def interpolation(interP,knots=None):
"""
Interpolates the given points and returns an object of the Spline class
Arguments:
* interP: interpolation points, (L x 2) matrix
* knotP: knot points, (L+4 x 1) matrix
* default: equidistant on [0,1]
"""
nip=len(interP)
ctrlP=np.zeros((nip,2))
if knots != None:
knots = np.array(knots,dtype='float')
if len(ctrlP) + 4 != len(knots):
raise ValueError('Knots is of the wrong size')
else:
knots = np.hstack((np.zeros(3), np.linspace(0,1,len(ctrlP)-2),
np.ones(3)))
xi=(knots[1:-3]+knots[2:-2]+knots[3:-1])/3
nMatrix=np.zeros((len(xi),len(xi)))
for i in xrange(len(xi)):
fun=basisFunction(i,knots)
nMatrix[:,i]=fun(xi,3)
ctrlP[:,0]=sl.solve(nMatrix,interP[:,0])
ctrlP[:,1]=sl.solve(nMatrix,interP[:,1])
return Spline(ctrlP)
def fit(self, X, y, **params):
"""
Fit Ridge regression model
Parameters
----------
X : numpy array of shape [n_samples,n_features]
Training data
y : numpy array of shape [n_samples]
Target values
Returns
-------
self : returns an instance of self.
"""
self._set_params(**params)
X = np.asanyarray(X, dtype=np.float)
y = np.asanyarray(y, dtype=np.float)
n_samples, n_features = X.shape
X, y, Xmean, ymean = self._center_data(X, y)
if n_samples > n_features:
# w = inv(X^t X + alpha*Id) * X.T y
self.coef_ = linalg.solve(np.dot(X.T, X) + self.alpha * np.eye(n_features), np.dot(X.T, y))
else:
# w = X.T * inv(X X^t + alpha*Id) y
self.coef_ = np.dot(X.T, linalg.solve(np.dot(X, X.T) + self.alpha * np.eye(n_samples), y))
self._set_intercept(Xmean, ymean)
return self
def get_inner_circle(A, B, C): xa, ya = A[0], A[1] xb, yb = B[0], B[1] xc, yc = C[0], C[1] ka = (yb - ya) / (xb - xa) if xb != xa else None kb = (yc - yb) / (xc - xb) if xc != xb else None alpha = np.arctan(ka) if ka != None else np.pi / 2 beta = np.arctan(kb) if kb != None else np.pi / 2 a = np.sqrt((xb - xc)**2 + (yb - yc)**2) b = np.sqrt((xa - xc)**2 + (ya - yc)**2) c = np.sqrt((xa - xb)**2 + (ya - yb)**2) ang_a = np.arccos((b**2 + c**2 - a**2) / (2 * b * c)) ang_b = np.arccos((a**2 + c**2 - b**2) / (2 * a * c)) # 两条角平分线的斜率 k1 = np.tan(alpha + ang_a / 2) k2 = np.tan(beta + ang_b / 2) kv = np.tan(alpha + np.pi / 2) # 求圆心 y, x = solve([[1.0, -k1], [1.0, -k2]], [ya - k1 * xa, yb - k2 * xb]) ym, xm = solve([[1.0, -ka], [1.0, -kv]], [ya - ka * xa, y - kv * x]) r1 = np.sqrt((x - xm)**2 + (y - ym)**2) return(x, y, r1)
def debias_nz(prob, reg=1e-3):
(w, X, y, lam, a_i, c_i, ipro) = prob
nd = X.shape[0]
#This is a fancy way of pruning zeros. You could also do this with loops without too much worry.
#You could also use numpy.nonzero to do this.
stn = array([0] + filter(lambda x : x != 0, list((w[1:] != 0) * range(1, len(w)))), dtype=int)
Hn = ipro[:, stn]
Hn = Hn[stn, :].copy()
#
nz = len(stn)
#
Xty = zeros(nz)
Xty[0] = y.sum()
for i in range(1, nz):
if sp.issparse(X):
Xty[i] = X[:, stn[i] - 1].T.dot(y)[0]
else:
Xty[i] = X[:, stn[i] - 1].dot(y)
Xty *= 2
#
try:
wdb = la.solve(Hn, Xty, sym_pos=True)
except la.LinAlgError:
print("Oh no! Matrix is Singular. Trying again using regularization.")
Hn[range(nz), range(nz)] += 2 * reg
wdb = la.solve(Hn, Xty, sym_pos=True)
#Update c_i
c_i -= ipro[:, stn].dot(wdb - w[stn])
c_i[stn] += a_i[stn] * (wdb - w[stn])
w[stn] = wdb
return
def _solve_cholesky(X, y, alpha, sample_weight=None):
# w = inv(X^t X + alpha*Id) * X.T y
n_samples, n_features = X.shape
n_targets = y.shape[1]
has_sw = sample_weight is not None
if has_sw:
sample_weight = sample_weight * np.ones(n_samples)
sample_weight_matrix = sparse.dia_matrix((sample_weight, 0),
shape=(n_samples, n_samples))
weighted_X = safe_sparse_dot(sample_weight_matrix, X)
A = safe_sparse_dot(weighted_X.T, X, dense_output=True)
Xy = safe_sparse_dot(weighted_X.T, y, dense_output=True)
else:
A = safe_sparse_dot(X.T, X, dense_output=True)
Xy = safe_sparse_dot(X.T, y, dense_output=True)
one_alpha = np.array_equal(alpha, len(alpha) * [alpha[0]])
if one_alpha:
A.flat[::n_features + 1] += alpha[0]
return linalg.solve(A, Xy, sym_pos=True,
overwrite_a=True).T
else:
coefs = np.empty([n_targets, n_features])
for coef, target, current_alpha in zip(coefs, Xy.T, alpha):
A.flat[::n_features + 1] += current_alpha
coef[:] = linalg.solve(A, target, sym_pos=True,
overwrite_a=False).ravel()
A.flat[::n_features + 1] -= current_alpha
return coefs
def prob6(N=10):
"""Time regular and sparse linear system solvers. Plot the system size
versus the execution times. As always, use log scales where appropriate.
"""
domain = 2**np.arange(2,N+1)
solve, spsolve = [], []
for n in domain:
A = prob5(n).tocsr()
b = np.random.random(n)
start = time()
spla.spsolve(A, b)
spsolve.append(time()-start)
A = A.toarray()
start = time()
la.solve(A, b)
solve.append(time()-start)
plt.subplot(121)
plt.plot(domain, spsolve, '.-', lw=2, label="spla.spsolve()")
plt.plot(domain, solve, '.-', lw=2, label="la.solve()")
plt.xlabel("n"); plt.ylabel("Seconds")
plt.legend(loc="upper left")
plt.subplot(122)
plt.loglog(domain, spsolve, '.-', basex=2, basey=2, lw=2)
plt.loglog(domain, solve, '.-', basex=2, basey=2, lw=2)
plt.xlabel("n")
plt.suptitle("Problem 6 Solution")
plt.show()
def planeintersect(self, other):
""" returns line of intersection of this plane and another
None is returned if planes are parallel
20110207 NEW VERSION: M. Krauss
"""
N1 = self.N
N2 = other.N
D1 = self.D
D2 = other.D
if (N1.cross(N2)).abs() == 0:
# Planes are parallel
return None
else:
v = N1.cross(N2)
b = np.array([[D1],[D2]])
p = Point(0,0,0)
try:
# intersection with the plane x=0
A = np.array([[N1.y , N1.z],[N2.y , N2.z]])
x = la.solve(A,b)
p = Point(0,x[0],x[1])
return Line(p, v)
except:
try:
# intersection with the plane y=0
A = np.array([[N1.x , N1.z],[N2.x , N2.z]])
x = la.solve(A,b)
p = Point(x[0],0,x[1])
return Line(p, v)
except:
# intersection with the plane z=0
A = np.array([[N1.x , N1.y],[N2.x , N2.y]])
x = la.solve(A,b)
p = Point(x[0],x[1],0)
return Line(p, v)
def update_values(self):
"""
This method is for updating in the finite horizon case. It shifts the
current value function
V_t(x) = x' P_t x + d_t
and the optimal policy F_t one step *back* in time, replacing the pair
P_t and d_t with P_{t-1} and d_{t-1}, and F_t with F_{t-1}
"""
# === Simplify notation === #
Q, R, A, B, C = self.Q, self.R, self.A, self.B, self.C
P, d = self.P, self.d
# == Some useful matrices == #
S1 = Q + self.beta * dot(B.T, dot(P, B))
S2 = self.beta * dot(B.T, dot(P, A))
S3 = self.beta * dot(A.T, dot(P, A))
# == Compute F as (Q + B'PB)^{-1} (beta B'PA) == #
self.F = solve(S1, S2)
# === Shift P back in time one step == #
new_P = R - dot(S2.T, solve(S1, S2)) + S3
# == Recalling that trace(AB) = trace(BA) == #
new_d = self.beta * (d + np.trace(dot(P, dot(C, C.T))))
# == Set new state == #
self.P, self.d = new_P, new_d
def matlabLDiv(a, b):
'''Implements the matlab \ operator on arrays (maybe works
on matrices also).
Solves the "left divide" equation: a * x = b for x
Inputs:
a,b arrays
Returns: a \ b
Results depend on dimensions of matrices. See documentation for
matlab's MLDIVIDE operator \ .
Theory is on the Numpy for Matlab user's page.
http://wiki.scipy.org/NumPy_for_Matlab_Users
See also this site, but beware of caveats
http://stackoverflow.com/questions/1001634/array-division-translating-from-matlab-to-python
'''
import scipy.linalg as LA
# Check if a is square.
try:
(r,c) = np.shape(a)
except ValueError: # In case a is of dim=1, cannot unpack two vals.
return LA.solve(a,b)
else:
if (r == c): # Square
return LA.solve(a,b)
else:
return LA.lstsq(a,b)
def predict(self, X_p):
'''
Predict the values of the GP at each position x_p
From Rasmussen & Williams "Gaussian Processes for Machine Learning",
pg. 19, algorithm 2.1
'''
(X, Y) = (self.X, self.Y)
sigma_n = self.hyper_params[0]
K = self.kernel.value(self.hyper_params[1:], X, X)
K_p = self.kernel.value(self.hyper_params[1:], X, X_p)
K_p_p = self.kernel.value(self.hyper_params[1:], X_p, X_p)
(self.K, self.K_p, self.K_p_p) = K, K_p, K_p_p
# since the kernel is (should be) pos. def. and symmetric,
# we can solve in a quick/stable way using the cholesky decomposition
L = np.matrix(linalg.cholesky(K + sigma_n**2 * np.eye(len(X)),
lower = True))
alpha = np.matrix(linalg.solve(L.T, linalg.solve(L, Y)))
f_p_mean = K_p.T * alpha.T
v = np.matrix(linalg.solve(L, K_p))
f_p_covariance = K_p_p - v.T*v
log_marginal = (-0.5*np.matrix(Y)*alpha.T - sum(np.log(np.diag(L))) -
len(X)/2.0*np.log(2*np.pi))
return (np.array(f_p_mean).flatten(),
f_p_covariance,
np.array(log_marginal).flatten())
def matrix_normal_density(X, M, U, V):
"""Sample from a matrix normal distribution"""
norm = - 0.5*np.log(la.det(2*np.pi*U)) - 0.5*np.log(la.det(2*np.pi*V))
XM = X-M
pptn = -0.5*np.trace( np.dot(la.solve(U,XM),la.solve(V,XM.T)) )
pdf = norm + pptn
return pdf
def get_absorption_variance(P, plain, absorbing):
"""
Get expected times to absorption.
Note that if an index is indicated as absorbing by its presence
in the sequence of absorbing state indices,
then it will be treated as absorbing
even if the transition matrix P indicates otherwise.
@param P: transition matrix
@param plain: sequence of plain state indices
@param absorbing: sequence of absorbing state indices
@return: variance of times to absorption or 0 from absorbing states
"""
# check that P is really a transition matrix
MatrixUtil.assert_transition_matrix(P)
# define some state lists
states = np.hstack((plain, absorbing))
# check that the index sequences match the size of P
if sorted(states) != range(len(P)):
raise ValueError('P is not conformant with the index sequences')
# compute the time to absorption
Q = P[plain, :][:, plain]
c = np.ones_like(plain)
I = np.eye(len(plain))
t = linalg.solve(I - Q, c)
# compute the variance
vplain = 2*linalg.solve(I - Q, t) - t*(t+1)
v = np.hstack((vplain, np.zeros_like(absorbing)))
return v[inverse_permutation(states)]
def test_care_g(self):
A = matrix([[-2, -1],[-1, -1]])
Q = matrix([[0, 0],[0, 1]])
B = matrix([[1, 0],[0, 4]])
R = matrix([[2, 0],[0, 1]])
S = matrix([[0, 0],[0, 0]])
E = matrix([[2, 1],[1, 2]])
X,L,G = care(A,B,Q,R,S,E)
# print("The solution obtained is", X)
assert_array_almost_equal(
A.T * X * E + E.T * X * A -
(E.T * X * B + S) * solve(R, B.T * X * E + S.T) + Q, zeros((2,2)))
assert_array_almost_equal(solve(R, B.T * X * E + S.T), G)
A = matrix([[-2, -1],[-1, -1]])
Q = matrix([[0, 0],[0, 1]])
B = matrix([[1],[0]])
R = 1
S = matrix([[1],[0]])
E = matrix([[2, 1],[1, 2]])
X,L,G = care(A,B,Q,R,S,E)
# print("The solution obtained is", X)
assert_array_almost_equal(
A.T * X * E + E.T * X * A -
(E.T * X * B + S) / R * (B.T * X * E + S.T) + Q , zeros((2,2)))
assert_array_almost_equal(dot( 1/R , dot(B.T,dot(X,E)) + S.T) , G)
def amp_int_one(self):
"""
Compute transition amplitude of wave function by one particle operator
integration form.
. A = 2k^(-1) <F, [E-H0]\psi>
"""
v1 = self.gtos.calc_mat_sto(self.v1)
(s, t, v0) = self.gtos.calc_mat_stv(1)
(sH, tH, v0H) = self.gtos_cc.calc_mat_stv(self.gtos, 1)
m = self.gtos.calc_vec_sto(self.s)
# solve driven eq
c0 = la.solve(self.energy*s-self.z*v0-t, m)
c1 = la.solve(self.energy*s-self.z*v0-v1-t, m)
# compute amplitude
k = np.sqrt(2.0*self.energy)
plmx_y0p = np.dot(m, c0)
j_plmx = np.sqrt(-k*plmx_y0p.imag)
# compute amplitude (E-H0)
y0p_esmh0_yp = np.dot(c0.conj(), np.dot(sH*self.energy-tH-v0H, c1))
y0m_esmh0_yp = np.dot(c0 , np.dot(s *self.energy-t -v0 , c1))
amp2 = (y0p_esmh0_yp - y0m_esmh0_yp)/(2.0j*j_plmx)
return 2.0/k*amp2
def fit(d):
title("polynomial model (dim=%d)" % d)
xlabel('x')
ylabel('y')
xlim(0,pi)
ylim(0,1)
scatter(xs, ys)
# 単純な最小二乗法
X = array([xs**k for k in range(d+1)]).T
a = LA.solve(X.T.dot(X), X.T.dot(ys))
# グラフ生成
x = linspace(0, pi, 100)
y = a.dot(array([x**k for k in range(d+1)]))
plot(x, y, label="linear regression")
# パラメータの事前分布の平均・分散の逆行列
a0 = zeros(d+1)
s0inv = LA.inv(diag([2**(-k) for k in range(d+1)]))
# 誤差分布の分散. 本当は何らかの推定で決めますが, とりあえず1に．
s = 1
# ベイズ線形回帰
a = LA.solve(X.T.dot(X)/s**2 + s0inv, X.T.dot(ys)/s**2 + s0inv.dot(a0))
x = linspace(0, pi, 100)
y = a.dot(array([x**k for k in range(d+1)]))
plot(x, y, label="bayesian linear regression")
legend(loc=3)
savefig("fig3-13-%d.png" % d)
def amp_int_two(self):
"""
Compute transition amplitude of wave function by two particle operator form.
. A = 2k^(-1)<F, s + v1psi>
In this function, F is evaluated by TIWP method and CBF method.
TIWP is based on F is proportional to the solution of H0 driven equation.
We put driven term as s.
. (E-H0)psi0 = s
. F = Im[psi0]/Sqrt(k Im<s, psi0>)
"""
v1 = self.gtos.calc_mat_sto(self.v1)
v1H = self.gtos_cc.calc_mat_sto(self.gtos, self.v1)
(s, t, v0) = self.gtos.calc_mat_stv(1)
v0 = self.z * v0
m = self.gtos.calc_vec_sto(self.s)
# solve driven eq
c0 = la.solve(self.energy*s-v0-t, m)
c1 = la.solve(self.energy*s-v0-v1-t, m)
# compute amplitude (V2psi + muphi)
# plmx = -m (wave packet)
k = np.sqrt(2.0*self.energy)
plmx_y0p = np.dot(m, c0)
y0p_muphi = plmx_y0p.conj()
y0p_v_yp = np.dot(c0.conj(), np.dot(v1H,c1))
y0m_v_yp = np.dot(c0, np.dot(v1, c1))
imy0p_v_yp = (y0p_v_yp - y0m_v_yp)/(2.0j)
j_plmx = np.sqrt(-np.pi*plmx_y0p.imag)
amp = (y0p_muphi.imag + imy0p_v_yp)/j_plmx
return amp
def _H(self):
r"""Continuous-time linear time invariant system.
This method is used to create a Continuous-time linear
time invariant system for the mdof system.
From this system we can obtain poles, impulse response,
generate a bode, etc.
"""
Z = np.zeros((self.n, self.n))
I = np.eye(self.n)
# x' = Ax + Bu
B2 = I
A = self.A()
B = np.vstack([Z, la.solve(self.M, B2)])
# y = Cx + Du
# Observation matrices
Cd = I
Cv = Z
Ca = Z
C = np.hstack((Cd - Ca @ la.solve(self.M, self.K), Cv - Ca @ la.solve(self.M, self.C)))
D = Ca @ la.solve(self.M, B2)
sys = signal.lti(A, B, C, D)
return sys
def A(self):
"""State space matrix"""
Z = np.zeros((self.n, self.n))
I = np.eye(self.n)
A = np.vstack([np.hstack([Z, I]), np.hstack([la.solve(-self.M, self.K), la.solve(-self.M, self.C)])])
return A
def _pade(A, m):
n = np.shape(A)[0]
c = _padecoeff(m)
if m != 13:
apows = [[] for jj in range(int(np.ceil((m + 1) / 2)))]
apows[0] = sp.eye(n, n, format='csr')
apows[1] = A * A
for jj in range(2, int(np.ceil((m + 1) / 2))):
apows[jj] = apows[jj - 1] * apows[1]
U = sp.lil_matrix((n, n)).tocsr()
V = sp.lil_matrix((n, n)).tocsr()
for jj in range(m, 0, -2):
U = U + c[jj] * apows[jj // 2]
U = A * U
for jj in range(m - 1, -1, -2):
V = V + c[jj] * apows[(jj + 1) // 2]
F = la.solve((-U + V).todense(), (U + V).todense())
return sp.lil_matrix(F).tocsr()
elif m == 13:
A2 = A * A
A4 = A2 * A2
A6 = A2 * A4
U = A * (A6 * (c[13] * A6 + c[11] * A4 + c[9] * A2) +
c[7] * A6 + c[5] * A4 + c[3] * A2 +
c[1] * sp.eye(n, n).tocsr())
V = A6 * (c[12] * A6 + c[10] * A4 + c[8] * A2) + c[6] * A6 + c[4] * \
A4 + c[2] * A2 + c[0] * sp.eye(n, n).tocsr()
F = la.solve((-U + V).todense(), (U + V).todense())
return sp.csr_matrix(F)
def test_trsv(self):
seed(1234)
for ind, dtype in enumerate(DTYPES):
n = 15
A = (rand(n, n)+eye(n)).astype(dtype)
x = rand(n).astype(dtype)
func, = get_blas_funcs(('trsv',), dtype=dtype)
y1 = func(a=A, x=x)
y2 = solve(triu(A), x)
assert_array_almost_equal(y1, y2)
y1 = func(a=A, x=x, lower=1)
y2 = solve(tril(A), x)
assert_array_almost_equal(y1, y2)
y1 = func(a=A, x=x, diag=1)
A[arange(n), arange(n)] = dtype(1)
y2 = solve(triu(A), x)
assert_array_almost_equal(y1, y2)
y1 = func(a=A, x=x, diag=1, trans=1)
y2 = solve(triu(A).T, x)
assert_array_almost_equal(y1, y2)
y1 = func(a=A, x=x, diag=1, trans=2)
y2 = solve(triu(A).conj().T, x)
assert_array_almost_equal(y1, y2)
def test_dare(self):
A = matrix([[-0.6, 0],[-0.1, -0.4]])
Q = matrix([[2, 1],[1, 0]])
B = matrix([[2, 1],[0, 1]])
R = matrix([[1, 0],[0, 1]])
X,L,G = dare(A,B,Q,R)
# print("The solution obtained is", X)
assert_array_almost_equal(
A.T * X * A - X -
A.T * X * B * solve(B.T * X * B + R, B.T * X * A) + Q, zeros((2,2)))
assert_array_almost_equal(solve(B.T * X * B + R, B.T * X * A), G)
# check for stable closed loop
lam = eigvals(A - B * G)
assert_array_less(abs(lam), 1.0)
A = matrix([[1, 0],[-1, 1]])
Q = matrix([[0, 1],[1, 1]])
B = matrix([[1],[0]])
R = 2
X,L,G = dare(A,B,Q,R)
# print("The solution obtained is", X)
assert_array_almost_equal(
A.T * X * A - X -
A.T * X * B * solve(B.T * X * B + R, B.T * X * A) + Q, zeros((2,2)))
assert_array_almost_equal(B.T * X * A / (B.T * X * B + R), G)
# check for stable closed loop
lam = eigvals(A - B * G)
assert_array_less(abs(lam), 1.0)
def test_tpsv(self):
seed(1234)
for ind, dtype in enumerate(DTYPES):
n = 10
x = rand(n).astype(dtype)
# Upper triangular array
A = triu(rand(n, n)) if ind < 2 else triu(rand(n, n)+rand(n, n)*1j)
A += eye(n)
# Form the packed storage
c, r = tril_indices(n)
Ap = A[r, c]
func, = get_blas_funcs(('tpsv',), dtype=dtype)
y1 = func(n=n, ap=Ap, x=x)
y2 = solve(A, x)
assert_array_almost_equal(y1, y2)
y1 = func(n=n, ap=Ap, x=x, diag=1)
A[arange(n), arange(n)] = dtype(1)
y2 = solve(A, x)
assert_array_almost_equal(y1, y2)
y1 = func(n=n, ap=Ap, x=x, diag=1, trans=1)
y2 = solve(A.T, x)
assert_array_almost_equal(y1, y2)
y1 = func(n=n, ap=Ap, x=x, diag=1, trans=2)
y2 = solve(A.conj().T, x)
assert_array_almost_equal(y1, y2)
def conditional(self,xb):
""" conditional(self,xb)
Calculates the mean and covariance of the conditional distribution
when the variables xb are observed.
Input: xb - The observed variables. It is assumed that the observed variables occupy the
last positions in the array of random variables, i.e. if x is the random variable
associated with the object, then it is partioned as x = [xa,xb]^T.
Output: mean_a_given_b - mean of the conditional distribution
cov_a_given_b - covariance of the conditional distribution
"""
xb = np.array(xb,ndmin=1)
nb = len(xb)
n_rand_var = len(self.mean)
if nb >= n_rand_var:
raise ValueError('The conditional vector should be smaller than the random variable!')
mean = self.mean
cov = self.cov
# Partition the mean and covariance
na = n_rand_var - nb
mean_a = self.mean[:na]
mean_b = self.mean[na:]
cov_a = self.cov[:na,:na]
cov_b = self.cov[na:,na:]
cov_ab = self.cov[:na,na:]
#Calculate the conditional mean and covariance
mean_a_given_b = mean_a.flatten() + np.dot(cov_ab,solve(cov_b,xb.flatten()-mean_b.flatten()))
cov_a_given_b = cov_a - np.dot(cov_ab,solve(cov_b,cov_ab.transpose()))
return mean_a_given_b, cov_a_given_b
def HODLRDirectSolver( K, F, minSize, tol ):
n = K.shape[0]
# Base Case
if (n <= minSize):
return li.solve(K, F)
else:
p = (n/2)
U1,V1 = lowRankApprox(K[:p,p:], -1, tol)
U2,V2 = lowRankApprox(K[p:,:p], -1, tol)
r1 = U1.shape[1]
r2 = U2.shape[1]
d1c1 = HODLRDirectSolver(K[:p,:p], np.concatenate((U1, F[:p,:]),1), minSize, tol)
d2c2 = HODLRDirectSolver(K[p:,p:], np.concatenate((U2, F[p:,:]),1), minSize, tol)
d1 = d1c1[:,:r1]
c1 = d1c1[:,r1:]
d2 = d2c2[:,:r2]
c2 = d2c2[:,r2:]
S1 = np.concatenate((np.eye(r2), np.dot(V2, d1)),1)
S2 = np.concatenate((np.dot(V1,d2), np.eye(r1)),1)
S = np.concatenate((S1,S2),0)
y = li.solve(S, np.concatenate((np.dot(V2,c1), np.dot(V1,c2)),0))
y1 = y[:r2,:]
y2 = y[r2:,:]
x1 = c1 - np.dot(d1,y2)
x2 = c2 - np.dot(d2,y1)
return np.concatenate((x1, x2), 0)
def computeProjectionVectors( self, P, L, U ) : eK = matrix( identity( self.dim, float64 )[ 0: ,( self.dim - 1 ) ] ).T U = matrix(U, float64) U[ self.dim - 1, self.dim - 1 ] = 1.0 # Sergio: I added this exception because in rare cases, the matrix # U is singular, which gives rise to a LinAlgError. try: x1 = matrix( solve( U, eK ), float64 ) except LinAlgError: print "Matrix U was singular, so we input a fake x1\n" print "U: ", U x1 = matrix(ones(self.dim)) #print "x1", x1 del U LT = matrix( L, float64, copy=False ).T PT = matrix( P, float64, copy=False ).T x2 = matrix( solve( LT*PT, eK ), float64 ) del L del P del LT del PT del eK return ( x1, x2 )
def computeHgg(self,f,g,Rg,Rgy,Rgyz):
# set up self.Hgg with correspond matrix
# also should give other two derivatives
D = np.exp(g)*self.D0
gp = np.gradient(g,self.dy)
jumpsq = np.zeros(self.bins)
driftsq = np.zeros(self.bins)
driftsum = np.zeros(self.bins)
for k in np.arange(self.bins):
jumpsq[k] = sum(self.jumps[self.binpos==k]**2)
driftsq[k] = sum( (D[k]*(f[k]+self.m[self.binpos==k])+gp[k]*D[k])**2)*self.dt*self.dt
driftsum[k] = sum( D[k]*(f[k]+self.m[self.binpos==k])+gp[k]*D[k])*self.dt
a1 = (jumpsq + driftsq)/D/4.0/self.dt
a2 = driftsum/2.
a4 = D*self.dt/2.
A1 = np.tile(a1,(self.bins,1))*Rg+np.tile(a2,(self.bins,1))*-Rgy
A2 = np.tile(a2,(self.bins,1))*Rg+np.tile(a4,(self.bins,1))*-Rgy
A3 = np.tile(a1,(self.bins,1))*Rgy+np.tile(a2,(self.bins,1))*Rgyz
A4 = np.tile(a2,(self.bins,1))*Rgy+np.tile(a4,(self.bins,1))*Rgyz
A5 = np.tile(a1,(self.bins,1))*Rgy+np.tile(a2,(self.bins,1))*Rgyz
A6 = np.tile(a2,(self.bins,1))*Rgy+np.tile(a4,(self.bins,1))*Rgyz
M = np.vstack((Rg,Rgy))
A = np.vstack((np.hstack((A1,A2)) ,np.hstack((A3,A4)) ))
Lambda = solve(np.eye(self.bins*2)+A.astype(np.float128),M)
Hggy = Lambda[self.bins:,:]
Hgg = Lambda[:self.bins,:]
Hggyz = solve(np.eye(self.bins)+A6,Rgyz-A5.dot(Hggy.T))
return 0.5*(Hgg+Hgg.T),Hggy,0.5*(Hggyz+Hggyz.T) # first and third should be symmetric. get rid of small errors
def invert_low_rank(Ainv, U, C, V, diag=False):
"""
Invert the matrix (A+UCV) where A^{-1} is known and C is lower rank than A
Let N be rank of A and K be rank of C where K << N
Then we can write the inverse,
(A+UCV)^{-1} = A^{-1} - A^{-1}U (C^{-1}+VA^{-1}U)^{-1} VA^{-1}
:param Ainv: NxN matrix A^{-1}
:param U: NxK matrix
:param C: KxK invertible matrix
:param V: KxN matrix
:return:
"""
N,K = U.shape
Cinv = inv(C)
if diag:
assert Ainv.shape == (N,)
tmp1 = einsum('ij,j,jk->ik', V, Ainv, U)
tmp2 = einsum('ij,j->ij', V, Ainv)
tmp3 = solve(Cinv + tmp1, tmp2)
# tmp4 = -U.dot(tmp3)
tmp4 = -einsum('ij,jk->ik', U, tmp3)
tmp4[diag_indices(N)] += 1
return einsum('i,ij->ij', Ainv, tmp4)
else:
tmp = solve(Cinv + V.dot(Ainv).dot(U), V.dot(Ainv))
return Ainv - Ainv.dot(U).dot(tmp)
def b_operator(self, P):
r"""
The B operator, mapping P into
.. math::
B(P) := R - beta^2 A'PB(Q + beta B'PB)^{-1}B'PA + beta A'PA
and also returning
.. math::
F := (Q + beta B'PB)^{-1} beta B'PA
Parameters
----------
P : array_like(float, ndim=2)
A matrix that should be n x n
Returns
-------
F : array_like(float, ndim=2)
The F matrix as defined above
new_p : array_like(float, ndim=2)
The matrix P after applying the B operator
"""
A, B, Q, R, beta = self.A, self.B, self.Q, self.R, self.beta
S1 = Q + beta * dot(B.T, dot(P, B))
S2 = beta * dot(B.T, dot(P, A))
S3 = beta * dot(A.T, dot(P, A))
F = solve(S1, S2)
new_P = R - dot(S2.T, solve(S1, S2)) + S3
return F, new_P
def _solve_cholesky_kernel(K, y, alpha, sample_weight=None, copy=False):
# dual_coef = inv(X X^t + alpha*Id) y
n_samples = K.shape[0]
n_targets = y.shape[1]
if copy:
K = K.copy()
alpha = np.atleast_1d(alpha)
one_alpha = (alpha == alpha[0]).all()
has_sw = isinstance(sample_weight, np.ndarray) \
or sample_weight not in [1.0, None]
if has_sw:
# Unlike other solvers, we need to support sample_weight directly
# because K might be a pre-computed kernel.
sw = np.sqrt(np.atleast_1d(sample_weight))
y = y * sw[:, np.newaxis]
K *= np.outer(sw, sw)
if one_alpha:
# Only one penalty, we can solve multi-target problems in one time.
K.flat[::n_samples + 1] += alpha[0]
try:
# Note: we must use overwrite_a=False in order to be able to
# use the fall-back solution below in case a LinAlgError
# is raised
dual_coef = linalg.solve(K, y, sym_pos=True,
overwrite_a=False)
except np.linalg.LinAlgError:
warnings.warn("Singular matrix in solving dual problem. Using "
"least-squares solution instead.")
dual_coef = linalg.lstsq(K, y)[0]
# K is expensive to compute and store in memory so change it back in
# case it was user-given.
K.flat[::n_samples + 1] -= alpha[0]
if has_sw:
dual_coef *= sw[:, np.newaxis]
return dual_coef
else:
# One penalty per target. We need to solve each target separately.
dual_coefs = np.empty([n_targets, n_samples])
for dual_coef, target, current_alpha in zip(dual_coefs, y.T, alpha):
K.flat[::n_samples + 1] += current_alpha
dual_coef[:] = linalg.solve(K, target, sym_pos=True,
overwrite_a=False).ravel()
K.flat[::n_samples + 1] -= current_alpha
if has_sw:
dual_coefs *= sw[np.newaxis, :]
return dual_coefs.T
def LU_solve(A,rhs,p): # Solve with straight LU (M,N) = A.shape C = dot(A.T,A) + p*eye(N) P,L,U = sp.linalg.lu(C) y = linalg.solve(dot(P,L),rhs) x = linalg.solve(U,y) return x
def _woodbury_algorithm(A, ur, ll, b, k):
'''
Solve a cyclic banded linear system with upper right
and lower blocks of size ``(k-1) / 2`` using
the Woodbury formula
Parameters
----------
A : 2-D array, shape(k, n)
Matrix of diagonals of original matrix(see
``solve_banded`` documentation).
ur : 2-D array, shape(bs, bs)
Upper right block matrix.
ll : 2-D array, shape(bs, bs)
Lower left block matrix.
b : 1-D array, shape(n,)
Vector of constant terms of the system of linear equations.
k : int
B-spline degree.
Returns
-------
c : 1-D array, shape(n,)
Solution of the original system of linear equations.
Notes
-----
This algorithm works only for systems with banded matrix A plus
a correction term U @ V.T, where the matrix U @ V.T gives upper right
and lower left block of A
The system is solved with the following steps:
1. New systems of linear equations are constructed:
A @ z_i = u_i,
u_i - columnn vector of U,
i = 1, ..., k - 1
2. Matrix Z is formed from vectors z_i:
Z = [ z_1 | z_2 | ... | z_{k - 1} ]
3. Matrix H = (1 + V.T @ Z)^{-1}
4. The system A' @ y = b is solved
5. x = y - Z @ (H @ V.T @ y)
Also, ``n`` should be greater than ``k``, otherwise corner block
elements will intersect with diagonals.
Examples
--------
Consider the case of n = 8, k = 5 (size of blocks - 2 x 2).
The matrix of a system: U: V:
x x x * * a b a b 0 0 0 0 1 0
x x x x * * c 0 c 0 0 0 0 0 1
x x x x x * * 0 0 0 0 0 0 0 0
* x x x x x * 0 0 0 0 0 0 0 0
* * x x x x x 0 0 0 0 0 0 0 0
d * * x x x x 0 0 d 0 1 0 0 0
e f * * x x x 0 0 e f 0 1 0 0
References
----------
.. [1] William H. Press, Saul A. Teukolsky, William T. Vetterling
and Brian P. Flannery, Numerical Recipes, 2007, Section 2.7.3
'''
k_mod = k - k % 2
bs = int((k - 1) / 2) + (k + 1) % 2
n = A.shape[1] + 1
U = np.zeros((n - 1, k_mod))
VT = np.zeros((k_mod, n - 1)) # V transpose
# upper right block
U[:bs, :bs] = ur
VT[np.arange(bs), np.arange(bs) - bs] = 1
# lower left block
U[-bs:, -bs:] = ll
VT[np.arange(bs) - bs, np.arange(bs)] = 1
Z = solve_banded((bs, bs), A, U)
H = solve(np.identity(k_mod) + VT @ Z, np.identity(k_mod))
y = solve_banded((bs, bs), A, b)
c = y - Z @ (H @ (VT @ y))
return c
def solve(self, v, tol=0):
"""Evaluate w = M^-1 v"""
if self.collapsed is not None:
return solve(self.collapsed, v)
return LowRankMatrix._solve(v, self.alpha, self.cs, self.ds)
def cheby_bvp_test(xa,
xb,
pts,
aderiv,
ideriv,
bderiv,
va,
vb,
F,
S,
xextra=None):
"""Test the convergence of the chebyshev spectral solver for a boundary
value problem
args:
xa - float, value of left (lower) domain boundary
xb - float, value of right (higher) domain boundary
pts - iterable, different numbers of points to use
aderiv - int, order of derivative for left boundary condition
ideriv - int, order of derivative for internal nodes
(must be 1 or 2)
bderiv - int, order of derivative for right boundary condition
va - value for the left boundary
vb - value for the right boundary
F - solution function
S - source function (whichever derivative of F)
optional args:
xextra - extra points within the domain to include as
collocation points
returns:
err - maximum relative error for each of the numerical solutions
xs - array of grid points for each solution
qexs - array of exact solution values for each solution
qsps - array of numerical solution values for each solution"""
#number of resolutions to test
L = len(pts)
#exact and spectral solution arrays
qexs, qsps = [], []
#grids
xs = []
#max relative error
err = np.empty((L, ))
#do the tests
for i in range(L):
#set up the grid/solver
xhat, theta, x, A = cheby_bvp_setup(xa,
xb,
pts[i],
aderiv,
ideriv,
bderiv,
xextra=xextra)
#create the source array
b = np.empty((len(x), ))
b[0] = va
b[1:-1] = np.array([S(_x) for _x in x[1:-1]])
b[-1] = vb
#solve for the coefficients of the Cheb expansion
coef = solve(A, b)
#evaluate the expansion
qsp = cheby_hat_sum(xhat, coef)
#evaluate the exact solution
qex = np.array([F(_x) for _x in x])
#store the error
err[i] = np.max(np.abs(qsp - qex) / np.abs(qex.max()))
#store the solution stuff
qexs.append(qex)
qsps.append(qsp)
xs.append(x)
return (err, xs, qexs, qsps)
def test_cantilever_beam(plot=False):
n = 100
L = 3 # total size of the beam along x
# Material Lastrobe Lescalloy
E = 203.e9 # Pa
rho = 7.83e3 # kg/m3
x = np.linspace(0, L, n)
# path
y = np.ones_like(x)
# tapered properties
b_root = 0.05 # m
b_tip = b_root # m
h_root = 0.05 # m
h_tip = h_root # m
A_root = h_root * b_root
A_tip = h_tip * b_tip
Izz_root = b_root * h_root**3 / 12
Izz_tip = b_tip * h_tip**3 / 12
# getting nodes
ncoords = np.vstack((x, y)).T
nids = 1 + np.arange(ncoords.shape[0])
nid_pos = dict(zip(nids, np.arange(len(nids))))
n1s = nids[0:-1]
n2s = nids[1:]
K = np.zeros((3 * n, 3 * n))
M = np.zeros((3 * n, 3 * n))
beams = []
for n1, n2 in zip(n1s, n2s):
pos1 = nid_pos[n1]
pos2 = nid_pos[n2]
x1, y1 = ncoords[pos1]
x2, y2 = ncoords[pos2]
A1 = A_root + (A_tip - A_root) * x1 * A_tip / (L - 0)
A2 = A_root + (A_tip - A_root) * x2 * A_tip / (L - 0)
Izz1 = Izz_root + (Izz_tip - Izz_root) * x1 * Izz_tip / (L - 0)
Izz2 = Izz_root + (Izz_tip - Izz_root) * x2 * Izz_tip / (L - 0)
beam = Beam2D()
beam.n1 = n1
beam.n2 = n2
beam.E = E
beam.rho = rho
beam.A1, beam.A2 = A1, A2
beam.Izz1, beam.Izz2 = Izz1, Izz2
update_K(beam, nid_pos, ncoords, K)
update_M(beam, nid_pos, M)
beams.append(beam)
# applying boundary conditions
bk = np.zeros(K.shape[0], dtype=bool) #array to store known DOFs
check = np.isclose(x, 0.) # locating node at root
# clamping at root
for i in range(DOF):
bk[i::DOF] = check
bu = ~bk # same as np.logical_not, defining unknown DOFs
# sub-matrices corresponding to unknown DOFs
Kuu = K[bu, :][:, bu]
Muu = M[bu, :][:, bu]
# test
Fy = 700
f = np.zeros(K.shape[0])
f[-2] = Fy
fu = f[bu]
# solving
uu = solve(Kuu, fu)
# vector u containing displacements for all DOFs
u = np.zeros(K.shape[0], dtype=float)
u[bu] = uu
u = u.reshape(n, -1)
a = L
deflection = Fy * a**3 / (3 * E * Izz_root) * (1 + 3 * (L - a) / 2 * a)
print('Theoretical deflection', deflection)
print('Numerical deflection', u[-1, 1])
assert np.isclose(deflection, u[-1, 1])
if plot:
# plotting
import matplotlib
matplotlib.use('TkAgg')
import matplotlib.pyplot as plt
plt.plot(x, y, '-k')
plt.plot(x + u[:, 0], y + u[:, 1], '--r')
plt.show()
def subsolv(m,n,epsimin,low,upp,alfa,beta,p0,q0,P,Q,a0,a,b,c,d):
"""
This function subsolv solves the MMA subproblem:
minimize SUM[p0j/(uppj-xj) +q0j/(xj-lowj)]+a0*z+SUM[ci*yi+0.5*di*(yi)^2],
subject to SUM[pij/(uppj-xj) + qij/(xj-lowj)] - ai*z - yi <= bi,
alfaj <= xj <= betaj, yi >= 0, z >= 0.
Input: m, n, low, upp, alfa, beta, p0, q0, P, Q, a0, a, b, c, d.
Output: xmma,ymma,zmma, slack variables and Lagrange multiplers.
"""
een = np.ones((n,1))
eem = np.ones((m,1))
epsi = 1
epsvecn = epsi*een
epsvecm = epsi*eem
x = 0.5*(alfa+beta)
y = eem.copy()
z = np.array([[1.0]])
lam = eem.copy()
xsi = een/(x-alfa)
xsi = np.maximum(xsi,een)
eta = een/(beta-x)
eta = np.maximum(eta,een)
mu = np.maximum(eem,0.5*c)
zet = np.array([[1.0]])
s = eem.copy()
itera = 0
# Start while epsi>epsimin
while epsi > epsimin:
epsvecn = epsi*een
epsvecm = epsi*eem
ux1 = upp-x
xl1 = x-low
ux2 = ux1*ux1
xl2 = xl1*xl1
uxinv1 = een/ux1
xlinv1 = een/xl1
plam = p0+np.dot(P.T,lam)
qlam = q0+np.dot(Q.T,lam)
gvec = np.dot(P,uxinv1)+np.dot(Q,xlinv1)
dpsidx = plam/ux2-qlam/xl2
rex = dpsidx-xsi+eta
rey = c+d*y-mu-lam
rez = a0-zet-np.dot(a.T,lam)
relam = gvec-a*z-y+s-b
rexsi = xsi*(x-alfa)-epsvecn
reeta = eta*(beta-x)-epsvecn
remu = mu*y-epsvecm
rezet = zet*z-epsi
res = lam*s-epsvecm
residu1 = np.concatenate((rex, rey, rez), axis = 0)
residu2 = np.concatenate((relam, rexsi, reeta, remu, rezet, res), axis = 0)
residu = np.concatenate((residu1, residu2), axis = 0)
residunorm = np.sqrt((np.dot(residu.T,residu)).item())
residumax = np.max(np.abs(residu))
ittt = 0
# Start while (residumax>0.9*epsi) and (ittt<200)
while (residumax > 0.9*epsi) and (ittt < 200):
ittt = ittt+1
itera = itera+1
ux1 = upp-x
xl1 = x-low
ux2 = ux1*ux1
xl2 = xl1*xl1
ux3 = ux1*ux2
xl3 = xl1*xl2
uxinv1 = een/ux1
xlinv1 = een/xl1
uxinv2 = een/ux2
xlinv2 = een/xl2
plam = p0+np.dot(P.T,lam)
qlam = q0+np.dot(Q.T,lam)
gvec = np.dot(P,uxinv1)+np.dot(Q,xlinv1)
GG = (diags(uxinv2.flatten(),0).dot(P.T)).T-(diags\
(xlinv2.flatten(),0).dot(Q.T)).T
dpsidx = plam/ux2-qlam/xl2
delx = dpsidx-epsvecn/(x-alfa)+epsvecn/(beta-x)
dely = c+d*y-lam-epsvecm/y
delz = a0-np.dot(a.T,lam)-epsi/z
dellam = gvec-a*z-y-b+epsvecm/lam
diagx = plam/ux3+qlam/xl3
diagx = 2*diagx+xsi/(x-alfa)+eta/(beta-x)
diagxinv = een/diagx
diagy = d+mu/y
diagyinv = eem/diagy
diaglam = s/lam
diaglamyi = diaglam+diagyinv
# Start if m<n
if m < n:
blam = dellam+dely/diagy-np.dot(GG,(delx/diagx))
bb = np.concatenate((blam,delz),axis = 0)
Alam = np.asarray(diags(diaglamyi.flatten(),0) \
+(diags(diagxinv.flatten(),0).dot(GG.T).T).dot(GG.T))
AAr1 = np.concatenate((Alam,a),axis = 1)
AAr2 = np.concatenate((a,-zet/z),axis = 0).T
AA = np.concatenate((AAr1,AAr2),axis = 0)
solut = solve(AA,bb)
dlam = solut[0:m]
dz = solut[m:m+1]
dx = -delx/diagx-np.dot(GG.T,dlam)/diagx
else:
diaglamyiinv = eem/diaglamyi
dellamyi = dellam+dely/diagy
Axx = np.asarray(diags(diagx.flatten(),0) \
+(diags(diaglamyiinv.flatten(),0).dot(GG).T).dot(GG))
azz = zet/z+np.dot(a.T,(a/diaglamyi))
axz = np.dot(-GG.T,(a/diaglamyi))
bx = delx+np.dot(GG.T,(dellamyi/diaglamyi))
bz = delz-np.dot(a.T,(dellamyi/diaglamyi))
AAr1 = np.concatenate((Axx,axz),axis = 1)
AAr2 = np.concatenate((axz.T,azz),axis = 1)
AA = np.concatenate((AAr1,AAr2),axis = 0)
bb = np.concatenate((-bx,-bz),axis = 0)
solut = solve(AA,bb)
dx = solut[0:n]
dz = solut[n:n+1]
dlam = np.dot(GG,dx)/diaglamyi-dz*(a/diaglamyi)\
+dellamyi/diaglamyi
# End if m<n
dy = -dely/diagy+dlam/diagy
dxsi = -xsi+epsvecn/(x-alfa)-(xsi*dx)/(x-alfa)
deta = -eta+epsvecn/(beta-x)+(eta*dx)/(beta-x)
dmu = -mu+epsvecm/y-(mu*dy)/y
dzet = -zet+epsi/z-zet*dz/z
ds = -s+epsvecm/lam-(s*dlam)/lam
xx = np.concatenate((y,z,lam,xsi,eta,mu,zet,s),axis = 0)
dxx = np.concatenate((dy,dz,dlam,dxsi,deta,dmu,dzet,ds),axis = 0)
#
stepxx = -1.01*dxx/xx
stmxx = np.max(stepxx)
stepalfa = -1.01*dx/(x-alfa)
stmalfa = np.max(stepalfa)
stepbeta = 1.01*dx/(beta-x)
stmbeta = np.max(stepbeta)
stmalbe = max(stmalfa,stmbeta)
stmalbexx = max(stmalbe,stmxx)
stminv = max(stmalbexx,1.0)
steg = 1.0/stminv
#
xold = x.copy()
yold = y.copy()
zold = z.copy()
lamold = lam.copy()
xsiold = xsi.copy()
etaold = eta.copy()
muold = mu.copy()
zetold = zet.copy()
sold = s.copy()
#
itto = 0
resinew = 2*residunorm
# Start: while (resinew>residunorm) and (itto<50)
while (resinew > residunorm) and (itto < 50):
itto = itto+1
x = xold+steg*dx
y = yold+steg*dy
z = zold+steg*dz
lam = lamold+steg*dlam
xsi = xsiold+steg*dxsi
eta = etaold+steg*deta
mu = muold+steg*dmu
zet = zetold+steg*dzet
s = sold+steg*ds
ux1 = upp-x
xl1 = x-low
ux2 = ux1*ux1
xl2 = xl1*xl1
uxinv1 = een/ux1
xlinv1 = een/xl1
plam = p0+np.dot(P.T,lam)
qlam = q0+np.dot(Q.T,lam)
gvec = np.dot(P,uxinv1)+np.dot(Q,xlinv1)
dpsidx = plam/ux2-qlam/xl2
rex = dpsidx-xsi+eta
rey = c+d*y-mu-lam
rez = a0-zet-np.dot(a.T,lam)
relam = gvec-np.dot(a,z)-y+s-b
rexsi = xsi*(x-alfa)-epsvecn
reeta = eta*(beta-x)-epsvecn
remu = mu*y-epsvecm
rezet = np.dot(zet,z)-epsi
res = lam*s-epsvecm
residu1 = np.concatenate((rex,rey,rez),axis = 0)
residu2 = np.concatenate((relam,rexsi,reeta,remu,rezet,res), \
axis = 0)
residu = np.concatenate((residu1,residu2),axis = 0)
resinew = np.sqrt(np.dot(residu.T,residu))
steg = steg/2
# End: while (resinew>residunorm) and (itto<50)
residunorm = resinew.copy()
residumax = max(abs(residu))
steg = 2*steg
# End: while (residumax>0.9*epsi) and (ittt<200)
epsi = 0.1*epsi
# End: while epsi>epsimin
xmma = x.copy()
ymma = y.copy()
zmma = z.copy()
lamma = lam
xsimma = xsi
etamma = eta
mumma = mu
zetmma = zet
smma = s
# Return values
return xmma,ymma,zmma,lamma,xsimma,etamma,mumma,zetmma,smma
def fdm_poisson_1d(N, bcs, pm):
'''The simplest 1D diffusion implementation.'''
# Gridsize
h = 1.0/N
# Define grid points
x = np.linspace(0, 1, N+1)
# Define zero matrix A of right size and insert
# non zero entries
A = np.zeros((N+1, N+1))
eps = pm['eps']
# Define tridiagonal part of A
for i in range(1, N):
A[i, i-1] = eps + h
A[i, i] = -2*eps - h
A[i, i+1] = eps
print("Hello Again")
# Compute rhs for f = sin(2 pi x)
# F = -h**2*np.sin(2*np.pi*x)
F = -h**2*x
# Now adapt matrix and rhs according to bc data
# Left boundary
bc0 = bcs[0]
if bc0[0] == "D":
A[0, 0] = 1
F[0] = bc0[1]
elif bc0[0] == "N":
# Apply a first order difference operator
A[0, 0] = 1
A[0, 1] = -1
F[0] = h*bc0[1]
# Should we add an improved variant?
# Right boundary
bc1 = bcs[1]
if bc1[0] == "D":
A[N, N] = 1
F[N] = bc1[1]
elif bc1[0] == "N":
# Apply a first order difference operator
A[N, N] = 1
A[N, N-1] = -1
F[N] = h*bc1[1]
# Solve AU = F
# (We will introduce a sparse solver when we look at 2D problems)
U = la.solve(A, F)
# Compute real solution and error at grid points
x_hr = np.linspace(0, 1, N+1)
u = 1/(2*np.pi)**2*np.sin(2*np.pi*x_hr)
err = np.abs(u - U)
print("Error |U - u|")
print(err)
print("Error max |U - u| ")
print(err.max())
# Clear figure first
plt.clf()
# Plot solution on a high resolution grid
# plt.plot(x_hr, u, "+-b")
# Plot discrete solution on chosen discretization grid
plt.plot(x, U, "x-r")
# Show figure (for non inline plotting)
plt.show()
def time_solve(self, size, contig, module):
if module == 'numpy':
nl.solve(self.a, self.b)
else:
sl.solve(self.a, self.b)
import numpy as np from scipy import linalg A = np.array([[3, 2, 0], [1, -1, 0], [0, 5, 1]]) b = np.array([2, 4, -1]) x = linalg.solve(A, b) print(x)
def asjacobian(J):
"""
Convert given object to one suitable for use as a Jacobian.
"""
spsolve = scipy.sparse.linalg.spsolve
if isinstance(J, Jacobian):
return J
elif inspect.isclass(J) and issubclass(J, Jacobian):
return J()
elif isinstance(J, np.ndarray):
if J.ndim > 2:
raise ValueError('array must have rank <= 2')
J = np.atleast_2d(np.asarray(J))
if J.shape[0] != J.shape[1]:
raise ValueError('array must be square')
return Jacobian(matvec=lambda v: dot(J, v),
rmatvec=lambda v: dot(J.conj().T, v),
solve=lambda v: solve(J, v),
rsolve=lambda v: solve(J.conj().T, v),
dtype=J.dtype,
shape=J.shape)
elif scipy.sparse.isspmatrix(J):
if J.shape[0] != J.shape[1]:
raise ValueError('matrix must be square')
return Jacobian(matvec=lambda v: J * v,
rmatvec=lambda v: J.conj().T * v,
solve=lambda v: spsolve(J, v),
rsolve=lambda v: spsolve(J.conj().T, v),
dtype=J.dtype,
shape=J.shape)
elif hasattr(J, 'shape') and hasattr(J, 'dtype') and hasattr(J, 'solve'):
return Jacobian(matvec=getattr(J, 'matvec'),
rmatvec=getattr(J, 'rmatvec'),
solve=J.solve,
rsolve=getattr(J, 'rsolve'),
update=getattr(J, 'update'),
setup=getattr(J, 'setup'),
dtype=J.dtype,
shape=J.shape)
elif callable(J):
# Assume it's a function J(x) that returns the Jacobian
class Jac(Jacobian):
def update(self, x, F):
self.x = x
def solve(self, v, tol=0):
m = J(self.x)
if isinstance(m, np.ndarray):
return solve(m, v)
elif scipy.sparse.isspmatrix(m):
return spsolve(m, v)
else:
raise ValueError("Unknown matrix type")
def matvec(self, v):
m = J(self.x)
if isinstance(m, np.ndarray):
return dot(m, v)
elif scipy.sparse.isspmatrix(m):
return m * v
else:
raise ValueError("Unknown matrix type")
def rsolve(self, v, tol=0):
m = J(self.x)
if isinstance(m, np.ndarray):
return solve(m.conj().T, v)
elif scipy.sparse.isspmatrix(m):
return spsolve(m.conj().T, v)
else:
raise ValueError("Unknown matrix type")
def rmatvec(self, v):
m = J(self.x)
if isinstance(m, np.ndarray):
return dot(m.conj().T, v)
elif scipy.sparse.isspmatrix(m):
return m.conj().T * v
else:
raise ValueError("Unknown matrix type")
return Jac()
elif isinstance(J, str):
return dict(broyden1=BroydenFirst,
broyden2=BroydenSecond,
anderson=Anderson,
diagbroyden=DiagBroyden,
linearmixing=LinearMixing,
excitingmixing=ExcitingMixing,
krylov=KrylovJacobian)[J]()
else:
raise TypeError('Cannot convert object to a Jacobian')
def _update_code_slow(X, subset, alpha, learning_rate,
offset,
Q, stat,
impute, debug):
"""Compute code for a mini-batch and update algorithm statistics accordingly
Parameters
----------
X: ndarray, (batch_size, len_subset)
Mini-batch of masked data to perform the update from
subset: ndarray (len_subset),
Mask used on X
alpha: float,
Regularization of the code (ridge penalty)
learning_rate: float in [0.5, 1],
Controls the sequence of weights in
the update of the surrogate function
offset: float,
Offset in the sequence of weights in
the update of the surrogate function
Q: ndarray (n_components, n_features):
Dictionary to perform ridge regression
stat: DictMFStats,
Statistics kept by the algorithm, to be updated by the function
impute: boolean,
Online update of the Gram matrix (Experimental)
debug: boolean,
Keeps track of the surrogate loss function
Returns
-------
P: ndarray,
Code for the mini-batch X
"""
batch_size, n_cols = X.shape
n_components = Q.shape[0]
Q_subset = Q[:, subset]
w_A, w_B = _get_weights(subset, stat.counter, batch_size,
learning_rate, offset)
stat.counter[0] += batch_size
stat.counter[subset + 1] += batch_size
if impute:
stat.T[:, 0] -= stat.T[:, subset + 1].sum(axis=1)
Qx = stat.T[:, 0][:, np.newaxis]
Qx += Q_subset.dot(X.T)
stat.T[:, subset + 1] = Q_subset * X.mean(axis=0)
stat.T[:, 0] += stat.T[:, subset + 1].sum(axis=1)
stat.G.flat[::n_components + 1] += alpha
else:
Qx = np.dot(Q_subset, X.T)
G = np.dot(Q_subset, Q_subset.T)
G.flat[::n_components + 1] += alpha
P = linalg.solve(G, Qx, sym_pos=True, overwrite_a=True, check_finite=False)
if debug:
dict_loss = .5 * np.sum(Q.dot(Q.T) * stat.A) - np.sum(Q * stat.B)
stat.loss_indep *= (1 - w_A)
stat.loss_indep += (.5 * np.sum(X ** 2) +
alpha * np.sum(P ** 2)) * w_A / batch_size
stat.loss[stat.n_iter] = stat.loss_indep + dict_loss
stat.A *= 1 - w_A
stat.A += P.dot(P.T) * w_A / batch_size
stat.B[:, subset] *= 1 - w_B
stat.B[:, subset] += P.dot(X) * w_B / batch_size
return P.T
def fit(self, X, t):
Xtil = np.c_[np.ones(X.shape[0]), X]
A = np.dot(Xtil.T, Xtil)
b = np.dot(Xtil.T, t)
self.w_ = linalg.solve(A, b)
def train(modelState, X, W, plan):
'''
Creates a new query state object for a topic model based on side-information.
This contains all those estimated parameters that are specific to the actual
date being queried - this must be used in conjunction with a model state.
The parameters are
modelState - the model state with all the model parameters
X - the D x F matrix of side information vectors
W - the D x V matrix of word **count** vectors.
iterations - how long to iterate for
epsilon - currently ignored, in future, allows us to stop early.
logInterval - the interval between iterations where we calculate and display
the log-likelihood bound
plotInterval - the interval between iterations we we display the log-likelihood
bound values calcuated at each log-interval
fastButInaccurate - if true, we may use a psedo-inverse instead of an inverse
when solving for Y when the true inverse is unavailable.
This returns a tuple of new model-state and query-state. The latter object will
contain X and W and also
s - A D-dimensional vector describing the offset in our bound on the true value of ln sum_k e^theta_dk
lxi - A DxK matrix used in the above bound, containing the negative Jakkola function applied to the
quadratic term xi
lambda - the topics we've inferred for the current batch of documents
nu - the variance of topics we've inferred (independent)
'''
# Unpack the model state tuple for ease of use and maybe speed improvements
K, Q, F, P, T, A, varA, Y, omY, sigY, sigT, U, V, vocab, sigmaSq, alphaSq, kappaSq, tauSq = modelState.K, modelState.Q, modelState.F, modelState.P, modelState.T, modelState.A, modelState.varA, modelState.Y, modelState.omY, modelState.sigY, modelState.sigT, modelState.U, modelState.V, modelState.vocab, modelState.topicVar, modelState.featVar, modelState.lowTopicVar, modelState.lowFeatVar
iterations, epsilon, logCount, plot, plotFile, plotIncremental, fastButInaccurate = plan.iterations, plan.epsilon, plan.logFrequency, plan.plot, plan.plotFile, plan.plotIncremental, plan.fastButInaccurate
mu0 = 0.0001
if W.dtype.kind == 'i': # for the sparseScalorQuotientOfDot() method to work
W = W.astype(DTYPE)
# Get ready to plot the evolution of the likelihood, with multiplicative updates (e.g. 1, 2, 4, 8, 16, 32, ...)
if logCount > 0:
multiStepSize = np.power(iterations, 1. / logCount)
logIter = 1
elbos = []
likes = []
iters = []
else:
logIter = iterations + 1
lastVarBoundValue = -sys.float_info.max
# We'll need the total word count per doc, and total count of docs
docLen = np.squeeze(
np.asarray(W.sum(axis=1))
) # Force to a one-dimensional array for np.newaxis trick to work
D = len(docLen)
# No need to recompute this every time
if X.dtype != DTYPE:
X = X.astype(DTYPE)
XTX = X.T.dot(X)
# Identity matrices that occur
I_P = ssp.eye(P, P, 0, DTYPE)
I_Q = ssp.eye(Q, Q, 0, DTYPE)
I_QP = ssp.eye(Q * P, Q * P, 0, DTYPE)
I_F = ssp.eye(
F, F, 0, DTYPE, "csc"
) # X is CSR, XTX is consequently CSC, sparse inverse requires CSC
T_QP = sp_vec_trans_matrix(Y.shape)
# Assign initial values to the query parameters
expLmda = np.exp(rd.random((D, K)).astype(DTYPE))
nu = np.ones((D, K), DTYPE)
s = np.zeros((D, ), DTYPE)
lxi = negJakkola(np.ones((D, K), DTYPE))
# If we don't bother optimising either tau or sigma we can just do all this here once only
tsq = tauSq
ssq = sigmaSq
overTsq = 1. / tsq
overSsq = 1. / ssq
overTsqSsq = 1. / (tsq * ssq)
# TODO the inverse being almost always dense means that it might
# be faster to convert to dense and use the normal solver, despite
# the size constraints.
# varA = 1./K * sla.inv (overTsq * I_F + overSsq * XTX)
tI_sXTX = (overTsq * I_F + overSsq * XTX).todense()
omA = la.inv(tI_sXTX)
scaledWordCounts = W.copy()
for iteration in range(iterations):
# =============================================================
# E-Step
# Model dists are q(Theta|A;Lambda;nu) q(A|Y) q(Y) and q(Z)....
# Where lambda is the posterior mean of theta.
# =============================================================
# Y, sigY, omY
#
# If U'U is invertible, use inverse to convert Y to a Sylvester eqn
# which has a much, much faster solver. Recall update for Y is of the form
# Y + AYB = C where A = U'U, B = V'V and C=U'AV
#
VTV = V.T.dot(V)
UTU = U.T.dot(U)
sigy = la.inv(I_QP + overTsqSsq * np.kron(VTV, UTU))
_quickPrintElbo("E-Step: q(Y) [sigY]", iteration, X, W, K, Q, F, P, T,
A, omA, Y, omY, sigY, sigT, U, V, vocab, tau, sigma,
expLmda, nu, lxi, s, docLen)
Y = mu0 + np.reshape(overTsqSsq * sigy.dot(vec(U.T.dot(A).dot(V))),
(Q, P),
order='F')
_quickPrintElbo("E-Step: q(Y) [Mean]", iteration, X, W, K, Q, F, P, T,
A, omA, Y, omY, sigY, sigT, U, V, vocab, tau, sigma,
expLmda, nu, lxi, s, docLen)
# A
#
# So it's normally A = (UYV' + L'X) omA with omA = inv(t*I_F + s*XTX)
# so A inv(omA) = UYV' + L'X
# so inv(omA)' A' = VY'U' + X'L
# at which point we can use a built-in solve
#
# A = (overTsq * U.dot(Y).dot(V.T) + X.T.dot(expLmda).T).dot(omA)
lmda = np.log(expLmda, out=expLmda)
A = la.solve(tI_sXTX, X.T.dot(lmda) + V.dot(Y.T).dot(U.T)).T
np.exp(expLmda, out=expLmda)
_quickPrintElbo("E-Step: q(A)", iteration, X, W, K, Q, F, P, T, A, omA,
Y, omY, sigY, sigT, U, V, vocab, tau, sigma, expLmda,
nu, lxi, s, docLen)
# lmda_dk, nu_dk, s_d, and xi_dk
#
XAT = X.dot(A.T)
query (VbSideTopicModelState (K, Q, F, P, T, A, omA, Y, omY, sigY, sigT, U, V, vocab, tau, sigma), \
X, W, \
VbSideTopicQueryState(expLmda, nu, lxi, s, docLen), \
scaledWordCounts=scaledWordCounts, \
XAT = XAT, \
iterations=10, \
logInterval = 0, plotInterval = 0)
# =============================================================
# M-Step
# Parameters for the softmax bound: lxi and s
# The projection used for A: U and V
# The vocabulary : vocab
# The variances: tau, sigma
# =============================================================
# U
#
try:
U = A.dot(V).dot(Y.T).dot (la.inv( \
Y.dot(V.T).dot(V).dot(Y.T) \
+ (vec_transpose_csr(T_QP, P).T.dot(np.kron(I_QP, VTV)).dot(vec_transpose(T_QP.dot(sigy), P))).T
))
except np.linalg.linalg.LinAlgError as e:
print(str(e))
print("Ruh-ro")
# order of last line above reversed to handle numpy bug preventing dot product from dense to sparse
_quickPrintElbo("M-Step: U", iteration, X, W, K, Q, F, P, T, A, omA, Y,
omY, sigY, sigT, U, V, vocab, tau, sigma, expLmda, nu,
lxi, s, docLen)
# V
#
# Temporarily this requires that we re-order sigY until I've implemented a fortran order
# vec transpose in Cython
sigY = sigY.T.copy()
V = A.T.dot(U).dot(Y).dot (la.inv ( \
Y.T.dot(U.T).dot(U).dot(Y) \
+ vec_transpose (sigY, Q).T.dot(np.kron(I_QP, UTU).dot(vec_transpose(I_QP, Q))) \
))
_quickPrintElbo("M-Step: V", iteration, X, W, K, Q, F, P, T, A, omA, Y,
omY, sigY, sigT, U, V, vocab, tau, sigma, expLmda, nu,
lxi, s, docLen)
# vocab
#
factor = (scaledWordCounts.T.dot(expLmda)
).T # Gets materialized as a dense matrix...
vocab *= factor
normalizerows_ip(vocab)
_quickPrintElbo("M-Step: \u03A6", iteration, X, W, K, Q, F, P, T, A,
omA, Y, omY, sigY, sigT, U, V, vocab, tau, sigma,
expLmda, nu, lxi, s, docLen)
# =============================================================
# Handle logging of variational bound, likelihood, etc.
# =============================================================
if iteration == logIter:
modelState = VbSideTopicModelState(K, Q, F, P, T, A, omA, Y, omY,
sigY, sigT, U, V, vocab,
sigmaSq, alphaSq, kappaSq,
tauSq)
queryState = VbSideTopicQueryState(expLmda, nu, lxi, s, docLen)
elbo = varBound(modelState, queryState, X, W, None, XAT, XTX)
likely = log_likelihood(
modelState, X, W,
queryState) #recons_error(modelState, X, W, queryState)
elbos.append(elbo)
iters.append(iteration)
likes.append(likely)
print("Iteration %5d ELBO %15f Log-Likelihood %15f" %
(iteration, elbo, likely))
logIter = min(np.ceil(logIter * multiStepSize), iterations - 1)
if elbo - lastVarBoundValue < epsilon:
break
else:
lastVarBoundValue = elbo
if plot and plotIncremental:
plot_bound(plotFile + "-iter-" + str(iteration),
np.array(iters), np.array(elbos), np.array(likes))
# Right before we end, plot the evolution of the bound and likelihood
# if we've been asked to do so.
if plot:
plot_bound(plotFile, iters, elbos, likes)
return VbSideTopicModelState (K, Q, F, P, T, A, omA, Y, omY, sigY, U, V, vocab, tau, sigma), \
VbSideTopicQueryState (expLmda, nu, lxi, s, docLen)
def ode2es(L, rho0):
"""Creates an exponential series that describes the time evolution for the
initial density matrix (or state vector) `rho0`, given the Liouvillian
(or Hamiltonian) `L`.
Parameters
----------
L : qobj
Liouvillian of the system.
rho0 : qobj
Initial state vector or density matrix.
Returns
-------
eseries : :class:`qutip.eseries`
``eseries`` represention of the system dynamics.
"""
if issuper(L):
# check initial state
if isket(rho0):
# Got a wave function as initial state: convert to density matrix.
rho0 = rho0 * rho0.dag()
# check if state is below error threshold
if abs(rho0.full()).sum() < 1e-10 + 1e-24:
# enforce zero operator
return eseries(qzero(rho0.dims[0]))
w, v = L.eigenstates()
v = np.hstack([ket.full() for ket in v])
# w[i] = eigenvalue i
# v[:,i] = eigenvector i
rlen = np.prod(rho0.shape)
r0 = mat2vec(rho0.full())
v0 = la.solve(v, r0)
vv = v * sp.spdiags(v0.T, 0, rlen, rlen)
out = None
for i in range(rlen):
qo = Qobj(vec2mat(vv[:, i]), dims=rho0.dims, shape=rho0.shape)
if out:
out += eseries(qo, w[i])
else:
out = eseries(qo, w[i])
elif isoper(L):
if not isket(rho0):
raise TypeError('Second argument must be a ket if first' +
'is a Hamiltonian.')
# check if state is below error threshold
if abs(rho0.full()).sum() < 1e-5 + 1e-20:
# enforce zero operator
dims = rho0.dims
return eseries(
Qobj(sp.csr_matrix((dims[0][0], dims[1][0]), dtype=complex)))
w, v = L.eigenstates()
v = np.hstack([ket.full() for ket in v])
# w[i] = eigenvalue i
# v[:,i] = eigenvector i
rlen = np.prod(rho0.shape)
r0 = rho0.full()
v0 = la.solve(v, r0)
vv = v * sp.spdiags(v0.T, 0, rlen, rlen)
out = None
for i in range(rlen):
qo = Qobj(np.matrix(vv[:, i]).T, dims=rho0.dims, shape=rho0.shape)
if out:
out += eseries(qo, -1.0j * w[i])
else:
out = eseries(qo, -1.0j * w[i])
else:
raise TypeError('First argument must be a Hamiltonian or Liouvillian.')
return estidy(out)
from scipy.linalg import solve import matplotlib.pyplot as plt X = np.array([0, 1, 2, 3, 4]) Y = np.array([0.98, -3.01, -6.99, -11.01, -15.0]) # Calcula os elementos das marizes a11 = np.sum(X**2) a12 = np.sum(X) a22 = len(X) b1 = np.sum(X * Y) b2 = np.sum(Y) # Monta e resolve o sistema A = np.array([[a11, a12], [a12, a22]]) B = np.array([b1, b2]) a = solve(A, B) print(a) # define a funcao g(x) para plotar g = lambda x: a[0] * x + a[1] # cria pontos (x, y) da reta Xr = np.arange(-1, 5, 0.5) # Plota os pontos e a reta plt.plot(X, Y, ".", Xr, g(Xr), "-") plt.grid() plt.show()
def get_fractional_coords(self, cartesian_coords):
return la.solve(self.uc.T, np.array(cartesian_coords).T).T
[901, 387, 738, 965, 972, 906],
[1063, 386, 655, 911, 969, 936],
[1089, 389, 679, 723, 268, 860],
])
n = []
for i in range(0, 6):
n.append(sum(m[i]))
s = [[], [], [], [], [], []]
for i in range(0, 6):
s[i].append(n[i] * a[i])
s = np.array(s)
s1 = np.array([[51775785], [58598322], [48705107], [62332394], [62399316],
[51069435]])
# print(s)
x = solve(m, s)
t = np.array([[11321], [12223], [15707], [14070], [12152], [11502]])
y = solve(m, s1)
for i in range(0, 6):
y[i] = trans(str(y[i][0]))
# print(x)
print(y)
plt.ylabel('时段内的平均等待时间/分钟', size=15)
plt.xlabel('时间段', size=15)
print(m.dot(t))
for i in range(0, 6):
print(s1[i][0] / sum(m[i]), end=',')
xnew = np.linspace(
def method_solve(A, b):
t51 = tm.time()
x5 = scl.solve(A, b)
t52 = tm.time() - t51
return x5, t52
def test_condition_pointswise(self):
"""
Generate samples and random field by conditioning on pointwise data
"""
#
# Initialize Gaussian Random Field
#
# Resolution
max_res = 10
n = 2**max_res + 1 # size
# Hurst parameter
H = 0.5 # Hurst parameter in [0.5,1]
# Form covariance and precision matrices
x = np.arange(1, n)
X, Y = np.meshgrid(x, x)
K = fbm_cov(X, Y, H)
# Compute the precision matrix
I = np.identity(n - 1)
Q = linalg.solve(K, I)
# Define mean
mean = np.random.rand(n - 1, 1)
# Define Gaussian field
u_cov = GaussianField(n - 1, mean=mean, K=K, mode='covariance')
u_prc = GaussianField(n - 1, mean=mean, K=Q, mode='precision')
# Define generating white noise
z = u_cov.iid_gauss(n_samples=10)
u_obs = u_cov.sample(z=z)
# Index of measured observations
A = np.arange(0, n - 1, 2)
# observed quantities
e = u_obs[A, 0][:, None]
#print('e shape', e.shape)
# change A into matrix
k = len(A)
rows = np.arange(k)
cols = A
vals = np.ones(k)
AA = sp.coo_matrix((vals, (rows, cols)), shape=(k, n - 1)).toarray()
AKAt = AA.dot(K.dot(AA.T))
KAt = K.dot(AA.T)
U, S, Vt = linalg.svd(AA)
#print(U)
#print(S)
#print(Vt)
#print(AA.dot(u_obs)-e)
k = e.shape[0]
Ko = 0.01 * np.identity(k)
# Debug
K = u_cov.covariance()
#U_spp = u_cov.support()
#A_cmp = A.dot(U_spp)
u_cond = u_cov.condition(A, e, Ko=Ko, n_samples=100)
"""
def locally_linear_embedding(X, k_neighbors, t_dimensions, reg_factor=1e-3):
"""
Parameters
----------
X : numpy array
input data, shape [n_samples, n_features], dtype must be numpy.float64.
k_neighbors : integer
number of nearest neighbors to consider for each point.
t_dimensions : integer
number of dimensions in the output data.
reg_factor : float
regularization factor, for the case k_neighbors > n_features.
Return
-------
Y : numpy array
dimension-reduced data, shape [n_samples, t_dimensions].
"""
# check X data: must be a 2-D numpy array, must be np.float64
if not isinstance(X, np.ndarray) or X.ndim != 2:
raise TypeError("Your input data is NOT a 2-D numpy array")
if X.dtype != np.float64:
raise TypeError("Your input data is NOT type: numpy.float64")
n_samples, n_features = X.shape
# check Parameters
if t_dimensions > n_features or t_dimensions < 1:
raise ValueError(
"Your input does NOT satisfy: 1 <= output dimension <= input dimension"
)
if k_neighbors >= n_samples or k_neighbors <= 0:
raise ValueError(
"Your input does NOT satisfy: 0 < k_neighbors < n_samples")
print("#### LLE algorithm started! ####")
k_take = k_neighbors + 1
# step 1, compute the k nearest neighbors of each point
print("\tStage 1: compute distance and find k-nearest neighbor")
idx = np.argpartition(cdist(X, X), (1, k_take), axis=0)[1:k_take].T
# step 1, compute the k-nn of each point (using scikit-learn)
# knn = NearestNeighbors(k_take).fit(X)
# idx = knn.kneighbors(X, return_distance=False)[:, 1:]
Z = X[idx].transpose(0, 2, 1) # own implementation
# step 2, compute co-variance matrix and then the weights
print("\tStage 2: construct the Weight matrix")
# the Matrix to contain the Weights:
Weights = np.empty((n_samples, k_neighbors), dtype=X.dtype)
# the ALL-ONE vector:
Ones = np.ones(k_neighbors, dtype=X.dtype)
for i, P in enumerate(Z):
# each neighbors - this point
D = P.T - X[i]
# Cov is the local covariance matrix
Cov = np.dot(D, D.T)
# regularization
# Cov = Cov + eye(K,K) * factor * (Cov.trace > 0 ? Cov.trace : 1)
r = reg_factor
trace = np.trace(Cov)
if trace > 0:
r *= trace
Cov.flat[::
k_take] += r # add the reg factor to the main diagonal of Cov
# find the weights of each neighbors
w = solve(Cov, Ones, overwrite_a=True, assume_a='pos')
# make sum(w) = 1
Weights[i, :] = w / np.sum(w)
# put the Weights in to a sparse matrix
W = csr_matrix((Weights.ravel(), idx.ravel(),
np.arange(0, n_samples * k_neighbors + 1, k_neighbors)),
shape=(n_samples, n_samples))
# Step 3 compute M = (I-W)'(I-W)
M = (W.T * W - W - W.T).toarray()
M.flat[::n_samples + 1] += 1
# Step 4 compute the eigen_values and eigen_vectors of M
print("\tStage 3: compute the eigenvectors and output")
eigen_values, eigen_vectors = eigh(M,
eigvals=(1, t_dimensions),
overwrite_a=True)
# Step 5 the 2nd to the d+1'th eigen_vectors is the output
print("#### LLE algorithm ended! ####")
return eigen_vectors[:, np.argsort(np.abs(eigen_values))]
def test_condition_ptwise(self):
#
# Initialize Gaussian Random Field
#
# Resolution
l_max = 9
n = 2**l_max + 1 # size
# Hurst parameter
H = 0.5 # Hurst parameter in [0.5,1]
# Form covariance and precision matrices
x = np.arange(1, n + 1)
X, Y = np.meshgrid(x, x)
K = fbm_cov(X, Y, H)
# Compute the precision matrix
I = np.identity(n)
Q = linalg.solve(K, I)
# Plot meshes
fig, ax = plt.subplots(1, 1)
n = 2**l_max + 1
for l in range(l_max):
nl = 2**l + 1
i_spp = [i * 2**(l_max - l) for i in range(nl)]
ax.plot(x[i_spp], l * np.ones(nl), '.')
#ax.plot(x,'.', markersize=0.1)
#plt.show()
# Plot conditioned field
fig, ax = plt.subplots(3, 3)
# Define original field
u = []
n = 2**(l_max) + 1
for l in range(l_max):
nl = 2**l + 1
i_spp = [i * 2**(l_max - l) for i in range(nl)]
V_spp = I[:, i_spp]
if l == 0:
u_fne = GaussianField(n, K=K, mode='covariance',\
support=V_spp)
u_obs = u_fne.sample()
i_obs = np.array(i_spp)
else:
u_fne = GaussianField(n, K=K, mode='covariance',\
support=V_spp)
u_cnd = u_fne.condition(i_obs, u_obs[i_obs], output='field')
u_obs = u_cnd.sample()
i_obs = np.array(i_spp)
u.append(u_obs)
# Plot
for ll in range(l, l_max):
i, j = np.unravel_index(ll, (3, 3))
if ll == l:
ax[i, j].plot(x[i_spp],
5 * np.exp(0.01 * u_obs[i_spp]),
linewidth=0.5)
else:
ax[i, j].plot(x[i_spp],
5 * np.exp(0.01 * u_obs[i_spp]),
'g',
linewidth=0.1,
alpha=0.1)
fig.savefig('successive_conditioning.pdf')
def calculate_lambda_ir(self):
lambd = []
for i, p in enumerate(self.p_r):
np.fill_diagonal(p, -1)
lambd.append(linalg.solve(p.T, -self.lambda_0_ir[:, i]))
return np.array(lambd).T
j = np.argmin((x[i] - x)**2 + (y[i] + dx - y)**2)
if i != j: G[i, j] = vy[i] / (2 * dx)
j = np.argmin((x[i] - x)**2 + (y[i] - dx - y)**2)
if i != j: G[i, j] = -vy[i] / (2 * dx)
# Form operator
A = L - G
# Boundary conditions
b = np.zeros(n)
for i in range(n):
if (x[i] == 0 or x[i] == 1 or y[i] == 0 or y[i] == 1):
A[i, :] = 0
A[i, i] = 1
if x[i] == 0:
b[i] = np.exp(-10 * (y[i] - 0.3)**2)
# Solve
from scipy.linalg import solve
u = solve(A, b)
# Plot
U = to_matrix(u)
plt.imshow(U, extent=(min(x), max(x), max(y), min(y)))
plt.colorbar()
plt.xlabel('x')
plt.ylabel('y')
plt.title('Temperature distriubtion of plate')
plt.show()
if Triangles[i][0] < 14 * kr and Triangles[i][
1] < 14 * kr and Triangles[i][2] < 14 * kr:
continue
Triangles2.append([Triangles[i][0], Triangles[i][1], Triangles[i][2]])
Triangles = np.array(Triangles2)
A, B = sestaviMatriki(Points, Triangles, 14 * kr)
#print(Points.shape)
#print(Triangles.shape)
#print(Points[30:])
fig, (ax1, ax2) = plt.subplots(1, 2)
fig.set_size_inches((10, 5))
kot = minkot(Triangles, Points)
ax1.triplot(Points[:, 0], Points[:, 1], Triangles)
#ax1.set_aspect(1)
#x = np.abs(SOR(A,B,1.5))
x = np.abs(lin.solve(A, B))
#x = np.abs(lin.lstsq(A,B)[0])
C = poisKoef(x, Triangles, Points, 14 * kr)
cs = ax2.tricontourf(Points[:, 0],
Points[:, 1],
np.concatenate((np.zeros(14 * kr), x.flatten())),
levels=np.linspace(0, x[np.argmax(x)], 50),
cmap=plt.get_cmap("hot"))
plt.colorbar(cs)
#ax2.set_aspect(1)
plt.suptitle(
"Random, Najmanjši kot je {} stopinj, Pois. koef. je {}".format(
round(kot, 2), round(C, 4)))
plt.savefig("druga/random5.pdf")
if 0:
kr = 30
def _make_interp_per_full_matr(x, y, t, k):
'''
Returns a solution of a system for B-spline interpolation with periodic
boundary conditions. First ``k - 1`` rows of matrix are condtions of
periodicity (continuity of ``k - 1`` derivatives at the boundary points).
Last ``n`` rows are interpolation conditions.
RHS is ``k - 1`` zeros and ``n`` ordinates in this case.
Parameters
----------
x : 1-D array, shape (n,)
Values of x - coordinate of a given set of points.
y : 1-D array, shape (n,)
Values of y - coordinate of a given set of points.
t : 1-D array, shape(n+2*k,)
Vector of knots.
k : int
The maximum degree of spline
Returns
-------
c : 1-D array, shape (n+k-1,)
B-spline coefficients
Notes
-----
``t`` is supposed to be taken on circle.
'''
x, y, t = map(np.asarray, (x, y, t))
n = x.size
# LHS: the collocation matrix + derivatives at edges
matr = np.zeros((n + k - 1, n + k - 1))
# derivatives at x[0] and x[-1]:
for i in range(k - 1):
bb = _bspl.evaluate_all_bspl(t, k, x[0], k, nu=i + 1)
matr[i, :k + 1] += bb
bb = _bspl.evaluate_all_bspl(t, k, x[-1], n + k - 1, nu=i + 1)[:-1]
matr[i, -k:] -= bb
# collocation matrix
for i in range(n):
xval = x[i]
# find interval
if xval == t[k]:
left = k
else:
left = np.searchsorted(t, xval) - 1
# fill a row
bb = _bspl.evaluate_all_bspl(t, k, xval, left)
matr[i + k - 1, left - k:left + 1] = bb
# RHS
b = np.r_[[0] * (k - 1), y]
c = solve(matr, b)
return c
def calculate_pi_alpha(self):
"""
Computes the equilibrium probability quantities "pi_alpha" used
in MMVT theory. The value self.pi_alpha gets set by this
function.
"""
if self.k_alpha_beta is None:
raise Exception("Unable to call calculate_pi_alpha(): "\
"No statistics present in Data Sample.")
# First, determine if there is a bulk anchor, and if so, use it as
# the "pivot", if not, make our own pivot.
# The "pivot" is the entry in the flux_matrix that
bulk_index = None
for alpha, anchor in enumerate(self.model.anchors):
if anchor.bulkstate:
assert bulk_index is None, "Only one bulk state is allowed "\
"in model"
bulk_index = alpha
if bulk_index is None:
# Then we have a model without a bulk anchor: we need to make our
# own pivot
pivot_index = self.model.num_anchors
flux_matrix_dimension = self.model.num_anchors + 1
else:
# If a bulk anchor exists, we can use it as the pivot
assert bulk_index == self.model.num_anchors - 1, "The bulk "\
"anchor must be the last one in the model."
pivot_index = bulk_index
flux_matrix_dimension = self.model.num_anchors
self.pi_alpha = np.zeros(flux_matrix_dimension)
flux_matrix = np.zeros((flux_matrix_dimension, flux_matrix_dimension))
column_sum = np.zeros(flux_matrix_dimension)
flux_matrix[pivot_index, pivot_index] = 1.0
for alpha, anchor1 in enumerate(self.model.anchors):
flux_matrix[alpha, pivot_index] = 1.0
if anchor1.bulkstate:
continue
id_alias = anchor1.alias_from_neighbor_id(pivot_index)
flux_matrix[pivot_index, alpha] = 0.0
dead_end_anchor = False
if len(anchor1.milestones) == 1:
dead_end_anchor = True
for beta, anchor2 in enumerate(self.model.anchors):
if beta == pivot_index:
continue
if alpha == beta:
pass
else:
id_alias = anchor1.alias_from_neighbor_id(anchor2.index)
if id_alias is None:
flux_matrix[alpha, beta] = 0.0
else:
if dead_end_anchor:
# This line was supposed to work for a 1D
# Smoluchowski system, but with a 3D
# spherical system, the 2.0 needs to be 1.0.
#flux_matrix[alpha, beta] = 2.0 *\
# self.k_alpha_beta[(alpha, beta)]
flux_matrix[alpha, beta] = 1.0 *\
self.k_alpha_beta[(alpha, beta)]
else:
flux_matrix[alpha, beta] = \
self.k_alpha_beta[(alpha, beta)]
column_sum[alpha] += flux_matrix[alpha, beta]
flux_matrix[alpha, alpha] = -column_sum[alpha]
flux_matrix[pivot_index, pivot_index - 1] = HIGH_FLUX
prob_equil = np.zeros((flux_matrix_dimension, 1))
prob_equil[pivot_index] = 1.0
self.pi_alpha = abs(la.solve(flux_matrix.T, prob_equil))
# refine pi_alpha
pi_alpha_slice = np.zeros((flux_matrix_dimension - 1, 1))
for i in range(flux_matrix_dimension - 1):
pi_alpha_slice[i, 0] = -self.pi_alpha[i, 0] * flux_matrix[i, i]
K = flux_matrix_to_K(flux_matrix)
K_inf = np.linalg.matrix_power(K, FLUX_MATRIX_K_EXPONENT)
stationary_dist = K_inf.T @ pi_alpha_slice
for i in range(flux_matrix_dimension - 1):
self.pi_alpha[i, 0] = -stationary_dist[i, 0] / flux_matrix[i, i]
self.pi_alpha[-1, 0] = 0.0
self.pi_alpha = self.pi_alpha / np.sum(self.pi_alpha)
return
def rsolve(self, v, tol=0):
"""Evaluate w = M^-H v"""
if self.collapsed is not None:
return solve(self.collapsed.T.conj(), v)
return LowRankMatrix._solve(v, np.conj(self.alpha), self.ds, self.cs)
# v = v0 # set initial eigenvector guess
# print('\n Eigenvectors: ', eigvecs_truth)
# print('\nStarting guess v: ', v)
# print(' ')
print('\nEigenvalues: ', eigvals_truth)
print('\nStarting guess lambda: ', lambda0)
print(' ')
# print(f'\ni = 0, lambda = {lambda0}')
for i in range(1, num_iterations):
B = A_symm - lam * np.eye(m)
try:
omega = la.solve(B, v)
except:
print("Matrix is singular! Converged solution")
break
v = omega / la.norm(omega, 2)
lam = v.T @ (A_symm @ v)
# print(f'i = {i}, lambda = {lam}')
eigval_guesses.append(lam)
eigvec_guesses.append(v)
error = np.abs(eigval_guesses[-1] - eigval_guesses[-2])
rqiter_eigval_approx.append(np.round(np.abs(eigval_guesses[-1]), 8))
print('\nEigenvalue approximation: ', eigval_guesses[-1])
if allUnique(rqiter_eigval_approx) == False:
def _phase_one(A, b, x0, callback, postsolve_args, maxiter, tol, disp,
maxupdate, mast, pivot):
"""
The purpose of phase one is to find an initial basic feasible solution
(BFS) to the original problem.
Generates an auxiliary problem with a trivial BFS and an objective that
minimizes infeasibility of the original problem. Solves the auxiliary
problem using the main simplex routine (phase two). This either yields
a BFS to the original problem or determines that the original problem is
infeasible. If feasible, phase one detects redundant rows in the original
constraint matrix and removes them, then chooses additional indices as
necessary to complete a basis/BFS for the original problem.
"""
m, n = A.shape
status = 0
# generate auxiliary problem to get initial BFS
A, b, c, basis, x, status = _generate_auxiliary_problem(A, b, x0, tol)
if status == 6:
residual = c.dot(x)
iter_k = 0
return x, basis, A, b, residual, status, iter_k
# solve auxiliary problem
phase_one_n = n
iter_k = 0
x, basis, status, iter_k = _phase_two(c, A, x, basis, callback,
postsolve_args,
maxiter, tol, disp,
maxupdate, mast, pivot,
iter_k, phase_one_n)
# check for infeasibility
residual = c.dot(x)
if status == 0 and residual > tol:
status = 2
# drive artificial variables out of basis
# TODO: test redundant row removal better
# TODO: make solve more efficient with BGLU? This could take a while.
keep_rows = np.ones(m, dtype=bool)
for basis_column in basis[basis >= n]:
B = A[:, basis]
try:
basis_finder = np.abs(solve(B, A)) # inefficient
pertinent_row = np.argmax(basis_finder[:, basis_column])
eligible_columns = np.ones(n, dtype=bool)
eligible_columns[basis[basis < n]] = 0
eligible_column_indices = np.where(eligible_columns)[0]
index = np.argmax(basis_finder[:, :n]
[pertinent_row, eligible_columns])
new_basis_column = eligible_column_indices[index]
if basis_finder[pertinent_row, new_basis_column] < tol:
keep_rows[pertinent_row] = False
else:
basis[basis == basis_column] = new_basis_column
except LinAlgError:
status = 4
# form solution to original problem
A = A[keep_rows, :n]
basis = basis[keep_rows]
x = x[:n]
m = A.shape[0]
return x, basis, A, b, residual, status, iter_k
def get_freq_modes_over_f(power_mat,
window_function,
frequency,
n_modes,
plots=False):
"""Fines the most correlated frequency modes and fits thier noise."""
n_f = len(frequency)
d_f = sp.mean(sp.diff(frequency))
dt = 1. / 2. / frequency[-1]
n_chan = power_mat.shape[-1]
n_time = window_function.shape[0]
# The threshold for assuming there isn't enough data to measure anything.
no_data_thres = 10. / n_time
# Initialize the dictionary that will hold all the parameters.
output_params = {}
# First take the low frequency part of the spetrum matrix and average over
# enough bins to get a well conditioned matrix.
low_f_mat = sp.mean(power_mat[:4 * n_chan, :, :].real, 0)
# Factor the matrix to get the most correlated modes.
e, v = linalg.eigh(low_f_mat)
# Make sure they are sorted.
if not sp.alltrue(sp.diff(e) >= 0):
raise RuntimeError("Eigenvalues not sorted")
# Power matrix striped of the biggest modes.
reduced_power = sp.copy(power_mat)
mode_list = []
# Solve for the spectra of these modes.
for ii in range(n_modes):
this_mode_params = {}
# Get power spectrum and window function for this mode.
mode = v[:, -1 - ii]
mode_power = sp.sum(mode * power_mat.real, -1)
mode_power = sp.sum(mode * mode_power, -1)
mode_window = sp.sum(mode[:, None]**2 * window_function, 1)
mode_window = sp.sum(mode_window * mode[None, :]**2, 1)
# Protect against no data.
if sp.mean(mode_window).real < no_data_thres:
this_mode_params['amplitude'] = 0.
this_mode_params['index'] = 0.
this_mode_params['f_0'] = 1.
this_mode_params['thermal'] = T_infinity**2 * dt
else:
# Fit the spectrum.
p = fit_overf_const(mode_power, mode_window, frequency)
# Put all the parameters we measured into the output.
this_mode_params['amplitude'] = p[0]
this_mode_params['index'] = p[1]
this_mode_params['f_0'] = p[2]
this_mode_params['thermal'] = p[3]
this_mode_params['mode'] = mode
output_params['over_f_mode_' + str(ii)] = this_mode_params
# Remove the mode from the power matrix.
tmp_amp = sp.sum(reduced_power * mode, -1)
tmp_amp2 = sp.sum(reduced_power * mode[:, None], -2)
tmp_amp3 = sp.sum(tmp_amp2 * mode, -1)
reduced_power -= tmp_amp[:, :, None] * mode
reduced_power -= tmp_amp2[:, None, :] * mode[:, None]
reduced_power += tmp_amp3[:, None, None] * mode[:, None] * mode
mode_list.append(mode)
# Initialize the compensation matrix, that will be used to restore thermal
# noise that gets subtracted out. See Jan 29, Feb 17th, 2012 of Kiyo's
# notes.
compensation = sp.eye(n_chan, dtype=float)
for mode1 in mode_list:
compensation.flat[::n_chan + 1] -= 2 * mode1**2
for mode2 in mode_list:
mode_prod = mode1 * mode2
compensation += mode_prod[:, None] * mode_prod[None, :]
# Now that we've striped the noisiest modes, measure the auto power
# spectrum, averaged over channels.
auto_spec_mean = reduced_power.view()
auto_spec_mean.shape = (n_f, n_chan**2)
auto_spec_mean = auto_spec_mean[:, ::n_chan + 1].real
auto_spec_mean = sp.mean(auto_spec_mean, -1)
diag_window = window_function.view()
diag_window.shape = (n_time, n_chan**2)
diag_window = diag_window[:, ::n_chan + 1]
auto_spec_window = sp.mean(diag_window, -1)
if sp.mean(auto_spec_window).real < no_data_thres:
auto_cross_over = 0.
auto_index = 0.
auto_thermal = 0
else:
auto_spec_params = fit_overf_const(auto_spec_mean, auto_spec_window,
frequency)
auto_thermal = auto_spec_params[3]
if (auto_spec_params[0] <= 0 or auto_spec_params[3] <= 0
or auto_spec_params[1] > -0.599):
auto_cross_over = 0.
auto_index = 0.
else:
auto_index = auto_spec_params[1]
auto_cross_over = auto_spec_params[2] * (
auto_spec_params[0] / auto_spec_params[3])**(-1. / auto_index)
#if auto_cross_over < d_f:
# auto_index = 0.
# auto_cross_over = 0.
# Plot the mean auto spectrum if desired.
if plots:
h = plt.gcf()
a = h.add_subplot(*h.current_subplot)
norm = sp.mean(auto_spec_window).real
auto_plot = auto_spec_mean / norm
plotable = auto_plot > 0
lines = a.loglog(frequency[plotable], auto_plot[plotable])
c = lines[-1].get_color()
# And plot the fit in a light color.
if auto_cross_over > d_f / 4.:
spec = npow.overf_power_spectrum(auto_thermal, auto_index,
auto_cross_over, dt, n_time)
else:
spec = sp.zeros(n_time, dtype=float)
spec += auto_thermal
spec[0] = 0
spec = npow.convolve_power(spec, auto_spec_window)
spec = npow.prune_power(spec)
spec = spec[1:].real
if norm > no_data_thres:
spec /= norm
plotable = spec > 0
a.loglog(frequency[plotable],
spec[plotable],
c=c,
alpha=0.4,
linestyle=':')
output_params['all_channel_index'] = auto_index
output_params['all_channel_corner_f'] = auto_cross_over
# Finally measure the thermal part of the noise in each channel.
cross_over_ind = sp.digitize([auto_cross_over * 4], frequency)[0]
cross_over_ind = max(cross_over_ind, n_f // 2)
cross_over_ind = min(cross_over_ind, int(9. * n_f / 10.))
thermal = reduced_power[cross_over_ind:, :, :].real
n_high_f = thermal.shape[0]
thermal.shape = (n_high_f, n_chan**2)
thermal = sp.mean(thermal[:, ::n_chan + 1], 0)
thermal_norms = sp.mean(diag_window, 0).real
bad_inds = thermal_norms < no_data_thres
thermal_norms[bad_inds] = 1.
# Compensate for power lost in mode subtraction.
compensation[:, bad_inds] = 0
compensation[bad_inds, :] = 0
for ii in xrange(n_chan):
if bad_inds[ii]:
compensation[ii, ii] = 1.
thermal = linalg.solve(compensation, thermal)
# Normalize
thermal /= thermal_norms
thermal[bad_inds] = T_infinity**2 * dt
# Occationally the compensation fails horribly on a few channels.
# When this happens, zero out the offending indices.
thermal[thermal < 0] = 0
output_params['thermal'] = thermal
# Now that we know what thermal is, we can subtract it out of the modes we
# already measured.
for ii in range(n_modes):
mode_params = output_params['over_f_mode_' + str(ii)]
thermal_contribution = sp.sum(mode_params['mode']**2 * thermal)
# Subtract a maximum of 90% of the white noise to keep things positive
# definate.
new_white = max(mode_params['thermal'] - thermal_contribution,
0.1 * mode_params['thermal'])
if mode_params['thermal'] < 0.5 * T_infinity**2 * dt:
mode_params['thermal'] = new_white
return output_params
def dirac_recon_joint_alg(G,
measurement,
num_dirac,
shape_b,
flatten_order='F',
num_band=1,
noise_level=0,
max_ini=100,
stop_cri='mse',
max_inner_iter=20,
max_num_same_x=1,
max_num_same_y=1):
"""
ALGORITHM that reconstructs 2D Dirac deltas jointly
min |a - Gb|^2
s.t. c_1 * b = 0
c_2 * b = 0
This is the exact form that we have in the paper without any alternations for
performance considerations, e.g., reusing intermediate results, etc.
The new formulation exploit the fact that c_1 and c_2 are linearly indepdnet.
Hence, the effective number of unknowns are less than the total size of the two filters.
:param G: the linear mapping that links the unknown uniformly sampled
sinusoids to the given measurements
:param measurement: the given measurements of the 2D Dirac deltas
:param num_dirac: number of Dirac deltas
:param shape_b: shape of the (2D) uniformly sampled sinusoids
:param flatten_order: flatten order to be used. This is related to how G is build.
If the dimension 0 of G is 'C' ordered, then flattern_order = 'C';
otherwise, flattern_order = 'F'.
:param noise_level: noise level present in the given measurements
:param max_ini: maximum number of random initializations
:param stop_cri: stopping criterion, either 'mse' or 'max_iter'
:param max_inner_iter: maximum number of inner iterations for each random initializations
:param max_num_same_x: maximum number of Dirac deltas that have the same horizontal locations.
This will impose the minimum dimension of the annihilating filter used.
:param max_num_same_y: maximum number of Dirac deltas that have the same vertical locations
This will impose the minimum dimension of the annihilating filter used.
:return:
"""
compute_mse = (stop_cri == 'mse')
measurement = measurement.flatten(flatten_order)
num_non_zero = num_dirac + 2
# choose the shapes of the 2D annihilating filters (as square as possible)
# total number of entries should be at least num_dirac + 1
shape_c_0 = int(np.ceil(np.sqrt(num_non_zero)))
shape_c_1 = int(np.ceil(num_non_zero / shape_c_0))
if shape_c_0 > shape_c_1:
shape_c_1, shape_c_0 = shape_c_0, shape_c_1
# sanity check
assert shape_c_0 * shape_c_1 >= num_non_zero
# in case of common roots, the filter has to satisfy a certain minimum dimension
shape_c_1 = max(shape_c_1, max_num_same_y + 1)
shape_c_0 = int(np.ceil(num_non_zero / shape_c_1))
shape_c_0 = max(shape_c_0, max_num_same_x + 1)
shape_c = (shape_c_0, shape_c_1)
# total number of coefficients in c1 and c2
num_coef = shape_c_0 * shape_c_1
if num_band > 1:
def func_build_R(coef1, coef2, shape_in):
R_mtx_band = R_mtx_joint(coef1, coef2, shape_in)
return linalg.block_diag(*[R_mtx_band for _ in range(num_band)])
else:
def func_build_R(coef1, coef2, shape_in):
return R_mtx_joint(coef1, coef2, shape_in)
# determine the effective row rank of the joint annihilation right-dual matrix
c1_test = np.random.randn(shape_c_0, shape_c_1) + \
1j * np.random.randn(shape_c_0, shape_c_1)
c2_test = np.random.randn(shape_c_0, shape_c_1) + \
1j * np.random.randn(shape_c_0, shape_c_1)
R_test = func_build_R(c1_test, c2_test, shape_b)
try:
s_test = linalg.svd(R_test, compute_uv=False)
shape_Tb0_effective = min(
R_test.shape) - np.where(np.abs(s_test) < 1e-12)[0].size
except ValueError:
# the effective number of equations as predicted by the derivation
shape_Tb0_effective = \
min(max(np.prod(shape_b) - compute_effective_num_eq_2d(shape_c, shape_c),
num_coef - 1 + num_coef - 1),
R_test.shape[0])
# sizes of various matrices / vectors
sz_coef = num_coef * 2 - 1 # -1 because of linear independence
sz_S0 = num_coef * 2 - 2 * num_non_zero
sz_R1 = np.prod(shape_b) * num_band
# a few indices that are fixed
idx_bg0_Tb = sz_coef
idx_end0_Tb = sz_coef + shape_Tb0_effective
idx_bg1_Tb = 0
idx_end1_Tb = sz_coef
idx_bg0_TbH = 0
idx_end0_TbH = sz_coef
idx_bg1_TbH = sz_coef
idx_end1_TbH = sz_coef + shape_Tb0_effective
idx_bg0_Rc = sz_coef
idx_end0_Rc = sz_coef + shape_Tb0_effective
idx_bg1_Rc = sz_coef + shape_Tb0_effective
idx_end1_Rc = sz_coef + shape_Tb0_effective + sz_R1
idx_bg0_RcH = sz_coef + shape_Tb0_effective
idx_end0_RcH = sz_coef + shape_Tb0_effective + sz_R1
idx_bg1_RcH = sz_coef
idx_end1_RcH = sz_coef + shape_Tb0_effective
# pre-compute a few things
GtG = np.dot(G.conj().T, G)
Gt_a = np.dot(G.conj().T, measurement)
try:
beta = linalg.lstsq(G, measurement)[0]
except np.linalg.linalg.LinAlgError:
beta = linalg.solve(GtG, Gt_a)
beta_reshaped = np.reshape(beta, (shape_b[0], shape_b[1], num_band),
order='F')
Tbeta0 = np.vstack([
T_mtx_joint(beta_reshaped[:, :, band_count], shape_c, shape_c)
for band_count in range(num_band)
])
# QR-decomposition of Tbeta0.T
Tbeta_band = np.vstack([
convmtx2_valid(beta_reshaped[:, :, band_count], shape_c[0], shape_c[1])
for band_count in range(num_band)
])
Qtilde_full = linalg.qr(Tbeta_band.conj().T,
mode='economic',
pivoting=False)[0]
Qtilde1 = Qtilde_full
Qtilde2 = Qtilde_full[:, 1:]
Qtilde_mtx = linalg.block_diag(Qtilde1, Qtilde2)
Tbeta0_Qtilde = np.dot(Tbeta0, Qtilde_mtx)
# initializations
min_error = np.inf
rhs = np.concatenate(
(np.zeros(sz_coef + shape_Tb0_effective + sz_R1 + sz_S0,
dtype=float), np.append(np.ones(2, dtype=float), 0)))
rhs_bl = np.concatenate(
(Gt_a, np.zeros(shape_Tb0_effective, dtype=Gt_a.dtype)))
c1_opt = None
c2_opt = None
# iterations over different random initializations of the annihilating filter coefficients
ini = 0
while ini < max_ini:
ini += 1
c1 = np.random.randn(shape_c_0, shape_c_1) + \
1j * np.random.randn(shape_c_0, shape_c_1)
c2 = np.random.randn(shape_c_0, shape_c_1) + \
1j * np.random.randn(shape_c_0, shape_c_1)
# build a selection matrix that chooses a subset of c1 and c2 to ZERO OUT
S = np.dot(
planar_sel_coef_subset((shape_c_0, shape_c_1),
(shape_c_0, shape_c_1),
num_non_zero=num_non_zero,
max_num_same_x=max_num_same_x,
max_num_same_y=max_num_same_y), Qtilde_mtx)
S_H = S.conj().T
# the initializations of the annihilating filter coefficients
c0 = np.column_stack(
(linalg.block_diag(
np.dot(Qtilde1.T, c1.flatten('F'))[:, np.newaxis],
np.dot(Qtilde2.T, c2.flatten('F'))[:, np.newaxis]),
np.concatenate((np.dot(Qtilde1.T, c2.flatten('F')),
np.dot(Qtilde2.T, c1.flatten('F'))))[:,
np.newaxis]))
mtx_S_row = np.hstack(
(S, np.zeros((sz_S0, shape_Tb0_effective + sz_R1 + sz_S0 + 3))))
# last row in mtx_loop
mtx_last_row = np.hstack(
(c0.T, np.zeros((3, shape_Tb0_effective + sz_R1 + sz_S0 + 3))))
R_loop = func_build_R(c1, c2, shape_b)
# use QR decomposition to extract effective lines of equations
Q_H = linalg.qr(R_loop, mode='economic',
pivoting=False)[0][:, :shape_Tb0_effective].conj().T
R_loop = np.dot(Q_H, R_loop)
Tbeta_loop = np.dot(Q_H, Tbeta0_Qtilde)
# inner loop for each random initialization
for inner in range(max_inner_iter):
if inner == 0:
mtx_loop = np.vstack(
(np.hstack((np.zeros(
(sz_coef, sz_coef)), Tbeta_loop.conj().T,
np.zeros((sz_coef, sz_R1)), S_H, c0.conj())),
np.hstack(
(Tbeta_loop,
np.zeros((shape_Tb0_effective, shape_Tb0_effective)),
-R_loop, np.zeros(
(shape_Tb0_effective, 3 + sz_S0)))),
np.hstack(
(np.zeros((sz_R1, sz_coef)), -R_loop.conj().T, GtG,
np.zeros(
(sz_R1, 3 + sz_S0)))), mtx_S_row, mtx_last_row))
else:
mtx_loop[idx_bg0_Tb:idx_end0_Tb,
idx_bg1_Tb:idx_end1_Tb] = Tbeta_loop
mtx_loop[idx_bg0_TbH:idx_end0_TbH,
idx_bg1_TbH:idx_end1_TbH] = Tbeta_loop.conj().T
mtx_loop[idx_bg0_Rc:idx_end0_Rc,
idx_bg1_Rc:idx_end1_Rc] = -R_loop
mtx_loop[idx_bg0_RcH:idx_end0_RcH,
idx_bg1_RcH:idx_end1_RcH] = -R_loop.conj().T
# solve annihilating filter coefficients
try:
coef = np.dot(Qtilde_mtx,
linalg.solve(mtx_loop, rhs)[:sz_coef])
except linalg.LinAlgError:
break
c1 = np.reshape(coef[:num_coef], shape_c, order='F')
c2 = np.reshape(coef[num_coef:], shape_c, order='F')
# update the right-dual matrix R and T based on the new coefficients
R_loop = func_build_R(c1, c2, shape_b)
# use QR decomposition to extract effective lines of equations
Q_H = linalg.qr(
R_loop, mode='economic',
pivoting=False)[0][:, :shape_Tb0_effective].conj().T
R_loop = np.dot(Q_H, R_loop)
Tbeta_loop = np.dot(Q_H, Tbeta0_Qtilde)
# reconstruct b
if inner == 0:
mtx_brecon = np.vstack(
(np.hstack((GtG, R_loop.conj().T)),
np.hstack(
(R_loop,
np.zeros(
(shape_Tb0_effective, shape_Tb0_effective))))))
else:
mtx_brecon[:sz_R1, sz_R1:] = R_loop.conj().T
mtx_brecon[sz_R1:, :sz_R1] = R_loop
try:
b_recon = linalg.solve(mtx_brecon, rhs_bl)[:sz_R1]
except linalg.LinAlgError:
break
# compute fitting error
error_loop = linalg.norm(measurement - np.dot(G, b_recon))
if 0 <= error_loop < min_error:
# check that the number of non-zero entries are
# indeed num_dirac + 1 (could be less)
c1[np.abs(c1) < 1e-2 * np.max(np.abs(c1))] = 0
c2[np.abs(c2) < 1e-2 * np.max(np.abs(c2))] = 0
nnz_cond = \
np.sum(1 - np.isclose(np.abs(c1), 0).astype(int)) == num_non_zero and \
np.sum(1 - np.isclose(np.abs(c2), 0).astype(int)) == num_non_zero
cord1_0, cord1_1 = np.nonzero(c1)
cord2_0, cord2_1 = np.nonzero(c2)
min_order_cond = \
(np.max(cord2_0) - np.min(cord2_0) + 1 >= max_num_same_x + 1) and \
(np.max(cord1_1) - np.min(cord1_1) + 1 >= max_num_same_y + 1)
if nnz_cond and min_order_cond:
min_error = error_loop
b_opt = b_recon
c1_opt = c1
c2_opt = c2
if compute_mse and min_error < noise_level:
break
if compute_mse and min_error < noise_level:
break
if c1_opt is None or c2_opt is None:
max_ini += 1
print('fitting SNR {:.2f}'.format(
20 * np.log10(linalg.norm(measurement) / min_error)))
return c1_opt, c2_opt, min_error, b_opt, ini