trace
normalized_np = np.array(normalized_mp.tolist(), dtype = np.complex)
dm_groundstate = Qobj(normalized_np, dims = 2 * [number_of_spins*[2]])
dephased = do_dephasing_dm(dm_groundstate, number_of_spins)
return H, dm_groundstate, dephased
def QSmatrix(val):
if QSMODE == MODE_NORM:
return np.matrix(val, dtype=np.complex128)
else:
return mpmath.matrix(val)
def __init__(self,rang=10,acc=100):
# Generates the coefficients of Legendre polynomial of n-th order.
# acc is the number of decimal characters of the coefficients.
# self.cf is the list with coefficients.
self.rang = rang
self.acc = mp.dps = acc
cn = mpf(0.0)
k = mpf(0)
n = mpf(rang)
m = mpf(n/2)
cf = []
for k in range(n+1):
cn = (-
1)**(n+k)*factorial(n+k)/(factorial(nk)*factorial(k)*factorial(k))
cf.append(cn)
cf.reverse()
# Generates the coefficients of of the
implicit Runge-Kutta scheme of Gauss-Legendre type.
# acc is the number of the decimal
characters of the coefficients.
# Gives back the cortege (r,b,a), the terms
of which correspond to Butcher scheme
#
# r1 | a11 . . . а1n
# . | . .
# . | . .
# . | . .
# rn | an1 . . . ann
# ---+--------------
# | b1 . . . bn
self.r = polyroots(cf)
A1 = matrix(rang)
for j in range(n):
for k in range(n):
A1[k,j] = self.r[j]**k
bn = []
for j in range(n):
bn.append(mpf(1.0)/mpf(j+1))
B = matrix(bn)
self.b = lu_solve(A1,B)
self.a = matrix(rang)
for i in range(1,n+1):
A1 = matrix(rang)
cil = []
for l in range(1,n+1):
cil.append(mpf(self.r[i-
1])**l/mpf(l))
for j in range(n):
A1[l-1,j] = self.r[j]**(l-1)
Cil = matrix(cil)
an = lu_solve(A1,Cil)
for k in range(n):
self.a[i-1,k] = an[k]
def init(self,f,t,h,initvalues):
self.size = len(initvalues)
self.f = f
31
self.t = t
self.h = h
self.yb = matrix(initvalues)
self.ks = matrix(self.size,self.rang)
for k in range(self.size):
for i in range(self.rang):
self.ks[k,i] = self.r[i]
self.tn = matrix(1,self.rang)
for i in range(self.rang):
self.tn[i] = t + h*self.r[i]
self.y = matrix(self.size,self.rang)
for k in range(self.size):
for i in range(self.rang):
self.y[k,i] = self.yb[k]
temp = mpf(0.0)
for j in range(self.rang):
temp += self.a[i,j]*self.ks[k,j]
self.y[k,i] += temp
self.yn = matrix(self.yb)
def iterate(self,tn,y,yn,ks):
# Generates the coefficients of the implicit
Runge-Kutta scheme for the given step
# with the method of the simple iteration
with an initial value, coinciding with the coefficients,
# calculated at the previous step. At
sufficiently small step this must
# work. There exists such a value of the
step, Under which convergence is guaranteed.
# No automatic re-setup of the step is
foreseen in this procedure.
mp.dps = self.acc
y0 = matrix(yn)
norme = mpf(1.0)
#eps0 = pow(eps,mpf(3.0)/mpf(4.0))
eps0 = sqrt(eps)
ks1 = matrix(self.size,self.rang)
yt = matrix(1,self.size)
count = 0
while True:
count += 1
for i in range(self.rang):
for k in range(self.size):
yt[k] = y[k,i]
for k in range(self.size):
ks1[k,i] = self.f(tn,yt)[k]
norme = mpf(0.0)
2
for k in range(self.size):
for i in range(self.rang):
norme += (ks1[k,i]-
ks[k,i])*(ks1[k,i]-ks[k,i])
norme = sqrt(norme)
for k in range(self.size):
for i in range(self.rang):
ks[k,i] = ks1[k,i]
for k in range(self.size):
for i in range(self.rang):
y[k,i] = y0[k]
for j in range(self.rang):
y[k,i] +=
self.h*self.a[i,j]*ks[k,j]
if norme <= eps0:
break
if count >= 100:
print unicode('No convergence','UTF-8')
exit(0)
return ks1
def step(self):
mp.dps = self.acc
self.ks =
self.iterate(self.tn,self.y,self.yn,self.ks)
for k in range(self.size):
for i in range(self.rang):
self.yn[k] +=
self.h*self.b[i]*self.ks[k,i]
for k in range(self.size):
for i in range(self.rang):
self.y[k,i] = self.yn[k]
for j in range(self.rang):
self.y[k,i] +=
self.a[i,j]*self.ks[k,j]
self.t += self.h
for i in range(self.rang):
self.tn[i] = self.t + self.h*self.r[i]
return self.yn
def convert_mpmath_mat( M): return mpmath.matrix([i for i in M.tolist()])
)
P[n, n+1] = \
((N-n)/N) * (epsilon * (1/2) +
(1 - epsilon) * ((n/(N-1) > p) + (n/(N-1) == p) * (1/2))
)
P[n, n] = 1 - P[n, n-1] - P[n, n+1]
P[N, N-1], P[N, N] = epsilon * (1/2), 1 - epsilon * (1/2)
return P
# Stochastic matrix instances
Ps = []
Ps.append(
mp.matrix([[0.9 , 0.075, 0.025],
[0.15, 0.8 , 0.05 ],
[0.25, 0.25 , 0.5 ]])
)
Ps.append(KMR_Markov_matrix_sequential(N=3, p=1./3, epsilon=1e-14))
Ps.append(KMR_Markov_matrix_sequential(N=27, p=1./3, epsilon=1e-2))
# Transition rate matrix instances
As = [P.copy() for P in Ps]
for A in As:
for i in range(A.rows):
A[i, i] = -mp.fsum((A[i, j] for j in range(A.cols) if j != i))
def run_stoch_eig(P, verbose=0):
"""
stoch_eig returns a stochastic vector x such that x P = x
def inv_trafo(self):
if self._inv_trafo is None:
M = mpmath.matrix(self.trafo.tolist())
self._inv_trafo = ndarray.Array((M**-1).tolist())
return self._inv_trafo