def solve_discrete_lyapunov(a, q, complex_step=False):
r"""
Solves the discrete Lyapunov equation using a bilinear transformation.
Notes
-----
This is a modification of the version in Scipy (see
https://github.com/scipy/scipy/blob/master/scipy/linalg/_solvers.py)
which allows passing through the complex numbers in the matrix a
(usually the transition matrix) in order to allow complex step
differentiation.
"""
eye = np.eye(a.shape[0])
if not complex_step:
aH = a.conj().transpose()
aHI_inv = np.linalg.inv(aH + eye)
b = np.dot(aH - eye, aHI_inv)
c = 2 * np.dot(np.dot(np.linalg.inv(a + eye), q), aHI_inv)
return solve_sylvester(b.conj().transpose(), b, -c)
else:
aH = a.transpose()
aHI_inv = np.linalg.inv(aH + eye)
b = np.dot(aH - eye, aHI_inv)
c = 2 * np.dot(np.dot(np.linalg.inv(a + eye), q), aHI_inv)
return solve_sylvester(b.transpose(), b, -c)
def solve_discrete_lyapunov(a, q, complex_step=False):
r"""
Solves the discrete Lyapunov equation using a bilinear transformation.
Notes
-----
This is a modification of the version in Scipy (see
https://github.com/scipy/scipy/blob/master/scipy/linalg/_solvers.py)
which allows passing through the complex numbers in the matrix a
(usually the transition matrix) in order to allow complex step
differentiation.
"""
eye = np.eye(a.shape[0])
if not complex_step:
aH = a.conj().transpose()
aHI_inv = np.linalg.inv(aH + eye)
b = np.dot(aH - eye, aHI_inv)
c = 2 * np.dot(np.dot(np.linalg.inv(a + eye), q), aHI_inv)
return solve_sylvester(b.conj().transpose(), b, -c)
else:
aH = a.transpose()
aHI_inv = np.linalg.inv(aH + eye)
b = np.dot(aH - eye, aHI_inv)
c = 2 * np.dot(np.dot(np.linalg.inv(a + eye), q), aHI_inv)
return solve_sylvester(b.transpose(), b, -c)
def dexpm_triu(
a11: np.ndarray,
a12: np.ndarray,
a22: np.ndarray,
da11: np.ndarray,
da12: np.ndarray,
da22: np.ndarray,
dt: float,
f11: np.ndarray,
f22: np.ndarray,
df11: np.ndarray,
df22: np.ndarray,
) -> Tuple[np.ndarray, np.ndarray]:
"""Compute the exponential of the upper triangular matrix A and its derivative using the
Parlett's method
.. math::
F = \\exp(A) = \\exp \\left(\\begin{bmatrix} a11 & a12 \\\\ 0 & a22 \\end{bmatrix} \\right)
= \\begin{bmatrix} f11 & f12 \\\\ 0 & f22 \\end{bmatrix}
Args:
a11: Upper left input matrix
a12: Upper right input matrix
a22: Lower right input matrix
da11: Partial derivative upper left input matrix
da12: Partial derivative upper right input matrix
da22: Partial derivative lower right input matrix
dt: Sampling time
f11: Upper left output matrix
f22: Lower right output matrix
df11: Partial derivative upper left output matrix
df22: Partial derivative lower right output matrix
Returns:
2-elements tuple containing
- **F**: Matrix exponential of A
- **dF**: Derivative of the matrix exponential of A
"""
nj, n11, n22 = da12.shape
f12 = solve_sylvester(a11, -a22, f11 @ a12 - a12 @ f22)
df12 = np.zeros((nj, n11, n22))
tmp = -da11 @ f12 + f12 @ da22 + df11 @ a12 + f11 @ da12 - da12 @ f22 - a12 @ df22
for n in range(nj):
if tmp[n].any():
df12[n] = solve_sylvester(a11, -a22, tmp[n])
F = np.zeros((n11 + n22, n11 + n22))
F[:n11, :n11] = f11
F[:n11, n11:] = f12
F[n11:, n11:] = f22
dF = np.zeros((nj, n11 + n22, n11 + n22))
dF[:, :n11, :n11] = df11
dF[:, :n11, n11:] = df12
dF[:, n11:, n11:] = df22
return F, dF
def test_simple(self):
a = np.array([[1, 2], [0, 4]])
b = np.array([[5, 6], [0, 8]])
c = np.array([[9, 10], [11, 12]])
x = solve_sylvester(a, b, c)
assert_array_almost_equal(np.dot(a, x) + np.dot(x, b), c)
# Test with a complex form, but effectively same matrices
ac = complex(0, 1)*a
bc = complex(0, 1)*b
cc = complex(0, 1)*c
x = solve_sylvester(a, b, c)
assert_array_almost_equal(np.dot(ac, x) + np.dot(x, bc), cc)
def expm_triu(
a11: np.ndarray, a12: np.ndarray, a22: np.ndarray, dt: float, f11: np.ndarray, f22: np.ndarray,
) -> np.ndarray:
"""Compute the exponential of the upper triangular matrix A using the Parlett's method
.. math::
F = \\exp(A) = \\exp \\left(\\begin{bmatrix} a11 & a12 \\\\ 0 & a22 \\end{bmatrix} \\right)
= \\begin{bmatrix} f11 & f12 \\\\ 0 & f22 \\end{bmatrix}
Args:
a11: Upper left input matrix
a12: Upper right input matrix
a22: Lower right input matrix
dt: Sampling time
f11: Upper left output matrix
f22: Lower right output matrix
Returns:
F: Matrix exponential of A
"""
F = block_diag(f11, f22)
dim = a12.shape[0]
F[:dim, dim:] = solve_sylvester(a11, -a22, f11 @ a12 - a12 @ f22)
return F
def test_nonsquare(self):
a = np.array([[8, 1, 6], [3, 5, 7], [4, 9, 2]])
b = np.array([[2, 3], [4, 5]])
c = np.array([[1, 2], [3, 4], [5, 6]])
x = solve_sylvester(a, b, c)
assert_array_almost_equal(np.dot(a, x) + np.dot(x, b), c)
def get_sdrgain_upd(amat, wnrm='fro', maxeps=None,
baseA=None, baseZ=None, baseGain=None,
maxfac=None):
deltaA = amat - baseA
# nda = npla.norm(deltaA, ord=wnrm)
# nz = npla.norm(baseZ, ord=wnrm)
# na = npla.norm(baseA, ord=wnrm)
# import ipdb; ipdb.set_trace()
epsP = spla.solve_sylvester(amat, -baseZ, -deltaA)
# print('debugging!!!')
# epsP = 0*amat
eps = npla.norm(epsP, ord=wnrm)
print('|amat - baseA|: {0} -- |E|: {1}'.
format(npla.norm(deltaA, ord=wnrm), eps))
if maxeps is not None:
if eps < maxeps:
updGaint = npla.solve(epsP+np.eye(epsP.shape[0]), baseGain.T)
return updGaint.T, True
elif maxfac is not None:
if (1+eps)/(1-eps) < maxfac and eps < 1:
updGaint = npla.solve(epsP+np.eye(epsP.shape[0]), baseGain.T)
return updGaint.T, True
return None, False
def _w_update(x, w, Omega, b, rho, gamma, maxepochs=25):
theta = w.copy()
n = x.shape[0]
z = w.copy() #numpy.zeros(shape=w.shape)
u = numpy.zeros(shape=w.shape)
# cache multiplications
xb = x.T.dot(b)
xx = x.T.dot(x)
for l in range(maxepochs):
zprev = z.copy()
# updates
theta = solve_sylvester(rho * xx + rho * numpy.eye(xx.shape[0]),
2 * Omega, rho * xb + rho * (z - u))
z = softthreshold(theta + u, 1. * gamma / rho)
u = u + theta - z
# compute residuals
dualresid = numpy.linalg.norm(-rho * (z - zprev), 'fro')
primalresid = numpy.linalg.norm(theta - z, 'fro')
epspri = n * EPSABS + EPSREL * numpy.max(
[numpy.linalg.norm(theta, 'fro'),
numpy.linalg.norm(z, 'fro'), 0])
epsdual = n * EPSABS + EPSREL * numpy.linalg.norm(rho * u, 'fro')
# check for convergence
if (dualresid < epsdual) and (primalresid < epspri):
break
return z
def subdivision_eigenvectors_lower_block(N): # U1 in Stam's paper
# Sylvester Equation:
# A * X + X * B = Q
# From Stam:
# u1 * sigma - s12 * u1 = s11 * u0
# Fit to Sylvester:
# (u1 * sigma - s12 * u1)^T = (s11 * u0)^T
# (u1 * sigma)^T + (-1 * s12 * u1)^T = (s11 * u0)^T
# Transpose distributes over multiplication: (A * B)^T = B^T * A^T
# sigma is diagonal : (sigma^T = sigma)
# sigma * u1^T + u1^T * (-1 * s12)^T = (s11 * u0)^T
# |___| |__| |__| |__________| |__________|
# A X X B Q
sigma = np.diag(np.asarray(subdivision_eigenvalues(N)))
s12 = np.asarray(s12_matrix())
s11 = np.asarray(s11_matrix(N))
u0 = np.asarray(subdivision_eigenvectors(N))
A = sigma # (M, M)
B = (-1 * s12).transpose() # (N, N)
Q = (s11.dot(u0)).transpose() # (M, N)
X = la.solve_sylvester(A, B, Q).transpose() # (M, N)
return X.tolist()
def sylvester_project(M, A, B, r):
"""
Project via SVD on error matrix + solving the Sylvester equation.
"""
t = time.time()
E = np.dot(A, M) - np.dot(M, B)
G,H,dr = get_GH(E)
G_r = G[:, 0:r]
H_r = H[:, 0:r]
lowrank = np.dot(G_r, H_r.T)
print('norm(E-lowrank): ', np.linalg.norm(E-lowrank))
# Sylvester solve
M_class = solve_sylvester(A, -B, lowrank)
print('rank(lowrank): ', np.linalg.matrix_rank(lowrank))
print('rank(A): ', np.linalg.matrix_rank(A))
print('rank(B): ', np.linalg.matrix_rank(B))
print('norm(M-M_class): ', np.linalg.norm(M-M_class))
E_class = np.dot(A, M_class) - np.dot(M_class, B)
print('rank of E_class',np.linalg.matrix_rank(E_class))
#print 'eigvals of E_class',np.linalg.eigvals(E_class)
print('time of sylv project: ', time.time() - t)
return M_class
def lyapunov(A, Q=None):
"""
Solve the equation :math:`A^T X + X A = -Q`.
default Q is set to I
:param A: system matrix
:type A: np.ndarray | np.matrix | List[List]
:param Q: matrix
:type Q: np.ndarray | np.matrix
:return: the matrix X if there is a solution
:rtype: np.ndarray | None
"""
A = np.array(A)
if Q is None:
Q = np.eye(A.shape[0])
try:
# continuous lyapunov equation is a sylvester equation
X = solve_sylvester(A.T, A, -Q)
if config['use_numpy_matrix']:
return np.mat(X)
else:
return X
except LinAlgError:
return None
def adjust_reads(mat, column_spread = column_spread, row_spread = row_spread, cutoff = c, r = n_rows, c = n_cols):
mat = np.array(mat).flatten()
mat = mat.reshape(r,c)
adjusted_reads = np.rint(solve_sylvester(column_spread,row_spread,mat))
# A lower bound cutoff removes false positives (unfortunately also remove true reads with low counts in that cell)
adjusted_reads[adjusted_reads < cutoff] = 0
return adjusted_reads
def dy_Z(z_i):
DFi = self.DF(z_i)[n:,:]
df = self.df(z_i)
A = np.linalg.solve(DFi[:,y] + DFi[:,e].dot(df),DFi[:,v].dot(Ivy))
B = np.linalg.inv(self.dZ_Z)
C = -np.linalg.solve(DFi[:,y] + DFi[:,e].dot(df),DFi[:,Y].dot(self.dY_Z)+DFi[:,Z].dot(V)).dot(B)
return solve_sylvester(A,B,C)
def graph_sc(X, D, Lc, beta=0.6, rho=0.01, T=None, sc_n_iter=10, **args):
# Update code
A = sparse_decode(X, D.T, sc_mode=3, lambda1=T).T
Z = A.copy()
U = 0
i = 0
Lm1 = np.dot(D.T, D) + rho*np.identity(D.shape[1])
Lm2 = beta*Lc.toarray() if sparse.issparse(Lc) else beta*Lc
P3 = np.dot(D.T, X)
while i < sc_n_iter:
i += 1
Rm = P3 + rho*(Z - U)
A = linalg.solve_sylvester(Lm1, Lm2, Rm)
Z = A + U
ids = np.argsort(np.abs(Z), 0)[::-1]
ids = ids[T:, :].T
for k in range(Z.shape[1]):
Z[ids[k], k] = 0
U = U + A - Z
for k in range(Z.shape[1]):
wk = np.argwhere(Z[:,k] != 0).squeeze()
Z[wk, k] = np.dot(linalg.pinv(D[:,wk]), X[:,k])
return Z
def ridge_regression_modified(X, Y, L, D):
"""
Computes the solution of the multivariate problem given in question 1.4.
Takes as input X and Y arrays, the parameter lambda, and the penalty
matrix D. Returns the optimal values for beta0 and b.
"""
# Define N, d, and q
N = X.shape[1]
d = X.shape[0]
q = Y.shape[0]
# Convert X and Y arrays
X = np.transpose(np.vstack((np.ones(N), X)))
Y = np.transpose(Y)
# Calculate the optimal parameters for b
b_opt = solve_sylvester(np.matmul(
np.transpose(X), X), L * D, np.matmul(np.transpose(X), Y))
# Calculate average of x1,...,xN
total_X = np.zeros(d + 1)
for i, j in enumerate(X):
total_X += j
x_bar = total_X / N
# Calculate average of y1,..,yN
total_Y = np.zeros(q)
for i, j in enumerate(Y):
total_Y += j
y_bar = total_Y / N
# Calculate the optimal parameters for beta0
beta_opt = y_bar - np.matmul(np.transpose(b_opt), x_bar)
return beta_opt, b_opt
def get_sdrefb_upd(amat, t, fbtype=None, wnrm=2,
B=None, R=None, Q=None, maxeps=None,
baseA=None, baseZ=None, baseP=None, maxfac=None, **kwargs):
if fbtype == 'sylvupdfb' or fbtype == 'singsylvupd':
if baseP is not None:
deltaA = amat - baseA
epsP = spla.solve_sylvester(amat, -baseZ, -deltaA)
eps = npla.norm(epsP, ord=wnrm)
print('|amat - baseA|: {0} -- |E|: {1}'.
format(npla.norm(deltaA, ord=wnrm), eps))
if maxeps is not None:
if eps < maxeps:
opepsPinv = npla.inv(epsP+np.eye(epsP.shape[0]))
return baseP.dot(opepsPinv), True
elif maxfac is not None:
if (1+eps)/(1-eps) < maxfac and eps < 1:
opepsPinv = npla.inv(epsP+np.eye(epsP.shape[0]))
return baseP.dot(opepsPinv), True
# otherwise: (SDRE feedback or `eps` too large already)
# curX = spla.solve_continuous_are(amat, B, Q, R)
# if fbtype == 'sylvupdfb' or fbtype == 'singsylvupd':
# logger.debug('in `get_fb_dict`: t={0}: eps={1} too large, switch!'.
# format(t, eps))
# else:
# logger.debug('t={0}: computed the SDRE feedback')
return None, False
def _w_update(x, w, Omega, b, rho, gamma, maxepochs=25):
theta = w.copy()
n = x.shape[0]
z = w.copy() #numpy.zeros(shape=w.shape)
u = numpy.zeros(shape=w.shape)
# cache multiplications
xb = x.T.dot(b)
xx = x.T.dot(x)
for l in range(maxepochs):
zprev = z.copy()
# updates
theta = solve_sylvester(rho*xx + rho * numpy.eye(xx.shape[0]), 2*Omega, rho*xb + rho*(z-u))
z = softthreshold(theta + u, 1.*gamma/rho)
u = u + theta - z
# compute residuals
dualresid = numpy.linalg.norm(-rho * (z - zprev), 'fro')
primalresid = numpy.linalg.norm(theta - z, 'fro')
epspri = n * EPSABS + EPSREL * numpy.max([numpy.linalg.norm(theta, 'fro'), numpy.linalg.norm(z, 'fro'), 0])
epsdual = n * EPSABS + EPSREL * numpy.linalg.norm(rho*u, 'fro')
# check for convergence
if (dualresid < epsdual) and (primalresid < epspri):
break
return z
def SAE(X, S, lamb):
A = S.dot(S.T)
B = lamb * (X.dot(X.T))
C = (1 + lamb) * (S.dot(X.T))
W = linalg.solve_sylvester(A, B, C)
return W
def sylvester_propagate(self, a, b, het, phi, y_left_bound, y_right_bound,
z_left_bound, z_right_bound):
matrix_c = a * phi @ self.matrix_z + a * self.matrix_y @ phi + phi
res = la.solve_sylvester(b * self.matrix_y,
b * self.matrix_z + np.eye(self.n_z),
matrix_c)
return res
def SAE(X, S, lamb):
A = np.nan_to_num(S.dot(S.T))
B = np.nan_to_num(lamb * (X.dot(X.T)))
C = np.nan_to_num((1 + lamb) * (S.dot(X.T)))
W = linalg.solve_sylvester(A, B, C)
return W
def _g2sSylvester(A, B, F, G, u, v):
"""
% Purpose : Solves the semi-definite Sylvester Equation of the form
% A'*A * Phi + Phi * B'*B - A'*F - G*B = 0,
% Where the null vectors of A and B are known to be
% A * u = 0
% B * v = 0
%
% Use (syntax):
% Phi = g2sSylvester( A, B, F, G, u, v )
%
% Input Parameters :
% A, B, F, G := Coefficient matrices of the Sylvester Equation
% u, v := Respective null vectors of A and B
%
% Return Parameters :
% Phi := The minimal norm solution to the Sylvester Equation
%
% Description and algorithms:
% The rank deficient Sylvester equation is solved by means of Householder
% reflections and the Bartels-Stewart algorithm. It uses the MATLAB
% function "lyap", in reference to Lyapunov Equations, a special case of
% the Sylvester Equation.
"""
# Household vectors (???)
m, n = len(u), len(v)
u[0] += norm(u)
u *= np.sqrt(2) / norm(u)
v[0] += norm(v)
v *= np.sqrt(2) / norm(v)
# Apply householder updates
A -= np.dot(np.dot(A, u), u.T)
B -= np.dot(np.dot(B, v), v.T)
F -= np.dot(np.dot(F, v), v.T)
G -= np.dot(u, (np.dot(u.T, G)))
# Solve the system of equations
phi = np.zeros((m, n))
phi[0, 1:] = mrdivide(G[0, :], B[:, 1:].T)
phi[1:, 0] = mldivide(A[:, 1:], F[:, 0].T)
phi[1:, 1:] = solve_sylvester(
np.dot(A[:, 1:].T, A[:, 1:]),
np.dot(B[:, 1:].T, B[:, 1:]),
-np.dot(-A[:, 1:].T, F[:, 1:]) + np.dot(G[1:, :], B[:, 1:]),
)
# Invert the householder updates
if u.dtype == "complex128":
u = np.sqrt(np.real(u) ** 2 + np.imag(u) ** 2)
if v.dtype == "complex128":
v = np.sqrt(np.real(v) ** 2 + np.imag(v) ** 2)
phi -= np.dot(u, (np.dot(u.T, phi)))
phi -= np.dot(np.dot(phi, v), v.T)
# phi += np.dot(u, np.dot(np.dot(u.T, (np.dot(phi, v))), v.T))
return phi
def fit_dim_red(self, data, r, eigen=None, verbose=False):
"""
Fits the emotion transfer model by a closed form solution
with low rank matrix A based on SVD of the data covariance.
If Y^T Y = V S V^T then uses A = V diag( eigen, 0) V^T
Parameters
----------
data: torch.Tensor of shape (n_samples, n_emotions, n_landmarks)
Input vector of data
r: Int
Rank of A
eigen: torch.Tensor of shape (r)
Eigenvalues of A to consider. Default None -> use all eigen=1
Returns
-------
Nothing
"""
if verbose:
print('Computing Gram matrices')
self.initialise(data)
self.model.compute_gram_train()
tmp = self.model.G_xt + self.lbda * self.model.n * self.model.m * \
torch.eye(self.model.n * self.model.m)
if verbose:
print('Computing SVD of empirical covariance')
cor = 1 / self.model.n * \
self.model.y_train.reshape(
-1, self.model.output_dim).T @ self.model.y_train.reshape(-1, self.model.output_dim)
u, d, v = torch.svd(cor)
if verbose:
print("Solving the associated linear system")
identity_r = torch.diag_embed(torch.Tensor([1 for i in range(r)] +
[0 for i in range(self.model.output_dim - r)]))
proj_r = identity_r[:, :r]
if eigen is None:
gamma_r, _ = torch.solve(
self.model.y_train.reshape(-1, self.model.output_dim) @ v @ proj_r, tmp)
self.model.alpha = gamma_r @ proj_r.T @ v.T
self.model.A = v @ identity_r @ v.T
else:
B = torch.inverse(torch.diag(eigen)).numpy()
Q = (self.model.y_train.reshape(-1, self.model.output_dim) @ v @ proj_r).numpy() @ B
gamma_r = solve_sylvester(self.model.G_xt,
self.lbda * self.model.n * self.model.m * B,
Q)
self.model.alpha = torch.from_numpy(
gamma_r) @ torch.diag(eigen) @ proj_r.T @ v.T
self.model.A = v @ proj_r @ torch.diag(eigen) @ proj_r.T @ v.T
if verbose:
print('Coefficients alpha fitted, empirical risk=',
self.training_risk())
def RW_NN_ALM(alpha, Z, n, m, W1, W2, flag, struc):
D_rec = lambda_k_1 = lambda_k_2 = E = A = D = np.zeros(
(n, m + 1)
) #Initiliaze lagrange multiplier and guesses for variables to 0 matrix
W1_sq_inv = np.linalg.inv(np.dot(W1, W1))
W2_sq = np.dot(W2, W2)
mu_init = 1.05
mu_k = mu_init
gamma = 1
k = 0
eps = 0.0001
step_A = step_E = np.matrix(np.identity(n))
E = np.zeros((n, m + 1))
while (
k < 700
): # and np.linalg.norm(step_E) > eps and np.linalg.norm(step_A) > eps ):
WAW = np.dot(np.dot(W1, A), W2)
D_new = soft_threshold(
np.subtract(WAW, lambda_k_2 * (1 / (1.0 * mu_k))),
1 / (1.0 * mu_k))
step_D = np.subtract(D, D_new)
D = D_new
E_new = new_Error(alpha, mu_k, lambda_k_1, Z, A)
# for i in range(np.shape(E)[0]):
# for j in range(np.shape(E)[1]):
# E_new[i][j]=soft_threshold(np.subtract(np.subtract(Z,A),1/(1.0*mu_k)*lambda_k_1), alpha/mu_k)[i][j]
E_new = np.matrix(E_new)
step_E = np.subtract(E, E_new)
E = E_new
if (struc == 1):
for i in range(np.shape(E)[0]):
for j in range(i):
E[i, j] = 0 # Projection
RHS = np.add((1 / (1.0 * mu_k)) *
np.add(lambda_k_1, np.dot(np.dot(W1, lambda_k_2), W2)),
np.add(np.subtract(Z, E), np.dot(np.dot(W1, D), W2)))
A_new = spla.solve_sylvester(W1_sq_inv, W2_sq, np.dot(W1_sq_inv, RHS))
#T1 = np.dot(np.dot(W1, np.add (np.subtract(np.subtract(Z, E) , A) , 1/(1.0*mu_k)*lambda_k_1) ), W2 )
#T2 = np.add (np.subtract(D,WAW) , 1/(1.0*mu_k)*lambda_k_2)
#A_new = A + gamma*np.add(T1,T2)
step_A = np.subtract(A, A_new)
A = A_new
lambda_k_1 = np.add(lambda_k_1,
mu_k * (np.subtract(np.subtract(Z, A), E)))
lambda_k_2 = np.add(lambda_k_2, mu_k * (np.subtract(D, WAW)))
mu_k = mu_k * mu_init
k = k + 1
# U,S,V=np.linalg.svd(A)
# s=S[len(S)-1]
# A=soft_threshold(A,s)
if (flag == 1):
print "alpha", alpha, "iterations", k, "error A", np.linalg.norm(
A), "error norm E", np.linalg.norm(
E), "rank A", np.linalg.matrix_rank(A)
WAW = np.dot(np.dot(W1, np.subtract(Z, E)), W2)
return A, E, flag, WAW, np.subtract(Z, E)
def surf_height(surf_normal,penalty,estimate=0):
""" this function reconstruct the surface using a regularized least-square
method.
surface_normal is the heightxwidthx3 array that stores the surface normal
at each point
penalty is the value of lambda"""
def D_matrix(size):
D_mat = np.zeros(shape=(size,size))
D_mat[0,0] = -3
D_mat[0,1] = 4
D_mat[0,2] = -1
D_mat[-1,-3] = 1
D_mat[-1,-2] = -4
D_mat[-1,3] = 3
for row in range(1,size-1):
D_mat[row,row-1] = -1
D_mat[row,row+1] = 1
D_mat = D_mat/2
return D_mat
normal_array[np.isnan(normal_array)]=0
y,x = np.nonzero(surf_normal[:,:,0])
left = np.min(x)
right = np.max(x)
top = np.min(y)
bottom = np.max(y)
surf_normal = surf_normal[top:bottom+1,left:right+1,:]
#return surf_normal
Zx = -surf_normal[:,:,0]
Zy = surf_normal[:,:,1]
height,width,useless = surf_normal.shape
if estimate==0:
guess=np.zeros(shape=(height,width))
Dy = D_matrix(height)
Dx = D_matrix(width)
A = np.dot(Dy.T,Dy) + penalty**2*np.identity(height)
B = np.dot(Dx.T,Dx) + penalty**2*np.identity(width)
Q = np.dot(Dy.T,Zy )+ np.dot(Zx,Dx)+2*penalty**2*estimate
Z = spl.solve_sylvester(A,B,Q)
min_height = np.min(Z)
Z = Z - min_height
Z = Z/masked_image[top:bottom+1,left:right+1]
Z[Z==float('inf')] = np.nan
Z[Z==(-float('inf'))] = np.nan
return Z
def get_mapping(ts_cls):
X = pd.DataFrame()
S = np.zeros((n_att, 1))
#labels = pd.DataFrame()
#print (ts_cls)
tr_cls = [x for x in all_cls if (x not in ts_cls)]
#print (tr_cls)
#print (tr_cls)
for cls in tr_cls:
#print ('class', cls)
df = data[cls]
#attribute_mat = np.array([])
#print (cls)
attribute_vec = aam[class_names[cls]]
#print (cls)
m1 = df.shape[0]
#print (m1)
attribute_mat = np.repeat(np.array(attribute_vec).reshape(n_att, 1),
m1,
axis=1)
#attribute_mat = attribute_mat.reshape(n_att, m1)
#y = np.ones((m1, 1)) * cls
#y[:, i] = 1
#y_df = pd.DataFrame(y)
#Y = Y.append(y_df, ignore_index = True)
# print (m)
S = np.concatenate((S, attribute_mat), axis=1)
X = X.append(df, ignore_index=True)
#labels = labels.append(y_df, ignore_index = True)
#print (X.shape)
X = X.T
X = np.array(X)
S = S[:, 1:]
#labels = np.array(labels)
(k, n) = S.shape
#print (k, n)
(d, n) = X.shape
#print (d, n)
#print (labels.shape)
labda = 1
A = np.matmul(S, S.T)
#print (A.shape)
B = labda * np.matmul(X, X.T)
#print (B.shape)
C = (1 + labda) * np.matmul(S, X.T)
#print (C.shape)
W = solve_sylvester(A, B, C)
return (W)
def transformation(users: List[int],
representation: np.ndarray, entity_embeddings: np.ndarray,
ratings: np.ndarray, alpha=1.0) -> np.ndarray:
try:
_A = representation.T @ representation
_B = alpha * inv(entity_embeddings.T @ entity_embeddings)
_Q = representation.T @ ratings[users] @ entity_embeddings @ inv(entity_embeddings.T @ entity_embeddings)
return solve_sylvester(_A, _B, _Q)
except LinAlgError as err:
return np.zeros((representation.shape[1], entity_embeddings.shape[1]))
def SAE(X, S, lamd):
A = np.dot(S.T, S)
B = lamd * np.dot(X.T, X)
C = (1 + lamd) * np.dot(S.T, X)
try:
W = solve_sylvester(A, B, C)
except:
print('Sovler failed')
return None
return W
def np_solve_sylvester(tensors, sign_matrices, d_sign_matrices):
res = np.empty_like(tensors)
for k in range(tensors.shape[0]):
tensor = tensors[k]
sign_matrix = sign_matrices[k]
d_sign_matrix = d_sign_matrices[k]
q = np.dot(tensor, d_sign_matrix) - np.dot(d_sign_matrix, tensor)
a = sign_matrix
b = -sign_matrix
res[k] = linalg.solve_sylvester(a, b, q)
return res
def sylvester(M, N, n, r):
# Generate random rank r error matrix
G = np.random.random((n, r))
H = np.random.random((n, r))
GH = np.dot(G, H.T)
# Solve Sylvester equation to recover A
# Such that MA - AN^T = GH^T
A = solve_sylvester(M, -N, GH)
E = np.dot(M, A) - np.dot(A, N)
return A, G, H
def _calculate_w(self, s, x, lambda_val):
"""
:param s: array, (n, k)
:param x: array, (n, d)
:param lambda_val: hyper-parameter
:return: w: array, (k, d)
"""
A = s.T @ s # k*k
B = lambda_val * x.T @ x # d*d
C = (1 + lambda_val) * s.T @ x # k*d
w = solve_sylvester(A, B, C) # k*d
return w
def sae(vis_data, sem_data, lambda_):
"""
Computes the weight matrix that estimates the latent space of the Semantic Auto-encoder.
@param vis_data: dxN data matrix
@param sem_data: kxN semantic matrix
@param lambda_: regularisation parameter
@return: kxd projection matrix
"""
a = sem_data.dot(sem_data.transpose())
b = lambda_ * vis_data.dot(vis_data.transpose())
c = (1 + lambda_) * sem_data.dot(vis_data.transpose())
return solve_sylvester(a, b, c)
def sylvester(A, B, n, r): # Generate random rank r error matrix G = np.random.random((n, r)) H = np.random.random((n, r)) GH = np.dot(G,H.T) # Solve Sylvester equation to recover M # Such that AM - MB = GH^T M = solve_sylvester(A, -B, GH) E = np.dot(A,M) - np.dot(M,B) assert np.linalg.norm(E - GH) <= 1e-10 return M
def SAE(X, S, lamb):
"""
SAE - Semantic Autoencoder
:param X: d x N data matrix
:param S: k x N semantic matrix
:param lamb: regularization parameter
:return: W -> k x d projection function
"""
A = np.dot(S, S.T)
B = lamb * np.dot(X, X.T)
C = (1 + lamb) * np.dot(S, X.T)
W = solve_sylvester(A, B, C)
return W
def SAE(X, S, lam):
#For the error:Memory Error
#We use csc_matrix to take place of ndarray
S = scipy.transpose(S)
S = sparse.csc_matrix(S, dtype='float16')
A = np.dot(S, S.T)
B = lam * np.dot(X, X.T)
C = (1 + lam) * np.dot(S, X.T)
A = sparse.csc_matrix(A).toarray()
B = sparse.csc_matrix(B).toarray()
C = sparse.csc_matrix(C).toarray()
W = linalg.solve_sylvester(A, B, C) #Solve the Sylvester equation
return W
def fit(self, X, y, sample_weight=None):
lambd = self.lambd
X, y, X_offset, y_offset, X_scale = self._preprocess_data(
X,
y,
fit_intercept=self.fit_intercept,
normalize=self.normalize,
copy=self.copy_X,
sample_weight=sample_weight)
A = np.dot(y.T, y)
B = lambd * np.dot(X.T, X)
C = (1 + lambd) * np.dot(y.T, X)
W = linalg.solve_sylvester(A, B, C)
self.coef_ = W
self._set_intercept(X_offset, y_offset, X_scale)
return self
def find_W(self, X, S, ld):
# INPUTS:
# X: d x N - data matrix
# S: Number of Attributes (k) x N - semantic matrix
# ld: regularization parameter
#
# Return :
# W: kxd projection matrix
A = np.dot(S, S.T)
B = ld * np.dot(X, X.T)
C = (1 + ld) * np.dot(S, X.T)
W = linalg.solve_sylvester(A, B, C)
return W
def remove_a12(As, n_stable):
r"""Basis change to remove the (1, 2) block of the block-ordered real Schur matrix :math:`\mathbf{A}`
Being :math:`\mathbf{A}_s\in\mathbb{R}^{m\times m}` a matrix of the form
.. math:: \mathbf{A}_s = \begin{bmatrix} A_{11} & A_{12} \\ 0 & A_{22} \end{bmatrix}
the (1,2) block is removed by solving the Sylvester equation
.. math:: \mathbf{A}_{11}\mathbf{X} - \mathbf{X}\mathbf{A}_{22} + \mathbf{A}_{12} = 0
used to build the change of basis
.. math:: \mathbf{T} = \begin{bmatrix} \mathbf{I}_{s,s} & -\mathbf{X}_{s,u} \\ \mathbf{0}_{u, s}
& \mathbf{I}_{u,u} \end{bmatrix}
where :math:`s` and :math:`u` are the respective number of stable and unstable eigenvalues, such that
.. math:: \mathbf{TA}_s\mathbf{T}^\top = \begin{bmatrix} A_{11} & \mathbf{0} \\ 0 & A_{22} \end{bmatrix}.
Args:
As (np.ndarray): Block-ordered real Schur matrix (can be built using
:func:`sharpy.rom.utils.krylovutils.schur_ordered`).
n_stable (int): Number of stable eigenvalues in ``As``.
Returns:
np.ndarray: Basis transformation :math:`\mathbf{T}\in\mathbb{R}^{m\times m}`.
References:
Jaimoukha, I. M., Kasenally, E. D.. Implicitly Restarted Krylov Subspace Methods for Stable Partial Realizations
SIAM Journal of Matrix Analysis and Applications, 1997.
"""
A11 = As[:n_stable, :n_stable]
A12 = As[:n_stable, n_stable:]
A22 = As[n_stable:, n_stable:]
n = As.shape[0]
X = sclalg.solve_sylvester(A11, -A22, -A12)
T = np.block([[np.eye(n_stable), -X],
[np.zeros((n - n_stable, n_stable)),
np.eye(n - n_stable)]])
T2 = np.eye(n, n_stable)
# App = T2.T.dot(T.dot(As.dot(np.linalg.inv(T).dot(T2))))
return T, X
def update(self, X, y, Omega):
# save the previous W
self.prev_w = self.values.copy()
# Initialize Updates
Theta = self.values.copy() # dxk
Z = self.values.copy() ## warm start
U = numpy.zeros(shape=Z.shape)
# Cache multiplication
XX = X.T.dot(X) # dxd
Xy = X.T.dot(y) # dxk
for j in range(self.w_epochs):
Z_prev = Z.copy()
# Update Theta
C = Xy + self.rho * (Z - U)
Theta = solve_sylvester(XX + self.rho * numpy.eye(XX.shape[0]),
2 * Omega, C)
Z = softthreshold(Theta + U, self.gamma / self.rho)
U = U + Theta - Z
# Compute residuals
dualresid = numpy.linalg.norm(-self.rho * (Z - Z_prev), 2)
primalresid = numpy.linalg.norm(Theta - Z, 2)
epspri = X.shape[0] * EPSABS + EPSREL * numpy.max(
[numpy.linalg.norm(Theta, 2),
numpy.linalg.norm(Z, 2), 0])
epsdual = X.shape[0] * EPSABS + EPSREL * numpy.linalg.norm(
self.rho * U, 2)
# Confirm the dual is decreasing
if (j > 0) and (prevdual < dualresid):
#break
pass
prevdual = dualresid
# Check for convergence
if (dualresid < epsdual) and (primalresid < epspri):
break
if not self.quiet:
print "Learning W, Iteration: %i, Rho: %2.2f, Dualresid: %2.4f, EPS_Dual: %2.4f, PriResid: %2.4f, EPS_Pri: %2.4f" % (
j, self.rho, dualresid, epsdual, primalresid, epspri)
self.values = Z
def SAE(self, X, S, lamb):
"""
SAE - Semantic Autoencoder
:param X: d x N data matrix
:param S: k x N semantic matrix
:param lamb: regularization parameter
:return: W -> k x d projection function
"""
print("X shape", X.shape)
print("S shape", S.shape)
print("Lambda", lamb)
A = np.dot(S.T, S)
B = lamb * np.dot(X.T, X)
C = (1 + lamb) * np.dot(S.T, X)
W = solve_sylvester(A, B, C)
return W
def update(self, X, y, Omega):
# save the previous W
self.prev_w = self.values.copy()
# Initialize Updates
Theta = self.values.copy() # dxk
Z = self.values.copy() ## warm start
U = numpy.zeros(shape=Z.shape)
# Cache multiplication
XX = X.T.dot(X) # dxd
Xy = X.T.dot(y) # dxk
for j in range(self.w_epochs):
Z_prev = Z.copy()
# Update Theta
C = Xy + self.rho * (Z - U)
Theta = solve_sylvester(XX + self.rho * numpy.eye(XX.shape[0]), 2*Omega, C)
Z = softthreshold(Theta + U, self.gamma/self.rho)
U = U + Theta - Z
# Compute residuals
dualresid = numpy.linalg.norm(-self.rho*(Z - Z_prev), 2)
primalresid = numpy.linalg.norm(Theta - Z, 2)
epspri = X.shape[0] * EPSABS + EPSREL * numpy.max([numpy.linalg.norm(Theta, 2), numpy.linalg.norm(Z, 2), 0])
epsdual = X.shape[0] * EPSABS + EPSREL * numpy.linalg.norm(self.rho*U,2)
# Confirm the dual is decreasing
if (j > 0) and (prevdual < dualresid):
#break
pass
prevdual = dualresid
# Check for convergence
if (dualresid < epsdual) and (primalresid < epspri):
break
if not self.quiet:
print "Learning W, Iteration: %i, Rho: %2.2f, Dualresid: %2.4f, EPS_Dual: %2.4f, PriResid: %2.4f, EPS_Pri: %2.4f" % (j, self.rho, dualresid, epsdual, primalresid, epspri)
self.values = Z
def check_case(self, a, b, c):
x = solve_sylvester(a, b, c)
assert_array_almost_equal(np.dot(a, x) + np.dot(x, b), c)
def LinApp_Solve(AA,BB,CC,DD,FF,GG,HH,JJ,KK,LL,MM,WWW,TT,NN,Z0,Sylv):
"""
This code takes Uhlig's original code and puts it in the form of a
function. This version outputs the policy function coefficients: PP,
QQ and UU for X, and RR, SS and VV for Y.
Inputs overview:
The matrices of derivatives: AA - TT.
The autoregression coefficient matrix NN from the law of motion for Z.
Z0 is the Z-point about which the linearization is taken. For
linearizing about the steady state this is Zbar and normally Zbar = 0.
Sylv is an indicator variable telling the program to use the built-in
function sylvester() to solve for QQ and SS, if possible. Default is
to use Sylv=1.
Parameters
----------
AA : array_like, dtype=float, shape=(ny, nx)
The matrix represented above by :math:`A`. It is the matrix of
derivatives of the Y equations with repsect to :math:`X_t`
BB : array_like, dtype=float, shape=(ny, nx)
The matrix represented above by :math:`B`. It is the matrix of
derivatives of the Y equations with repsect to
:math:`X_{t-1}`.
CC : array_like, dtype=float, shape=(ny, ny)
The matrix represented above by :math:`C`. It is the matrix of
derivatives of the Y equations with repsect to :math:`Y_t`
DD : array_like, dtype=float, shape=(ny, nz)
The matrix represented above by :math:`C`. It is the matrix of
derivatives of the Y equations with repsect to :math:`Z_t`
FF : array_like, dtype=float, shape=(nx, nx)
The matrix represetned above by :math:`F`. It is the matrix of
derivatives of the model's characterizing equations with
respect to :math:`X_{t+1}`
GG : array_like, dtype=float, shape=(nx, nx)
The matrix represetned above by :math:`G`. It is the matrix of
derivatives of the model's characterizing equations with
respect to :math:`X_t`
HH : array_like, dtype=float, shape=(nx, nx)
The matrix represetned above by :math:`H`. It is the matrix of
derivatives of the model's characterizing equations with
respect to :math:`X_{t-1}`
JJ : array_like, dtype=float, shape=(nx, ny)
The matrix represetned above by :math:`J`. It is the matrix of
derivatives of the model's characterizing equations with
respect to :math:`Y_{t+1}`
KK : array_like, dtype=float, shape=(nx, ny)
The matrix represetned above by :math:`K`. It is the matrix of
derivatives of the model's characterizing equations with
respect to :math:`Y_t`
LL : array_like, dtype=float, shape=(nx, nz)
The matrix represetned above by :math:`L`. It is the matrix of
derivatives of the model's characterizing equations with
respect to :math:`Z_{t+1}`
MM : array_like, dtype=float, shape=(nx, nz)
The matrix represetned above by :math:`M`. It is the matrix of
derivatives of the model's characterizing equations with
respect to :math:`Z_t`
WWW : array, dtype=float, shape=(ny,)
The vector of the numerical errors of first ny characterizing
equations
TT : array, dtype=float, shape=(nx,)
The vector of the numerical errors of the next nx characterizing
equations following the first ny equations
NN : array_like, dtype=float, shape=(nz, nz)
The autocorrelation matrix for the exogenous state vector z.
Z0 : array, dtype=float, shape=(nz,)
The Z-point about which the linearization is taken. For linearizing
about the steady state this is Zbar and normally Zbar = 0.
QQ if true.
Sylv: binary, dtype=int
An indicator variable telling the program to use the built-in
function sylvester() to solve for QQ and SS, if possible. Default is
to use Sylv=1.
Returns
-------
P : 2D-array, dtype=float, shape=(nx, nx)
The matrix :math:`P` in the law of motion for endogenous state
variables described above.
Q : 2D-array, dtype=float, shape=(nx, nz)
The matrix :math:`Q` in the law of motion for exogenous state
variables described above.
U : array, dtype=float, shape=(nx,)
The vector of the constant term of the policy function for X,
the endogenous state variables
R : 2D-array, dtype=float, shape=(ny, nx)
The matrix :math:`R` in the law of motion for endogenous state
variables described above.
S : 2D-array, dtype=float, shape=(ny, nz)
The matrix :math:`S` in the law of motion for exogenous state
variables described above.
V : array, dtype=float, shape=(ny,)
The vector of the constant term of the policy function for Y,
the endogenous non-state variables
References
----------
.. [1] Uhlig, H. (1999): "A toolkit for analyzing nonlinear dynamic
stochastic models easily," in Computational Methods for the Study
of Dynamic Economies, ed. by R. Marimon, pp. 30-61. Oxford
University Press.
"""
#The original coding we did used the np.matrix form for our matrices so we
#make sure to set our inputs to numpy matrices.
AA = np.matrix(AA)
BB = np.matrix(BB)
CC = np.matrix(CC)
DD = np.matrix(DD)
FF = np.matrix(FF)
GG = np.matrix(GG)
HH = np.matrix(HH)
JJ = np.matrix(JJ)
KK = np.matrix(KK)
LL = np.matrix(LL)
MM = np.matrix(MM)
NN = np.matrix(NN)
WWW = np.array(WWW)
TT = np.array(TT)
Z0 = np.array(Z0)
#Tolerance level to use
TOL = .000001
# Here we use matrices to get pertinent dimensions.
nx = FF.shape[1]
l_equ = CC.shape[0]
ny = CC.shape[1]
nz = min(NN.shape)
# The following if and else blocks form the
# Psi, Gamma, Theta Xi, Delta mats
if l_equ == 0:
if CC.any():
# This blcok makes sure you don't throw an error with an empty CC.
CC_plus = la.pinv(CC)
CC_0 = _nullSpaceBasis(CC.T)
else:
CC_plus = np.mat([])
CC_0 = np.mat([])
Psi_mat = FF
Gamma_mat = -GG
Theta_mat = -HH
Xi_mat = np.mat(vstack((hstack((Gamma_mat, Theta_mat)),
hstack((eye(nx), zeros((nx, nx)))))))
Delta_mat = np.mat(vstack((hstack((Psi_mat, zeros((nx, nx)))),
hstack((zeros((nx, nx)), eye(nx))))))
else:
CC_plus = la.pinv(CC)
CC_0 = _nullSpaceBasis(CC.T)
if l_equ != ny:
Psi_mat = vstack((zeros((l_equ - ny, nx)), FF \
- dot(dot(JJ, CC_plus), AA)))
Gamma_mat = vstack((dot(CC_0, AA), dot(dot(JJ, CC_plus), BB) \
- GG + dot(dot(KK, CC_plus), AA)))
Theta_mat = vstack((dot(CC_0, BB), dot(dot(KK, CC_plus), BB) - HH))
else:
CC_inv = la.inv(CC)
Psi_mat = FF - dot(JJ.dot(CC_inv), AA)
Gamma_mat = dot(JJ.dot(CC_inv), BB) - GG + dot(dot(KK, CC_inv), AA)
Theta_mat = dot(KK.dot(CC_inv), BB) - HH
Xi_mat = vstack((hstack((Gamma_mat, Theta_mat)), \
hstack((eye(nx), zeros((nx, nx))))))
Delta_mat = vstack((hstack((Psi_mat, np.mat(zeros((nx, nx))))),\
hstack((zeros((nx, nx)), eye(nx)))))
# Now we need the generalized eigenvalues/vectors for Xi with respect to
# Delta. That is eVals and eVecs below.
eVals, eVecs = la.eig(Xi_mat, Delta_mat)
if npla.matrix_rank(eVecs) < nx:
print("Error: Xi is not diagonalizable, stopping...")
# From here to line 158 we Diagonalize Xi, form Lambda/Omega and find P.
else:
Xi_sortabs = np.sort(abs(eVals))
Xi_sortindex = np.argsort(abs(eVals))
Xi_sortedVec = np.array([eVecs[:, i] for i in Xi_sortindex]).T
Xi_sortval = eVals[Xi_sortindex]
Xi_select = np.arange(0, nx)
if np.imag(Xi_sortval[nx - 1]).any():
if (abs(Xi_sortval[nx - 1] - sp.conj(Xi_sortval[nx])) < TOL):
drop_index = 1
cond_1 = (abs(np.imag(Xi_sortval[drop_index-1])) > TOL)
cond_2 = drop_index < nx
while cond_1 and cond_2:
drop_index += 1
if drop_index >= nx:
print("There is an error. Too many complex eigenvalues."
+" Quitting...")
else:
print("Droping the lowest real eigenvalue. Beware of" +
" sunspots!")
Xi_select = np.array([np.arange(0, drop_index - 1),\
np.arange(drop_index, nx + 1)])
# Here Uhlig computes stuff if user chose "Manual roots" I skip it.
if max(abs(Xi_sortval[Xi_select])) > 1 + TOL:
print("It looks like we have unstable roots. This might not work...")
if abs(max(abs(Xi_sortval[Xi_select])) - 1) < TOL:
print("Check the model to make sure you have a unique steady" +
" state we are having problems with convergence.")
Lambda_mat = np.diag(Xi_sortval[Xi_select])
Omega_mat = Xi_sortedVec[nx:2 * nx, Xi_select]
if npla.matrix_rank(Omega_mat) < nx:
print("Omega matrix is not invertible, Can't solve for P; we" +
" proceed with the alternative, QZ-method, to get P...")
#~~~~~~~~~ QZ-method codes from SOLVE_QZ ~~~~~~~~#
Delta_up,Xi_up,UUU,VVV=la.qz(Delta_mat,Xi_mat, output='complex')
UUU=UUU.T
Xi_eigval = np.diag( np.diag(Xi_up)/np.maximum(np.diag(Delta_up),TOL))
Xi_sortabs= np.sort(abs(np.diag(Xi_eigval)))
Xi_sortindex= np.argsort(abs(np.diag(Xi_eigval)))
Xi_sortval = Xi_eigval[Xi_sortindex, Xi_sortindex]
Xi_select = np.arange(0, nx)
stake = max(abs(Xi_sortval[Xi_select])) + TOL
Delta_up, Xi_up, UUU, VVV = qzdiv(stake,Delta_up,Xi_up,UUU,VVV)
#Check conditions from line 49-109
if np.imag(Xi_sortval[nx - 1]).any():
if (abs(Xi_sortval[nx - 1] - sp.conj(Xi_sortval[nx])) < TOL):
print("Problem: You have complex eigenvalues! And this means"+
" PP matrix will contain complex numbers by this method." )
drop_index = 1
cond_1 = (abs(np.imag(Xi_sortval[drop_index-1])) > TOL)
cond_2 = drop_index < nx
while cond_1 and cond_2:
drop_index += 1
if drop_index >= nx:
print("There is an error. Too many complex eigenvalues."
+" Quitting...")
else:
print("Dropping the lowest real eigenvalue. Beware of" +
" sunspots!")
for i in xrange(drop_index,nx+1):
Delta_up,Xi_up,UUU,VVV = qzswitch(i,Delta_up,Xi_up,UUU,VVV)
Xi_select1 = np.arange(0,drop_index-1)
Xi_select = np.append(Xi_select1, np.arange(drop_index,nx+1))
if Xi_sortval[max(Xi_select)] < 1 - TOL:
print('There are stable roots NOT used. Proceeding with the' +
' smallest root.')
if max(abs(Xi_sortval[Xi_select])) > 1 + TOL:
print("It looks like we have unstable roots. This might not work...")
if abs(max(abs(Xi_sortval[Xi_select])) - 1) < TOL:
print("Check the model to make sure you have a unique steady" +
" state we are having problems with convergence.")
#End of checking conditions
#Lambda_mat = np.diag(Xi_sortval[Xi_select]) # to help sol_out.m
VVV=VVV.conj().T
VVV_2_1 = VVV[nx : 2*nx, 0 : nx]
VVV_2_2 = VVV[nx : 2*nx, nx :2*nx]
UUU_2_1 = UUU[nx : 2*nx, 0 : nx]
VVV = VVV.conj().T
if abs(la.det(UUU_2_1))< TOL:
print("One necessary condition for computing P is NOT satisfied,"+
" but we proceed anyways...")
if abs(la.det(VVV_2_1))< TOL:
print("VVV_2_1 matrix, used to compute for P, is not invertible; we"+
" are in trouble but we proceed anyways...")
PP = np.matrix( la.solve(- VVV_2_1, VVV_2_2) )
PP_imag = np.imag(PP)
PP = np.real(PP)
if (sum(sum(abs(PP_imag))) / sum(sum(abs(PP))) > .000001).any():
print("A lot of P is complex. We will continue with the" +
" real part and hope we don't lose too much information.")
#~~~~~~~~~ End of QZ-method ~~~~~~~~~#
#This follows the original uhlig.py file
else:
PP = dot(dot(Omega_mat, Lambda_mat), la.inv(Omega_mat))
PP_imag = np.imag(PP)
PP = np.real(PP)
if (sum(sum(abs(PP_imag))) / sum(sum(abs(PP))) > .000001).any():
print("A lot of P is complex. We will continue with the" +
" real part and hope we don't lose too much information.")
# The code from here to the end was from the Uhlig file calc_qrs.m.
# I think for python it fits better here than in a separate file.
# The if and else below make RR and VV depending on our model's setup.
if l_equ == 0:
RR = zeros((0, nx))
VV = hstack((kron(NN.T, FF) + kron(eye(nz), \
(dot(FF, PP) + GG)), kron(NN.T, JJ) + kron(eye(nz), KK)))
else:
RR = - dot(CC_plus, (dot(AA, PP) + BB))
VV = sp.vstack((hstack((kron(eye(nz), AA), \
kron(eye(nz), CC))), hstack((kron(NN.T, FF) +\
kron(eye(nz), dot(FF, PP) + dot(JJ, RR) + GG),\
kron(NN.T, JJ) + kron(eye(nz), KK)))))
# Now we use LL, NN, RR, VV to get the QQ, RR, SS, VV matrices.
# first try using Sylvester equation solver
if ny>0:
PM = (FF-la.solve(JJ.dot(CC),AA))
if npla.matrix_rank(PM)< nx+ny:
Sylv=0
print("Sylvester equation solver condition is not satisfied;"\
+" proceed with the original method...")
else:
if npla.matrix_rank(FF)< nx:
Sylv=0
print("Sylvester equation solver condition is not satisfied;"\
+" proceed with the original method...")
if Sylv:
print("Using Sylvester equation solver...")
if ny>0:
Anew = la.solve(PM, (FF.dot(PP)+GG+JJ.dot(RR)-\
la.solve(KK.dot(CC), AA)) )
Bnew = NN
Cnew1 = la.solve(JJ.dot(CC),DD.dot(NN))+la.solve(KK.dot(CC), DD)-\
LL.dot(NN)-MM
Cnew = la.solve(PM, Cnew1)
QQ = la.solve_sylvester(Anew,Bnew,Cnew)
SS = la.solve(-CC, (AA.dot(QQ)+DD))
else:
Anew = la.solve(FF, (FF.dot(PP)+GG))
Bnew = NN
Cnew = la.solve(FF, (-LL.dot(NN)-MM))
QQ = la.solve_sylvester(Anew,Bnew,Cnew)
SS = np.zeros((0,nz)) #empty matrix
# then the Uhlig's way
else:
if (npla.matrix_rank(VV) < nz * (nx + ny)):
print("Sorry but V is not invertible. Can't solve for Q and S;"+
" but we proceed anyways...")
LL = sp.mat(LL)
NN = sp.mat(NN)
LLNN_plus_MM = dot(LL, NN) + MM
if DD.any():
impvec = vstack([DD.T, np.reshape(LLNN_plus_MM,
(nx * nz, 1), 'F')])
else:
impvec = np.reshape(LLNN_plus_MM, (nx * nz, 1), 'F')
QQSS_vec = np.matrix(la.solve(-VV, impvec))
if (max(abs(QQSS_vec)) == sp.inf).any():
print("We have issues with Q and S. Entries are undefined." +
" Probably because V is no inverible.")
#Build QQ SS
QQ = np.reshape(np.matrix(QQSS_vec[0:nx * nz, 0]),
(nx, nz), 'F')
SS = np.reshape(QQSS_vec[(nx * nz):((nx + ny) * nz), 0],\
(ny, nz), 'F')
#Build WW - WW has the property [x(t)',y(t)',z(t)']=WW [x(t)',z(t)'].
WW = sp.vstack((
hstack((eye(nx), zeros((nx, nz)))),
hstack((dot(RR, la.pinv(PP)), (SS - dot(dot(RR, la.pinv(PP)), QQ)))),
hstack((zeros((nz, nx)), eye(nz)))))
# find constant terms
# redefine matrices to be 2D-arrays for generating vector UU and VVV
AA = np.array(AA)
CC = np.array(CC)
FF = np.array(FF)
GG = np.array(GG)
JJ = np.array(JJ)
KK = np.array(KK)
LL = np.array(LL)
NN = np.array(NN)
RR = np.array(RR)
QQ = np.array(QQ)
SS = np.array(SS)
if ny>0:
UU1 = -(FF.dot(PP)+GG+JJ.dot(RR)+FF-(JJ+KK).dot(la.solve(CC,AA)))
UU2 = (TT+(FF.dot(QQ)+JJ.dot(SS)+LL).dot(NN.dot(Z0)-Z0)- \
(JJ+KK).dot(la.solve(CC,WWW)))
UU = la.solve(UU1, UU2)
VVV = la.solve(- CC, (WWW+AA.dot(UU)) )
else:
UU = la.solve( -(FF.dot(PP)+FF+GG), (TT+(FF.dot(QQ)+LL).dot(NN.dot(Z0)-Z0)) )
VVV = np.array([])
return np.array(PP), np.array(QQ), np.array(UU), np.array(RR), np.array(SS),\
np.array(VVV)
def MMAPPH1NPPR(D, sigma, S, *argv):
"""
Returns various performane measures of a continuous time
MMAP[K]/PH[K]/1 non-preemptive priority queue, see [1]_.
Parameters
----------
D : list of matrices of shape (N,N), length (K+1)
The D0...DK matrices of the arrival process.
D1 corresponds to the lowest, DK to the highest priority.
sigma : list of row vectors, length (K)
The list containing the initial probability vectors of the service
time distributions of the various customer types. The length of the
vectors does not have to be the same.
S : list of square matrices, length (K)
The transient generators of the phase type distributions representing
the service time of the jobs belonging to various types.
further parameters :
The rest of the function parameters specify the options
and the performance measures to be computed.
The supported performance measures and options in this
function are:
+----------------+--------------------+----------------------------------------+
| Parameter name | Input parameters | Output |
+================+====================+========================================+
| "ncMoms" | Number of moments | The moments of the number of customers |
+----------------+--------------------+----------------------------------------+
| "ncDistr" | Upper limit K | The distribution of the number of |
| | | customers from level 0 to level K-1 |
+----------------+--------------------+----------------------------------------+
| "stMoms" | Number of moments | The sojourn time moments |
+----------------+--------------------+----------------------------------------+
| "stDistr" | A vector of points | The sojourn time distribution at the |
| | | requested points (cummulative, cdf) |
+----------------+--------------------+----------------------------------------+
| "prec" | The precision | Numerical precision used as a stopping |
| | | condition when solving the Riccati and |
| | | the matrix-quadratic equations |
+----------------+--------------------+----------------------------------------+
| "erlMaxOrder" | Integer number | The maximal Erlang order used in the |
| | | erlangization procedure. The default |
| | | value is 200. |
+----------------+--------------------+----------------------------------------+
| "classes" | Vector of integers | Only the performance measures |
| | | belonging to these classes are |
| | | returned. If not given, all classes |
| | | are analyzed. |
+----------------+--------------------+----------------------------------------+
(The quantities related to the number of customers in
the system include the customer in the server, and the
sojourn time related quantities include the service
times as well)
Returns
-------
Ret : list of the performance measures
Each entry of the list corresponds to a performance
measure requested. Each entry is a matrix, where the
columns belong to the various job types.
If there is just a single item,
then it is not put into a list.
References
----------
.. [1] G. Horvath, "Efficient analysis of the MMAP[K]/PH[K]/1
priority queue", European Journal of Operational
Research, 246(1), 128-139, 2015.
"""
K = len(D)-1
# parse options
eaten = []
erlMaxOrder = 200;
precision = 1e-14;
classes = np.arange(0,K)
for i in range(len(argv)):
if argv[i]=="prec":
precision = argv[i+1]
eaten.append(i)
eaten.append(i+1)
elif argv[i]=="erlMaxOrder":
erlMaxOrder = argv[i+1]
eaten.append(i)
eaten.append(i+1)
elif argv[i]=="classes":
classes = np.array(argv[i+1])-1
eaten.append(i)
eaten.append(i+1)
if butools.checkInput and not CheckMMAPRepresentation(D):
raise Exception('MMAPPH1PRPR: The arrival process is not a valid MMAP representation!')
if butools.checkInput:
for k in range(K):
if not CheckPHRepresentation(sigma[k],S[k]):
raise Exception('MMAPPH1PRPR: the vector and matrix describing the service times is not a valid PH representation!')
# some preparation
D0 = D[0]
N = D0.shape[0]
I = ml.eye(N)
sD = ml.zeros((N,N))
for Di in D:
sD += Di
s = []
M = np.empty(K)
for i in range(K):
s.append(np.sum(-S[i],1))
M[i] = sigma[i].size
# step 1. solution of the workload process of the joint queue
# ===========================================================
sM = np.sum(M)
Qwmm = ml.matrix(D0)
Qwpm = ml.zeros((N*sM, N))
Qwmp = ml.zeros((N, N*sM))
Qwpp = ml.zeros((N*sM, N*sM))
kix = 0
for i in range(K):
Qwmp[:,kix:kix+N*M[i]] = np.kron(D[i+1], sigma[i])
Qwpm[kix:kix+N*M[i],:] = np.kron(I,s[i])
Qwpp[kix:kix+N*M[i],:][:,kix:kix+N*M[i]] = np.kron(I,S[i])
kix += N*M[i]
# calculate fundamental matrices
Psiw, Kw, Uw = FluidFundamentalMatrices (Qwpp, Qwpm, Qwmp, Qwmm, 'PKU', precision)
# calculate boundary vector
Ua = ml.ones((N,1)) + 2*np.sum(Qwmp*(-Kw).I,1)
pm = Linsolve (ml.hstack((Uw,Ua)).T, ml.hstack((ml.zeros((1,N)),ml.ones((1,1)))).T).T
ro = ((1.0-np.sum(pm))/2.0)/(np.sum(pm)+(1.0-np.sum(pm))/2.0) # calc idle time with weight=1, and the busy time with weight=1/2
kappa = pm/np.sum(pm)
pi = CTMCSolve (sD)
lambd = []
for i in range(K):
lambd.append(np.sum(pi*D[i+1]))
Psiw = []
Qwmp = []
Qwzp = []
Qwpp = []
Qwmz = []
Qwpz = []
Qwzz = []
Qwmm = []
Qwpm = []
Qwzm = []
for k in range(K):
# step 2. construct a workload process for classes k...K
# ======================================================
Mlo = np.sum(M[:k])
Mhi = np.sum(M[k:])
Qkwpp = ml.zeros((N*Mlo*Mhi+N*Mhi, N*Mlo*Mhi+N*Mhi))
Qkwpz = ml.zeros((N*Mlo*Mhi+N*Mhi, N*Mlo))
Qkwpm = ml.zeros((N*Mlo*Mhi+N*Mhi, N))
Qkwmz = ml.zeros((N, N*Mlo))
Qkwmp = ml.zeros((N, N*Mlo*Mhi+N*Mhi))
Dlo = ml.matrix(D0)
for i in range(k):
Dlo = Dlo + D[i+1]
Qkwmm = Dlo
Qkwzp = ml.zeros((N*Mlo, N*Mlo*Mhi+N*Mhi))
Qkwzm = ml.zeros((N*Mlo, N))
Qkwzz = ml.zeros((N*Mlo, N*Mlo))
kix = 0
for i in range(k,K):
kix2 = 0
for j in range(k):
bs = N*M[j]*M[i]
bs2 = N*M[j]
Qkwpp[kix:kix+bs,kix:kix+bs] = np.kron(I,np.kron(ml.eye(M[j]),S[i]))
Qkwpz[kix:kix+bs,kix2:kix2+bs2] = np.kron(I,np.kron(ml.eye(M[j]),s[i]))
Qkwzp[kix2:kix2+bs2,kix:kix+bs] = np.kron(D[i+1],np.kron(ml.eye(M[j]), sigma[i]))
kix += bs
kix2 += bs2
for i in range(k,K):
bs = N*M[i]
Qkwpp[kix:kix+bs,:][:,kix:kix+bs] = np.kron(I,S[i])
Qkwpm[kix:kix+bs,:] = np.kron(I,s[i])
Qkwmp[:,kix:kix+bs] = np.kron(D[i+1],sigma[i])
kix += bs
kix = 0
for j in range(k):
bs = N*M[j]
Qkwzz[kix:kix+bs,kix:kix+bs] = np.kron(Dlo, ml.eye(M[j])) + np.kron(I, S[j])
Qkwzm[kix:kix+bs,:] = np.kron(I, s[j])
kix += bs
if Qkwzz.shape[0]>0:
Psikw = FluidFundamentalMatrices (Qkwpp+Qkwpz*(-Qkwzz).I*Qkwzp, Qkwpm+Qkwpz*(-Qkwzz).I*Qkwzm, Qkwmp, Qkwmm, 'P', precision)
else:
Psikw = FluidFundamentalMatrices (Qkwpp, Qkwpm, Qkwmp, Qkwmm, 'P', precision)
Psiw.append(Psikw)
Qwzp.append(Qkwzp)
Qwmp.append(Qkwmp)
Qwpp.append(Qkwpp)
Qwmz.append(Qkwmz)
Qwpz.append(Qkwpz)
Qwzz.append(Qkwzz)
Qwmm.append(Qkwmm)
Qwpm.append(Qkwpm)
Qwzm.append(Qkwzm)
# step 3. calculate Phi vectors
# =============================
lambdaS = sum(lambd)
phi = [(1-ro)*kappa*(-D0) / lambdaS]
q0 = [[]]
qL = [[]]
for k in range(K-1):
sDk = ml.matrix(D0)
for j in range(k+1):
sDk = sDk + D[j+1]
# pk
pk = sum(lambd[:k+1])/lambdaS - (1-ro)*kappa*np.sum(sDk,1)/lambdaS
# A^(k,1)
Qwzpk = Qwzp[k+1]
vix = 0
Ak = []
for ii in range(k+1):
bs = N*M[ii]
V1 = Qwzpk[vix:vix+bs,:]
Ak.append (np.kron(I,sigma[ii]) * (-np.kron(sDk,ml.eye(M[ii]))-np.kron(I,S[ii])).I * (np.kron(I,s[ii]) + V1*Psiw[k+1]))
vix += bs
# B^k
Qwmpk = Qwmp[k+1]
Bk = Qwmpk * Psiw[k+1]
ztag = phi[0]*((-D0).I*D[k+1]*Ak[k] - Ak[0] + (-D0).I*Bk)
for i in range(k):
ztag += phi[i+1]*(Ak[i]-Ak[i+1]) + phi[0]*(-D0).I*D[i+1]*Ak[i]
Mx = ml.eye(Ak[k].shape[0])-Ak[k]
Mx[:,0] = ml.ones((N,1))
phi.append(ml.hstack((pk, ztag[:,1:]))*Mx.I) # phi(k) = Psi^(k)_k * p(k). Psi^(k)_i = phi(i) / p(k)
q0.append(phi[0]*(-D0).I)
qLii = []
for ii in range(k+1):
qLii.append((phi[ii+1] - phi[ii] + phi[0]*(-D0).I*D[ii+1]) * np.kron(I,sigma[ii]) * (-np.kron(sDk,ml.eye(M[ii]))-np.kron(I,S[ii])).I)
qL.append(ml.hstack(qLii))
# step 4. calculate performance measures
# ======================================
Ret = []
for k in classes:
sD0k = ml.matrix(D0)
for i in range(k):
sD0k += D[i+1]
if k<K-1:
# step 4.1 calculate distribution of the workload process right
# before the arrivals of class k jobs
# ============================================================
if Qwzz[k].shape[0]>0:
Kw = Qwpp[k]+Qwpz[k]*(-Qwzz[k]).I*Qwzp[k] + Psiw[k]*Qwmp[k]
else:
Kw = Qwpp[k] + Psiw[k]*Qwmp[k]
BM = ml.zeros((0,0))
CM = ml.zeros((0,N))
DM = ml.zeros((0,0))
for i in range(k):
BM = la.block_diag(BM,np.kron(I,S[i]))
CM = ml.vstack((CM, np.kron(I,s[i])))
DM = la.block_diag(DM,np.kron(D[k+1],ml.eye(M[i])))
if k>0:
Kwu = ml.vstack((ml.hstack((Kw, (Qwpz[k]+Psiw[k]*Qwmz[k])*(-Qwzz[k]).I*DM)), ml.hstack((ml.zeros((BM.shape[0],Kw.shape[1])), BM))))
Bwu = ml.vstack((Psiw[k]*D[k+1], CM))
iniw = ml.hstack((q0[k]*Qwmp[k]+qL[k]*Qwzp[k], qL[k]*DM))
pwu = q0[k]*D[k+1]
else:
Kwu = Kw
Bwu = Psiw[k]*D[k+1]
iniw = pm*Qwmp[k]
pwu = pm*D[k+1]
norm = np.sum(pwu) + np.sum(iniw*(-Kwu).I*Bwu)
pwu = pwu / norm
iniw = iniw / norm
# step 4.2 create the fluid model whose first passage time equals the
# WAITING time of the low prioroity customers
# ==================================================================
KN = Kwu.shape[0]
Qspp = ml.zeros((KN+N*np.sum(M[k+1:]), KN+N*np.sum(M[k+1:])))
Qspm = ml.zeros((KN+N*np.sum(M[k+1:]), N))
Qsmp = ml.zeros((N, KN+N*np.sum(M[k+1:])))
Qsmm = sD0k + D[k+1]
kix = 0
for i in range(k+1,K):
bs = N*M[i]
Qspp[KN+kix:KN+kix+bs,:][:,KN+kix:KN+kix+bs] = np.kron(I,S[i])
Qspm[KN+kix:KN+kix+bs,:] = np.kron(I,s[i])
Qsmp[:,KN+kix:KN+kix+bs] = np.kron(D[i+1],sigma[i])
kix += bs
Qspp[:KN,:][:,:KN] = Kwu
Qspm[:KN,:] = Bwu
inis = ml.hstack((iniw, ml.zeros((1,N*np.sum(M[k+1:])))))
# calculate fundamental matrix
Psis = FluidFundamentalMatrices (Qspp, Qspm, Qsmp, Qsmm, 'P', precision)
# step 4.3. calculate the performance measures
# ==========================================
argIx = 0
while argIx<len(argv):
if argIx in eaten:
argIx += 1
continue
elif type(argv[argIx]) is str and argv[argIx]=="stMoms":
# MOMENTS OF THE SOJOURN TIME
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfSTMoms = argv[argIx+1]
# calculate waiting time moments
Pn = [Psis]
wtMoms = []
for n in range(1,numOfSTMoms+1):
A = Qspp + Psis*Qsmp
B = Qsmm + Qsmp*Psis
C = -2*n*Pn[n-1]
bino = 1
for i in range(1,n):
bino = bino * (n-i+1) / i
C += bino * Pn[i]*Qsmp*Pn[n-i]
P = la.solve_sylvester(A, B, -C)
Pn.append(P)
wtMoms.append(np.sum(inis*P*(-1)**n) / 2**n)
# calculate RESPONSE time moments
Pnr = [np.sum(inis*Pn[0])*sigma[k]]
rtMoms = []
for n in range(1,numOfSTMoms+1):
P = n*Pnr[n-1]*(-S[k]).I + (-1)**n*np.sum(inis*Pn[n])*sigma[k] / 2**n
Pnr.append(P)
rtMoms.append(np.sum(P)+np.sum(pwu)*math.factorial(n)*np.sum(sigma[k]*(-S[k]).I**n))
Ret.append(rtMoms)
argIx += 1
elif type(argv[argIx]) is str and argv[argIx]=="stDistr":
# DISTRIBUTION OF THE SOJOURN TIME
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
stCdfPoints = argv[argIx+1]
res = []
for t in stCdfPoints:
L = erlMaxOrder
lambdae = L/t/2
Psie = FluidFundamentalMatrices (Qspp-lambdae*ml.eye(Qspp.shape[0]), Qspm, Qsmp, Qsmm-lambdae*ml.eye(Qsmm.shape[0]), 'P', precision)
Pn = [Psie]
pr = (np.sum(pwu) + np.sum(inis*Psie)) * (1-np.sum(sigma[k]*(ml.eye(S[k].shape[0])-S[k]/2/lambdae).I**L))
for n in range(1,L):
A = Qspp + Psie*Qsmp - lambdae*ml.eye(Qspp.shape[0])
B = Qsmm + Qsmp*Psie - lambdae*ml.eye(Qsmm.shape[0])
C = 2*lambdae*Pn[n-1]
for i in range(1,n):
C += Pn[i]*Qsmp*Pn[n-i]
P = la.solve_sylvester(A, B, -C)
Pn.append(P)
pr += np.sum(inis*P) * (1-np.sum(sigma[k]*(np.eye(S[k].shape[0])-S[k]/2/lambdae).I**(L-n)))
res.append(pr)
Ret.append(np.array(res))
argIx += 1
elif type(argv[argIx]) is str and (argv[argIx]=="ncMoms" or argv[argIx]=="ncDistr"):
W = (-np.kron(sD-D[k+1],ml.eye(M[k]))-np.kron(I,S[k])).I*np.kron(D[k+1],ml.eye(M[k]))
iW = (ml.eye(W.shape[0])-W).I
w = np.kron(ml.eye(N),sigma[k])
omega = (-np.kron(sD-D[k+1],ml.eye(M[k]))-np.kron(I,S[k])).I*np.kron(I,s[k])
if argv[argIx]=="ncMoms":
# MOMENTS OF THE NUMBER OF JOBS
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfQLMoms = argv[argIx+1]
# first calculate it at departure instants
Psii = [Psis]
QLDPn = [inis*Psii[0]*w*iW]
for n in range(1,numOfQLMoms+1):
A = Qspp + Psis*Qsmp
B = Qsmm + Qsmp*Psis
C = n*Psii[n-1]*D[k+1]
bino = 1
for i in range(1,n):
bino = bino * (n-i+1) / i
C = C + bino * Psii[i]*Qsmp*Psii[n-i]
P = la.solve_sylvester(A, B, -C)
Psii.append(P)
QLDPn.append(n*QLDPn[n-1]*iW*W + inis*P*w*iW)
for n in range(numOfQLMoms+1):
QLDPn[n] = (QLDPn[n] + pwu*w*iW**(n+1)*W**n)*omega
# now calculate it at random time instance
QLPn = [pi]
qlMoms = []
iTerm = (ml.ones((N,1))*pi - sD).I
for n in range(1,numOfQLMoms+1):
sumP = np.sum(QLDPn[n]) + n*np.sum((QLDPn[n-1] - QLPn[n-1]*D[k+1]/lambd[k])*iTerm*D[k+1])
P = sumP*pi + n*(QLPn[n-1]*D[k+1] - QLDPn[n-1]*lambd[k])*iTerm
QLPn.append(P)
qlMoms.append(np.sum(P))
qlMoms = MomsFromFactorialMoms(qlMoms)
Ret.append(qlMoms)
argIx += 1
elif argv[argIx]=="ncDistr":
# DISTRIBUTION OF THE NUMBER OF JOBS
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfQLProbs = argv[argIx+1]
Psid = FluidFundamentalMatrices (Qspp, Qspm, Qsmp, sD0k, 'P', precision)
Pn = [Psid]
XDn = inis*Psid*w
dqlProbs = (XDn+pwu*w)*omega
for n in range(1,numOfQLProbs):
A = Qspp + Psid*Qsmp
B = sD0k + Qsmp*Psid
C = Pn[n-1]*D[k+1]
for i in range(1,n):
C += Pn[i]*Qsmp*Pn[n-i]
P = la.solve_sylvester(A, B, -C)
Pn.append(P)
XDn = XDn*W + inis*P*w
dqlProbs = ml.vstack((dqlProbs, (XDn+pwu*w*W**n)*omega))
# now calculate it at random time instance
iTerm = -(sD-D[k+1]).I
qlProbs = lambd[k]*dqlProbs[0,:]*iTerm
for n in range(1,numOfQLProbs):
P = (qlProbs[n-1,:]*D[k+1]+lambd[k]*(dqlProbs[n,:]-dqlProbs[n-1,:]))*iTerm
qlProbs = ml.vstack((qlProbs, P))
qlProbs = np.sum(qlProbs,1).A.flatten()
Ret.append(qlProbs)
argIx += 1
else:
raise Exception("MMAPPH1NPPR: Unknown parameter "+str(argv[argIx]))
argIx += 1
elif k==K-1:
# step 3. calculate the performance measures
# ==========================================
argIx = 0
while argIx<len(argv):
if argIx in eaten:
argIx += 1
continue
elif type(argv[argIx]) is str and (argv[argIx]=="stMoms" or argv[argIx]=="stDistr"):
Kw = Qwpp[k]+Qwpz[k]*(-Qwzz[k]).I*Qwzp[k] + Psiw[k]*Qwmp[k]
AM = ml.zeros((0,0))
BM = ml.zeros((0,0))
CM = ml.zeros((0,1))
DM = ml.zeros((0,0))
for i in range(k):
AM = la.block_diag(AM,np.kron(ml.ones((N,1)),np.kron(ml.eye(M[i]),s[k])))
BM = la.block_diag(BM,S[i])
CM = ml.vstack((CM, s[i]))
DM = la.block_diag(DM,np.kron(D[k+1],ml.eye(M[i])))
Z = ml.vstack((ml.hstack((Kw, ml.vstack((AM,ml.zeros((N*M[k],AM.shape[1])))))), ml.hstack((ml.zeros((BM.shape[0],Kw.shape[1])), BM))))
z = ml.vstack((ml.zeros((AM.shape[0],1)), np.kron(ml.ones((N,1)),s[k]), CM))
iniw = ml.hstack((q0[k]*Qwmp[k]+qL[k]*Qwzp[k], ml.zeros((1,BM.shape[0]))))
zeta = iniw/np.sum(iniw*(-Z).I*z)
if argv[argIx]=="stMoms":
# MOMENTS OF THE SOJOURN TIME
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfSTMoms = argv[argIx+1]
rtMoms = []
for i in range(1,numOfSTMoms+1):
rtMoms.append(np.sum(math.factorial(i)*zeta*(-Z).I**(i+1)*z))
Ret.append(rtMoms)
argIx += 1
if argv[argIx]=="stDistr":
# DISTRIBUTION OF THE SOJOURN TIME
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
stCdfPoints = argv[argIx+1]
rtDistr = []
for t in stCdfPoints:
rtDistr.append (np.sum(zeta*(-Z).I*(ml.eye(Z.shape[0])-la.expm(Z*t))*z))
Ret.append(np.array(rtDistr))
argIx += 1
elif type(argv[argIx]) is str and (argv[argIx]=="ncMoms" or argv[argIx]=="ncDistr"):
L = ml.zeros((N*np.sum(M),N*np.sum(M)))
B = ml.zeros((N*np.sum(M),N*np.sum(M)))
F = ml.zeros((N*np.sum(M),N*np.sum(M)))
kix = 0
for i in range(K):
bs = N*M[i]
F[kix:kix+bs,:][:,kix:kix+bs] = np.kron(D[k+1],ml.eye(M[i]))
L[kix:kix+bs,:][:,kix:kix+bs] = np.kron(sD0k,ml.eye(M[i])) + np.kron(I,S[i])
if i<K-1:
L[kix:kix+bs,:][:,N*np.sum(M[:k]):] = np.kron(I,s[i]*sigma[k])
else:
B[kix:kix+bs,:][:,N*np.sum(M[:k]):] = np.kron(I,s[i]*sigma[k])
kix += bs
R = QBDFundamentalMatrices (B, L, F, 'R', precision)
p0 = ml.hstack((qL[k], q0[k]*np.kron(I,sigma[k])))
p0 = p0/np.sum(p0*(ml.eye(R.shape[0])-R).I)
if argv[argIx]=="ncMoms":
# MOMENTS OF THE NUMBER OF JOBS
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfQLMoms = argv[argIx+1]
qlMoms = []
for i in range(1,numOfQLMoms+1):
qlMoms.append(np.sum(math.factorial(i)*p0*R**i*(ml.eye(R.shape[0])-R).I**(i+1)))
Ret.append(MomsFromFactorialMoms(qlMoms))
elif argv[argIx]=="ncDistr":
# DISTRIBUTION OF THE NUMBER OF JOBS
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
numOfQLProbs = argv[argIx+1]
qlProbs = [np.sum(p0)]
for i in range(1,numOfQLProbs):
qlProbs.append(np.sum(p0*R**i))
Ret.append(np.array(qlProbs))
argIx += 1
else:
raise Exception("MMAPPH1NPPR: Unknown parameter "+str(argv[argIx]))
argIx += 1
if len(Ret)==1:
return Ret[0]
else:
return Ret
def test_trivial(self):
a = np.array([[1.0, 0.0], [0.0, 1.0]])
b = np.array([[1.0]])
c = np.array([2.0, 2.0]).reshape(-1,1)
x = solve_sylvester(a, b, c)
assert_array_almost_equal(x, np.array([1.0, 1.0]).reshape(-1,1))
def MMAPPH1FCFS(D, sigma, S, *argv):
"""
Returns various performane measures of a MMAP[K]/PH[K]/1
first-come-first-serve queue, see [1]_.
Parameters
----------
D : list of matrices of shape (N,N), length (K+1)
The D0...DK matrices of the arrival process.
sigma : list of row vectors, length (K)
The list containing the initial probability vectors of the service
time distributions of the various customer types. The length of the
vectors does not have to be the same.
S : list of square matrices, length (K)
The transient generators of the phase type distributions representing
the service time of the jobs belonging to various types.
further parameters :
The rest of the function parameters specify the options
and the performance measures to be computed.
The supported performance measures and options in this
function are:
+----------------+--------------------+----------------------------------------+
| Parameter name | Input parameters | Output |
+================+====================+========================================+
| "ncMoms" | Number of moments | The moments of the number of customers |
+----------------+--------------------+----------------------------------------+
| "ncDistr" | Upper limit K | The distribution of the number of |
| | | customers from level 0 to level K-1 |
+----------------+--------------------+----------------------------------------+
| "stMoms" | Number of moments | The sojourn time moments |
+----------------+--------------------+----------------------------------------+
| "stDistr" | A vector of points | The sojourn time distribution at the |
| | | requested points (cummulative, cdf) |
+----------------+--------------------+----------------------------------------+
| "stDistrME" | None | The vector-matrix parameters of the |
| | | matrix-exponentially distributed |
| | | sojourn time distribution |
+----------------+--------------------+----------------------------------------+
| "stDistrPH" | None | The vector-matrix parameters of the |
| | | matrix-exponentially distributed |
| | | sojourn time distribution, converted |
| | | to a continuous PH representation |
+----------------+--------------------+----------------------------------------+
| "prec" | The precision | Numerical precision used as a stopping |
| | | condition when solving the Riccati |
| | | equation |
+----------------+--------------------+----------------------------------------+
| "classes" | Vector of integers | Only the performance measures |
| | | belonging to these classes are |
| | | returned. If not given, all classes |
| | | are analyzed. |
+----------------+--------------------+----------------------------------------+
(The quantities related to the number of customers in
the system include the customer in the server, and the
sojourn time related quantities include the service
times as well)
Returns
-------
Ret : list of the performance measures
Each entry of the list corresponds to a performance
measure requested. Each entry is a matrix, where the
columns belong to the various job types.
If there is just a single item,
then it is not put into a list.
References
----------
.. [1] Qiming He, "Analysis of a continuous time
SM[K]/PH[K]/1/FCFS queue: Age process, sojourn times,
and queue lengths", Journal of Systems Science and
Complexity, 25(1), pp 133-155, 2012.
"""
K = len(D)-1
# parse options
eaten = []
precision = 1e-14;
classes = np.arange(0,K)
for i in range(len(argv)):
if argv[i]=="prec":
precision = argv[i+1]
eaten.append(i)
eaten.append(i+1)
elif argv[i]=="classes":
classes = np.array(argv[i+1])-1
eaten.append(i)
eaten.append(i+1)
if butools.checkInput and not CheckMMAPRepresentation(D):
raise Exception('MMAPPH1FCFS: The arrival process is not a valid MMAP representation!')
if butools.checkInput:
for k in range(K):
if not CheckPHRepresentation(sigma[k],S[k]):
raise Exception('MMAPPH1FCFS: the vector and matrix describing the service times is not a valid PH representation!')
# some preparation
D0 = D[0]
N = D0.shape[0]
Ia = ml.eye(N);
Da = ml.zeros((N,N))
for q in range(K):
Da += D[q+1]
theta = CTMCSolve(D0+Da)
beta = [CTMCSolve(S[k]+ml.sum(-S[k],1)*sigma[k]) for k in range(K)]
lambd = [np.sum(theta*D[k+1]) for k in range(K)]
mu = [np.sum(beta[k]*(-S[k])) for k in range(K)]
Nsk = [S[k].shape[0] for k in range(K)]
ro = np.sum(np.array(lambd)/np.array(mu))
alpha = theta*Da/sum(lambd)
D0i = (-D0).I
Sa = S[0];
sa = [ml.zeros(sigma[0].shape)]*K
sa[0] = sigma[0]
ba = [ml.zeros(beta[0].shape)]*K
ba[0] = beta[0]
sv = [ml.zeros((Nsk[0],1))]*K
sv[0] = ml.sum(-S[0],1)
Pk = [D0i*D[q+1] for q in range(K)]
for k in range(1,K):
Sa = la.block_diag(Sa, S[k])
for q in range(K):
if q==k:
sa[q] = ml.hstack((sa[q], sigma[k]))
ba[q] = ml.hstack((ba[q], beta[k]))
sv[q] = ml.vstack((sv[q], -np.sum(S[k],1)))
else:
sa[q] = ml.hstack((sa[q], ml.zeros(sigma[k].shape)))
ba[q] = ml.hstack((ba[q], ml.zeros(beta[k].shape)))
sv[q] = ml.vstack((sv[q], ml.zeros((Nsk[k],1))))
Sa = ml.matrix(Sa)
P = D0i*Da
iVec = ml.kron(D[1],sa[0])
for k in range(1,K):
iVec += ml.kron(D[k+1],sa[k])
Ns = Sa.shape[0]
Is = ml.eye(Ns)
# step 1. solve the age process of the queue
# ==========================================
# solve Y0 and calculate T
Y0 = FluidFundamentalMatrices (ml.kron(Ia,Sa), ml.kron(Ia,-ml.sum(Sa,1)), iVec, D0, "P", precision)
T = ml.kron(Ia,Sa) + Y0 * iVec
# calculate pi0 and v0
pi0 = ml.zeros((1,T.shape[0]))
for k in range(K):
pi0 += ml.kron(theta*D[k+1],ba[k]/mu[k])
pi0 = - pi0 * T
iT = (-T).I
oa = ml.ones((N,1))
# step 2. calculate performance measures
# ======================================
Ret = []
for k in classes:
argIx = 0
clo = iT*ml.kron(oa,sv[k])
while argIx<len(argv):
if argIx in eaten:
argIx += 1
continue
elif type(argv[argIx]) is str and argv[argIx]=="stMoms":
numOfSTMoms = argv[argIx+1]
rtMoms = []
for m in range(1,numOfSTMoms+1):
rtMoms.append(math.factorial(m) * np.sum(pi0 * iT**m * clo / (pi0*clo)))
Ret.append(rtMoms)
argIx += 1
elif type(argv[argIx]) is str and argv[argIx]=="stDistr":
stCdfPoints = argv[argIx+1]
cdf = [];
for t in stCdfPoints:
pr = 1 - np.sum(pi0 * la.expm(T*t) * clo / (pi0*clo))
cdf.append(pr)
Ret.append(np.array(cdf))
argIx += 1
elif type(argv[argIx]) is str and argv[argIx]=="stDistrME":
Bm = SimilarityMatrixForVectors(clo/(pi0*clo),ml.ones((N*Ns,1)))
Bmi = Bm.I
A = Bm * T * Bmi
alpha = pi0 * Bmi
Ret.append(alpha)
Ret.append(A)
elif type(argv[argIx]) is str and argv[argIx]=="stDistrPH":
vv = pi0*iT
ix = np.arange(N*Ns)
nz = ix[vv.flat>precision]
delta = Diag(vv[:,nz])
cl = -T*clo/(pi0*clo)
alpha = cl[nz,:].T*delta
A = delta.I*T[nz,:][:,nz].T*delta
Ret.append(alpha)
Ret.append(A)
elif type(argv[argIx]) is str and argv[argIx]=="ncDistr":
numOfQLProbs = argv[argIx+1]
argIx += 1
values = np.empty(numOfQLProbs)
jm = ml.zeros((Ns,1))
jm[np.sum(Nsk[0:k]):np.sum(Nsk[0:k+1]),:] = 1
jmc = ml.ones((Ns,1))
jmc[np.sum(Nsk[0:k]):np.sum(Nsk[0:k+1]),:] = 0
LmCurr = la.solve_sylvester(T, ml.kron(D0+Da-D[k+1],Is), -ml.eye(N*Ns))
values[0] = 1-ro+np.sum(pi0*LmCurr*ml.kron(oa,jmc))
for i in range(1,numOfQLProbs):
LmPrev = LmCurr
LmCurr = la.solve_sylvester(T, ml.kron(D0+Da-D[k+1],Is), -LmPrev*ml.kron(D[k+1],Is))
values[i] = np.sum(pi0*LmCurr*ml.kron(oa,jmc) + pi0*LmPrev*ml.kron(oa,jm));
Ret.append(values)
elif type(argv[argIx]) is str and argv[argIx]=="ncMoms":
numOfQLMoms = argv[argIx+1]
argIx += 1
jm = ml.zeros((Ns,1))
jm[np.sum(Nsk[0:k]):np.sum(Nsk[0:k+1]),:] = 1
ELn = [la.solve_sylvester(T, ml.kron(D0+Da,Is), -ml.eye(N*Ns))]
qlMoms = []
for n in range(1,numOfQLMoms+1):
bino = 1
Btag = ml.zeros((N*Ns,N*Ns))
for i in range(n):
Btag += bino * ELn[i]
bino *= (n-i) / (i+1)
ELn.append(la.solve_sylvester(T, ml.kron(D0+Da,Is), -Btag*ml.kron(D[k+1],Is)))
qlMoms.append(np.sum(pi0*ELn[n]) + np.sum(pi0*Btag*ml.kron(oa,jm)))
Ret.append(qlMoms)
else:
raise Exception("MMAPPH1FCFS: Unknown parameter "+str(argv[argIx]))
argIx += 1
if len(Ret)==1:
return Ret[0]
else:
return Ret
