def estimate_m_matrix(self):
if self.m_matrix is None:
self.m_matrix = np.zeros(shape=(len(self.albedos), 3, 10))
counter = 0
for k, pixel_list in self.rgb_clusters.items():
if self.albedos[counter][0] > 0 and self.albedos[counter][1] > 0 and self.albedos[counter][2] > 0:
A = []
bs = []
for pixel in pixel_list:
i, j = pixel
n1, n2, n3 = self.depth_normals[i][j]
if n1 != 0 or n2 != 0 or n3 != 0:
bs.append(self.color_map[i][j] / self.albedos[counter])
A.append([n1 * n1, 2 * n1 * n2, 2 * n1 * n3, n1,
n2 * n2, 2 * n2 * n3, n2,
n3 * n3, n3,
1])
if not bs:
counter += 1
continue
A = np.array(A)
bs = np.array(bs)
red_lighting = lsq_linear(A, bs[:, 0])
self.m_matrix[counter][0] = red_lighting.x
green_lighting = lsq_linear(A, bs[:, 1])
self.m_matrix[counter][1] = green_lighting.x
blue_lighting = lsq_linear(A, bs[:, 2])
self.m_matrix[counter][2] = blue_lighting.x
counter += 1
def test_dense_rank_deficient(self):
A = np.array([[-0.307, -0.184]])
b = np.array([0.773])
lb = [-0.1, -0.1]
ub = [0.1, 0.1]
for lsq_solver in self.lsq_solvers:
res = lsq_linear(A,
b, (lb, ub),
method=self.method,
lsq_solver=lsq_solver)
assert_allclose(res.x, [-0.1, -0.1])
A = np.array([
[0.334, 0.668],
[-0.516, -1.032],
[0.192, 0.384],
])
b = np.array([-1.436, 0.135, 0.909])
lb = [0, -1]
ub = [1, -0.5]
for lsq_solver in self.lsq_solvers:
res = lsq_linear(A,
b, (lb, ub),
method=self.method,
lsq_solver=lsq_solver)
assert_allclose(res.optimality, 0, atol=1e-11)
def AICSP(self):
#A CSP approach to the problem... empty board, make first move.
global matrixOne
global vectorSolutions
global next_Button_Open
global vectorSolutions
global outcome
global terminate
if (self.newBoard() == True):
vectorSolutions = np.array([])
self.open_button(numRows // 2, numCols // 2)
#set up matrix for first time...
for row in range(0, numRows):
for col in range(0, numCols):
if (
but[row][col]['state'] == 'disabled'
and but[row][col]["text"] != ""
): #a button that is opened and is not useless (numbered)
vectorSolutions = np.append(
vectorSolutions, [int(but[row][col]["text"])])
temp = np.zeros(numRows * numCols)
temp = self.add_3by3_constraint(temp, row, col)
matrixOne = np.append(matrixOne,
np.array([temp]),
axis=0)
myAns = lsq_linear(matrixOne,
vectorSolutions,
bounds=(-0.0000, 1.0001)).x
myAns = np.reshape(myAns, (numRows, numCols))
next_Button_Open = self.findCurrMin(myAns)
but[next_Button_Open[0][0]][next_Button_Open[0][1]]["bg"] = "blue"
#append... solve... display... find min/solve... click...
# click...
else:
self.open_button(next_Button_Open[0][0], next_Button_Open[0][1])
if (terminate == True):
return
for row in range(0, numRows):
for col in range(0, numCols):
if (but[row][col]['state'] == 'disabled'
and but[row][col]["text"] != ""):
vectorSolutions = np.append(
vectorSolutions, [int(but[row][col]["text"])])
temp = np.zeros(numRows * numCols)
temp = self.add_3by3_constraint(temp, row, col)
matrixOne = np.append(matrixOne,
np.array([temp]),
axis=0)
myAns = lsq_linear(matrixOne,
vectorSolutions,
bounds=(-0.0000, 1.0001)).x
myAns = np.reshape(myAns, (numRows, numCols))
next_Button_Open = self.findCurrMin(myAns)
if (next_Button_Open == [-1, -1]):
next_Button_Open = self.moveLeftIssue(myAns)
but[next_Button_Open[0][0]][next_Button_Open[0][1]]["bg"] = "blue"
def test_option_lsmr_maxiter(self):
# Should work with positive integers or None
_ = lsq_linear(A, b, lsq_solver='lsmr', lsmr_maxiter=1)
_ = lsq_linear(A, b, lsq_solver='lsmr', lsmr_maxiter=None)
# Should raise error with 0 or negative max iter
err_message = "`lsmr_maxiter` must be None or positive integer."
with pytest.raises(ValueError, match=err_message):
_ = lsq_linear(A, b, lsq_solver='lsmr', lsmr_maxiter=0)
with pytest.raises(ValueError, match=err_message):
_ = lsq_linear(A, b, lsq_solver='lsmr', lsmr_maxiter=-1)
def test_sparse_and_LinearOperator(self):
m = 5000
n = 1000
A = rand(m, n, random_state=0)
b = self.rnd.randn(m)
res = lsq_linear(A, b)
assert_allclose(res.optimality, 0, atol=1e-6)
A = aslinearoperator(A)
res = lsq_linear(A, b)
assert_allclose(res.optimality, 0, atol=1e-6)
def test_sparse_and_LinearOperator(self):
m = 5000
n = 1000
A = rand(m, n, random_state=0)
b = self.rnd.randn(m)
res = lsq_linear(A, b)
assert_allclose(res.optimality, 0, atol=1e-6)
A = aslinearoperator(A)
res = lsq_linear(A, b)
assert_allclose(res.optimality, 0, atol=1e-6)
def update_weight_map(bone_transforms, rest_bones_t, poses, rest_pose, sparseness):
"""
Update the bone-vertex weight map W by fixing bone transformations and using a least squares
solver subject to non-negativity constraint, affinity constraint, and sparseness constraint.
inputs: bone_transforms |num_bones| x |num_poses| x 4 x 3 matrix representing the stacked
Rotation and Translation for each pose, for each bone.
rest_bones_t |num_bones| x 3 matrix representing the translations of the rest bones
poses |num_poses| x |num_verts| x 3 matrix representing coordinates of vertices of each pose
rest_pose |num_verts| x 3 numpy matrix representing the coordinates of vertices in rest pose
sparseness Maximum number of bones allowed to influence a particular vertex
return: A |num_verts| x |num_bones| weight map representing the influence of the jth bone on the ith vertex
"""
num_verts = rest_pose.shape[0]
num_poses = poses.shape[0]
num_bones = bone_transforms.shape[0]
W = np.empty((num_verts, num_bones))
for v in range(num_verts):
# For every vertex, solve a least squares problem
Rp = np.empty((num_bones, num_poses, 3))
for bone in range(num_bones):
Rp[bone] = bone_transforms[bone,:,:3,:].dot(rest_pose[v] - rest_bones_t[bone]) # |num_bones| x |num_poses| x 3
# R * p + T
Rp_T = Rp + bone_transforms[:, :, 3, :] # |num_bones| x |num_poses| x 3
A = Rp_T.transpose((1, 2, 0)).reshape((3 * num_poses, num_bones)) # 3 * |num_poses| x |num_bones|
b = poses[:, v, :].reshape(3 * num_poses) # 3 * |num_poses| x 1
# Bounds ensure non-negativity constraint and kind of affinity constraint
w = lsq_linear(A, b, bounds=(0, 1), method='bvls').x # |num_bones| x 1
w /= np.sum(w) # Ensure that w sums to 1 (affinity constraint)
# Remove |B| - |K| bone weights with the least "effect"
effect = np.linalg.norm((A * w).reshape(num_poses, 3, num_bones), axis=1) # |num_poses| x |num_bones|
effect = np.sum(effect, axis=0) # |num_bones| x 1
num_discarded = max(num_bones - sparseness, 0)
effective = np.argpartition(effect, num_discarded)[num_discarded:] # |sparseness| x 1
# Run least squares again, but only use the most effective bones
A_reduced = A[:, effective] # 3 * |num_poses| x |sparseness|
w_reduced = lsq_linear(A_reduced, b, bounds=(0, 1), method='bvls').x # |sparseness| x 1
w_reduced /= np.sum(w_reduced) # Ensure that w sums to 1 (affinity constraint)
w_sparse = np.zeros(num_bones)
w_sparse[effective] = w_reduced
w_sparse /= np.sum(w_sparse) # Ensure that w_sparse sums to 1 (affinity constraint)
W[v] = w_sparse
return W
def fit_linear_nnls(self, X, y, sample_weight=None):
if not isinstance(self.model, LinearRegression):
raise ValueError(
'Model is not linearRegression, could not call fit for linear nnls'
)
n_jobs_ = self.model.n_jobs
self.model.coef_ = []
X, y = check_X_y(X,
y,
accept_sparse=['csr', 'csc', 'coo'],
y_numeric=True,
multi_output=True)
if sample_weight is not None and np.atleast_1d(sample_weight).ndim > 1:
raise ValueError("Sample weights must be 1D array or scalar")
X, y, X_offset, y_offset, X_scale = self.model._preprocess_data(
X,
y,
fit_intercept=self.model.fit_intercept,
normalize=self.model.normalize,
copy=self.model.copy_X,
sample_weight=sample_weight)
if sample_weight is not None:
# Sample weight can be implemented via a simple rescaling.
X, y = _rescale_data(X, y, sample_weight)
if sp.issparse(X):
if y.ndim < 2:
# out = sparse_lsqr(X, y)
out = lsq_linear(X, y, bounds=(0, np.Inf))
self.model.coef_ = out[0]
self.model._residues = out[3]
else:
# sparse_lstsq cannot handle y with shape (M, K)
outs = Parallel(n_jobs=n_jobs_)(
delayed(lsq_linear)(X, y[:, j].ravel())
for j in range(y.shape[1]))
self.model.coef_ = np.vstack(out[0] for out in outs)
self.model._residues = np.vstack(out[3] for out in outs)
else:
# self.model.coef_, self.model.cost_, self.model.fun_, self.model.optimality_, self.model.active_mask_,
# self.model.nit_, self.model.status_, self.model.message_, self.model.success_\
out = lsq_linear(X, y, bounds=(0, np.Inf))
self.model.coef_ = out.x
self.model.coef_ = self.model.coef_.T
if y.ndim == 1:
self.model.coef_ = np.ravel(self.model.coef_)
self.model._set_intercept(X_offset, y_offset, X_scale)
return self.model
def update_weight_map(self, bone_transforms, rest_bones_t, poses,
rest_pose, sparseness):
num_verts = rest_pose.shape[0]
num_poses = poses.shape[0]
num_bones = bone_transforms.shape[0]
W = np.empty((num_verts, num_bones))
for v in range(num_verts):
# For every vertex, solve a least squares problem
Rp = np.empty((num_bones, num_poses, 3))
for bone in range(num_bones):
Rp[bone] = bone_transforms[bone, :, :3, :].dot(
rest_pose[v] -
rest_bones_t[bone]) # |num_bones| x |num_poses| x 3
# R * p + T
Rp_T = Rp + bone_transforms[:, :,
3, :] # |num_bones| x |num_poses| x 3
A = Rp_T.transpose((1, 2, 0)).reshape(
(3 * num_poses, num_bones)) # 3 * |num_poses| x |num_bones|
b = poses[:, v, :].reshape(3 * num_poses) # 3 * |num_poses| x 1
# Bounds ensure non-negativity constraint and kind of affinity constraint
w = lsq_linear(A, b, bounds=(0, 1),
method='bvls').x # |num_bones| x 1
w /= np.sum(w) # Ensure that w sums to 1 (affinity constraint)
# Remove |B| - |K| bone weights with the least "effect"
effect = np.linalg.norm((A * w).reshape(num_poses, 3, num_bones),
axis=1) # |num_poses| x |num_bones|
effect = np.sum(effect, axis=0) # |num_bones| x 1
num_discarded = max(num_bones - sparseness, 0)
effective = np.argpartition(
effect, num_discarded)[num_discarded:] # |sparseness| x 1
# Run least squares again, but only use the most effective bones
A_reduced = A[:, effective] # 3 * |num_poses| x |sparseness|
w_reduced = lsq_linear(A_reduced, b, bounds=(0, 1),
method='bvls').x # |sparseness| x 1
w_reduced /= np.sum(
w_reduced) # Ensure that w sums to 1 (affinity constraint)
w_sparse = np.zeros(num_bones)
w_sparse[effective] = w_reduced
w_sparse /= np.sum(
w_sparse
) # Ensure that w_sparse sums to 1 (affinity constraint)
W[v] = w_sparse
return W
def test_convergence_small_matrix(self):
A = np.array([[49.0, 41.0, -32.0], [-19.0, -32.0, -8.0],
[-13.0, 10.0, 69.0]])
b = np.array([-41.0, -90.0, 47.0])
bounds = np.array([[31.0, -44.0, 26.0], [54.0, -32.0, 28.0]])
x_bvls = lsq_linear(A, b, bounds=bounds, method='bvls').x
x_trf = lsq_linear(A, b, bounds=bounds, method='trf').x
cost_bvls = np.sum((A @ x_bvls - b)**2)
cost_trf = np.sum((A @ x_trf - b)**2)
assert_(abs(cost_bvls - cost_trf) < cost_trf * 1e-10)
def SparsifyDynamics(Theta, dxdt, llambda):
# Theta = [ x(0) x(1) ... x(n)
# y(0) y(1) ... y(n)
# z(0) z(1) ... z(n) ]
xdim = len(dxdt)
par_num = len(Theta[0])
opt_num = 0 # [0: lsq_linear, 1: leastsq]
# 初期解
for i in range(xdim):
# 最適化手法の選択
if opt_num == 0:
xi_i = lsq_linear(Theta.T, dxdt[i])
xi_i = xi_i.x
elif opt_num == 1:
xi_i = leastsq_for_matrix(Theta, dxdt[i])
xi_i = xi_i[0]
print(xi_i)
if i == 0:
Xi = np.array([xi_i])
else:
Xi = np.append(Xi, np.array([xi_i]), axis=0)
# 閾値以下は無視して、残ったものでもう一度回帰
for _ in range(10): # 10回回す様にする
for i in range(xdim):
useID = [n for n in range(len(Xi[i])) if abs(Xi[i][n]) > llambda]
zeroID = [n for n in range(len(Xi[i])) if abs(Xi[i][n]) <= llambda]
# Xiの不要な部分を全てゼロに
for j in zeroID:
Xi[i][j] = 0
# Thetaの中で必要なものだけをまとめる
Theta_tmp = np.array([Theta[n] for n in useID])
print(Theta_tmp.shape, dxdt[i].shape)
# もう一度回帰
if opt_num == 0:
xi_i = lsq_linear(Theta_tmp.T, dxdt[i])
xi_i = xi_i.x
if opt_num == 1:
xi_i = leastsq_for_matrix(Theta_tmp, dxdt[i])
xi_i = xi_i[0]
print(xi_i)
# Xiに代入
for n, xi_n in enumerate(useID):
Xi[i][xi_n] = xi_i[n]
return Xi
def test_sparse_bounds(self):
m = 5000
n = 1000
A = rand(m, n, random_state=0)
b = self.rnd.randn(m)
lb = self.rnd.randn(n)
ub = lb + 1
res = lsq_linear(A, b, (lb, ub))
assert_allclose(res.optimality, 0.0, atol=1e-6)
res = lsq_linear(A, b, (lb, ub), lsmr_tol=1e-13)
assert_allclose(res.optimality, 0.0, atol=1e-6)
res = lsq_linear(A, b, (lb, ub), lsmr_tol='auto')
assert_allclose(res.optimality, 0.0, atol=1e-6)
def test_sparse_bounds(self):
m = 5000
n = 1000
A = rand(m, n, random_state=0)
b = self.rnd.randn(m)
lb = self.rnd.randn(n)
ub = lb + 1
res = lsq_linear(A, b, (lb, ub))
assert_allclose(res.optimality, 0.0, atol=1e-8)
res = lsq_linear(A, b, (lb, ub), lsmr_tol=1e-13)
assert_allclose(res.optimality, 0.0, atol=1e-8)
res = lsq_linear(A, b, (lb, ub), lsmr_tol='auto')
assert_allclose(res.optimality, 0.0, atol=1e-8)
def _step(self, use_cached_xw):
#search for FW vertex and compute line search
f = self._search()
#check to make sure value to add is not in the current set (error should be ortho to current subspace)
if self.wts[f] > 0:
warnings.warn(self.alg_name +
'.run(): search selected a nonzero weight to update')
#run least squares optimal weight update
active_idcs = self.wts > 0
active_idcs[f] = True
X = self.x[active_idcs, :]
res = lsq_linear(X.T,
self.snorm * self.xs,
bounds=(0., np.inf),
max_iter=max(1000, 10 * self.xs.shape[0]))
#if the optimizer failed or our cost increased, stop
prev_cost = self.error()
if not res.success or np.sqrt(2. * res.cost) >= prev_cost:
self.reached_numeric_limit = True
return False
#update weights, xw, and prev_cost
self.wts[active_idcs] = res.x
self.xw = self.wts.dot(self.x)
return True
def infer(self, Ms, ys, scale_factors=None):
''' Either:
1) Ms is a single M and ys is a single y
(scale_factors ignored) or
2) Ms and ys are lists of M matrices and y vectors
and scale_factors is a list of the same length.
'''
A, y = self._apply_scales(Ms, ys, scale_factors)
if self.method == 'AS':
assert isinstance(
A, numpy.ndarray), "method 'AS' only works with dense matrices"
x_est, _ = optimize.nnls(A, y)
elif self.method == 'LB':
if self.lasso is None:
x_est, info = nls_lbfgs_b(A, y)
if self.lasso:
lasso = max(
0,
lsmr(A, y)[0].sum()) if self.lasso is True else self.lasso
x_est = nls_slsqp(A, y, lasso)
elif self.method == 'TRF':
x_est = optimize.lsq_linear(A, y, bounds=(0, numpy.inf),
tol=1e-3)['x']
elif self.method == 'new':
x_est, info = nnls(A, y, 1e-6, 1e-6)
x_est = x_est.reshape(A.shape[1]) # reshape to match shape of x
return x_est
def enforce_consistency(self):
"""Make the tree consistent by solving a least square optimization problem.
See 'Answering Range Queries Under Local Differential Privacy. Graham
Cormode, Tejas Kulkarni, Divesh Srivastava' for details. Improve the
accuracy of range queries. When this function is invoked, `use_efficient`
will be automatically set to `False` because using both optimizations does
not further improve the accuracy.
"""
# As consistency enforcement and Honaker trick together does not further
# improve the query accuracy. Honaker trick will be automatically disabled
# if this function is called.
self._use_efficient = False
if self._check_consistency():
return
ls_matrix = self._construct_matrix()
ls_rhs = self._flatten_hierarhical_hist()
consistent_hist = optimize.lsq_linear(ls_matrix,
ls_rhs,
bounds=(0, np.inf)).x
self._hierarchical_histogram = build_tree_from_leaf.create_hierarchical_histogram(
consistent_hist, self._arity)
def NLSQ(T10, cp, signal):
'''
Implementation of linear least-squares(LLS), "Murase K: Efficient method for calculating kinetic parameters using T1-weighted dynamic contrast-enhanced magnetic resonance imaging. Magnetic Resonance in Medicine 2004; 51:858–862."
:param T10:
:param cp:
:param signal:
:return:
'''
R10 = 1 / T10
s0 = (1 - np.exp(-TR * R10)) * np.sin(alpha) / (
1 - np.exp(-TR * R10) * np.cos(alpha))
M = np.mean(signal[:6]) / s0
R1t = -np.log((signal - M * np.sin(alpha)) /
(signal * np.cos(alpha) - M * np.sin(alpha))) / TR
ctis = (R1t - R10) / r1
cp_intergral = np.zeros_like(cp, dtype=np.float)
ctis_intergral = np.zeros_like(cp, dtype=np.float)
cp_length = np.size(cp)
for t in range(cp_length):
cp_intergral[t] = np.sum(cp[:t + 1]) * deltt
for t in range(cp_length):
ctis_intergral[t] = np.sum(ctis[:t + 1]) * deltt
matrixA = np.concatenate((cp_intergral.reshape(
(cp_length, 1)), -ctis_intergral.reshape(
(cp_length, 1)), cp.reshape((cp_length, 1))),
axis=1)
matrixC = ctis
bounds = [(1e-5, 1e-5, 0.0005), (0.7, 5, 0.1)]
matrixB = lsq_linear(matrixA, matrixC, bounds=bounds).x
vp = matrixB[2]
k2 = matrixB[1]
ktrans = matrixB[0] - k2 * vp
ve = ktrans / k2
return np.array([ktrans, vp, ve])
def perform_regr_L2(indiv, comb, val_cur):
""" Perform bounded linear regression using lsq_linear.
"""
indiv, comb = indiv.copy(), comb.copy()
indiv = indiv.reset_index()
a = []
b = []
### create systems of equations
# condition that total funds stay the same
a.append(list(np.ones(len(indiv))))
b.append(indiv.val.sum())
# relating individual currencies to their total target val
for cur_id in comb.index:
# skip bitcoin to prevent regression from overshooting targets
# if allocation is unbalanceable given exchange distribution
if cur_id == val_cur:
continue
# a is list of ilocs for each currency
a.append((indiv.cur_id == cur_id).astype(int).tolist())
# b is currency total target val
b.append(comb.val_tgt.loc[cur_id])
# regression
a = np.array(a)
b = np.array(b)
s = lsq_linear(a, b, bounds=(0, np.inf))
s = [s['x']]
indiv['val_nnls'] = s[0]
return indiv
def estimate_fluxes(S, AeqCons=None, beqCons=None, bndCons=None):
'''
Parameters
S: df, stoichiometric matrix, balanced metabolites in rows, total reactions in columns
AeqCons: df, A of equality constraints
beqCons: ser, b of equality constraints
bndCons: df, boundary constraints of flux
Returns
V: ser, net flux distribution
'''
# prepare A and b
A = pd.concat((S, AeqCons), sort=False)
A = A[S.columns]
A.replace(np.nan, 0, inplace=True)
b = pd.concat((pd.Series(np.zeros(S.shape[0]),
index=S.index,
dtype=np.float), beqCons))
if bndCons is None:
bndCons = pd.DataFrame(np.full((len(rxns), 2), [0, np.inf]),
index=rxns,
columns=['lb', 'ub'])
bnds = (bndCons['lb'].values, bndCons['ub'].values)
res = lsq_linear(A, b, bounds=bnds).x
V = pd.Series(res, index=S.columns, dtype=np.float)
return V
def _fit_rc(idx: int, td: np.ndarray, rs: np.ndarray, c_in: float,
cl: np.ndarray, res: np.ndarray, cs: np.ndarray,
cd: np.ndarray, r_unit: float, c_unit: float,
t_unit: float) -> None:
c_min = 1.0e-18
rs_flat = rs.flatten()
cl_flat = cl.flatten()
a = np.empty((rs_flat.size, 3))
a[:, 0] = rs_flat * c_unit / t_unit
a[:, 1] = cl_flat * r_unit / t_unit
a[:, 2] = 1
b = (td.flatten() - rs_flat * (cl_flat + c_in)) / t_unit
x = lstsq(a, b)[0]
rp = x[1] * r_unit
cd0 = x[2] * t_unit / rp
cs0 = x[0] * c_unit - cd0
if cs0 < 0 or cd0 < 0:
# we got negative capacitance, which causes problems.
# Now, assume the rp we got is correct, do another least square fit to enforce
# cs and cd are positive
a2 = np.empty((rs_flat.size, 2))
a2[:, 0] = a[:, 0]
a2[:, 1] = a[:, 0] + (rp * c_unit / t_unit)
b -= rp * cl_flat / t_unit
# noinspection PyUnresolvedReferences
opt_res = lsq_linear(a2, b, bounds=(c_min, float('inf'))).x
cs0 = opt_res[0] * c_unit
cd0 = opt_res[1] * c_unit
res[idx] = rp
cs[idx] = cs0
cd[idx] = cd0
def fromPhaseCurve(phiObs,
flux,
nSlices,
phiIn=None,
priorStd=1e3,
G=None,
brightnessMin=0,
brightnessMax=np.inf,
fullOutput=False):
'''Computes the slice brightnesses from a given phase curve and prior uncertainty.'''
# Compute G if it was not provided
if G is None:
G = getG(phiObs, nSlices, phiIn)
# Compute with bounded ridge regression using the pseudo-observations method.
X = np.vstack([G, np.eye(nSlices) / priorStd])
y = np.concatenate(
[flux, np.mean(flux) * np.ones(nSlices) / (2 * priorStd)])
f = lsq_linear(X, y, bounds=[brightnessMin, brightnessMax])
if not fullOutput:
return f.x
# Compute the log posterior probability
sigma = np.sqrt(2. * f.cost / (len(flux) - nSlices))
logLike = -len(flux) * np.log(
2 * np.pi) / 2. - len(flux) * sigma - f.cost / sigma**2
# Get the uncertainties on the brightness
errors = sigma**2 * np.linalg.inv(np.matmul(X.T, X))
return {
'brightness': f.x,
'brightnessCov': errors,
'logLike': logLike,
'residStd': sigma
}
def get_model(A, y, lamb=0, regularization='ridge', bounds=False, verbose=0):
n_col = A.shape[1]
if isinstance(regularization, str):
if regularization == 'ridge':
#
L = np.identity(n_col)
elif regularization == 'smooth':
# fill with a finite difference matrix
L = np.identity(n_col) * 2 + np.diag(
-np.ones(n_col - 1), k=1) + np.diag(-np.ones(n_col - 1), k=-1)
else:
L = regularization
if not bounds:
return linalg.solve(A.T.dot(A) + lamb * L, A.T.dot(y))
else:
lower_bnd = np.ones(n_unknowns)
upper_bnd = np.ones_like(lower_bnd)
lower_bnd[:nev + n_stations] *= -np.inf
lower_bnd[nev + n_stations:] = 0
upper_bnd *= np.inf
return lsq_linear(A.T.dot(A) + lamb * L,
A.T.dot(y), (lower_bnd, upper_bnd),
verbose=verbose)
def get_phase_compositions(self, point, simplex_id=None):
"""Compute phase contributions given a composition
input:
------
point : composition as a numpy array (dim,)
output:
-------
x. : Phase compositions as a numpy array of shape (dim, )
vertices : Compositions of coexisting phases. Each row correspond to
an entry in x with the same index
"""
from scipy.optimize import lsq_linear
if not self.is_solved:
raise RuntimeError(
'Phase diagram is not computed\n'
'Use .compute() before requesting phase compositions')
assert len(
point
) == self.dimension, 'Expected {}-component composition got {}'.format(
self.dimension, len(point))
if self.is_boundary_point(point):
raise RuntimeError(
'Boundary points are not considered in the computation.')
if simplex_id is None:
inside = np.asarray([
self.in_simplex(point, s)
for s in self.simplices[~self.coplanar]
],
dtype=bool)
in_simplices_ids = np.where(inside)[0]
else:
in_simplices_ids = [simplex_id]
for i in in_simplices_ids:
simplex = self.simplices[i]
num_comps = np.asarray(self.num_comps)[i]
vertices = self.grid[:, simplex]
energies = np.asarray([self.energy_func(x) for x in vertices.T])
if num_comps == 1:
# no-phase splits if the simplex is labelled 1-phase
continue
A = np.vstack(
(vertices.T[:-1, :], energies, np.ones(self.dimension)))
B = np.hstack((point[:-1], self.energy_func(point), 1))
lb = np.zeros(self.dimension)
ub = np.ones(self.dimension)
res = lsq_linear(A, B, bounds=(lb, ub), lsmr_tol='auto', verbose=0)
x = res.x
if not (x < 0).any():
# STOP if you found a simplex that has x>0
break
return x, vertices.T, num_comps
def gauss_newton(f, x0, J, atol: float = 1e-4, max_iter: int = 100):
"""Implements the Gauss-Newton method for NLLS
:param f: function to compute the residual vector
:param x0: array corresponding to initial guess
:param J: function to compute the jacobian of f
:param atol: stopping criterion for the root mean square
of the squared norm of the gradient of f
:param max_iter: maximum number of iterations to run before
terminating
"""
iterates = [
x0,
]
rms = lambda x: np.sqrt(np.mean(np.square(x)))
costs = [
rms(f(x0)),
]
cnt = 0
grad_rms = np.inf
while cnt < max_iter and grad_rms > atol:
x_k = iterates[-1]
A = J(x_k)
b = A @ x_k - f(x_k)
result = lsq_linear(A, b)
iterates.append(result.x)
costs.append(rms(f(result.x)))
grad_rms = rms(A.T * f(x_k))
cnt += 1
return iterates, np.asarray(costs)
def perform_regr(indiv, comb, val_cur):
""" Perform bounded linear regression using lsq_linear.
"""
indiv, comb = indiv.copy(), comb.copy()
# determine exchange val totals
ex_tot = indiv.reset_index().groupby(['ex_id'])[['val']].sum()
a = []
b = []
indiv = indiv.reset_index()
# create systems of equations
# relating exchange currencies to exchange total
for ex_id in ex_tot.index:
# a is a list of ilocs for each exchange
a.append((indiv.ex_id == ex_id).astype(int).tolist())
# b is exchange total
b.append(ex_tot.val.loc[ex_id])
# relating individual currencies to their total target val
for cur_id in comb.index:
# skip bitcoin to prevent regression from overshooting targets
# if allocation is unbalanceable given exchange distribution
if cur_id == val_cur:
continue
# a is list of ilocs for each currency
a.append((indiv.cur_id == cur_id).astype(int).tolist())
# b is currency total target val
b.append(comb.val_tgt.loc[cur_id])
# regression
a = np.array(a)
b = np.array(b)
s = lsq_linear(a, b, bounds=(0, np.inf))
# pprint(s)
s = [s['x']]
# print(['%.8f' % v for v in s[0]])
indiv['val_nnls'] = s[0]
return indiv
def calcN_Waq(self):
res = self.res
pvt = self.pvt
aquif = self.aquif
# Create dataframe to be returned
r = pd.DataFrame(columns=['WeCalc', 'Pb', 'So', 'Sg', 'Sw', 'Vp', \
'Bo', 'Bg', 'Bw', 'Rs', 'Eo', 'Eg', 'Efw', 'N'])
r = pd.concat([self.hist.data.dropna(subset=['p']), r], axis=1)
# Solve MBE for VOIP and Aquifer size using least square method
A = []
B = []
for index, row in r.iterrows():
Pb = self.updatePb(row['Np'], row['Gp'], row['Gi'])
p = row['p']
E = self.Eo(p, Pb) + res.m * self.Eg(p) + self.Efw(p)
Fi = row['Wi'] * pvt.Bw(p) + row['Gi'] * pvt.Bg(p) # Fi - Bw2*We
Fp = row['Np']*(pvt.Bo(p, Pb)+(row['Gp']/row['Np']-pvt.Rs(p, Pb)) \
*pvt.Bg(p))+row['Wp']*pvt.Bw(p)
A.append([E, (aquif.cr + pvt.cw) * (res.p0 - p)])
B.append(Fp - Fi)
Waqmin = r.tail(1)['Wp'] / (aquif.cr + pvt.cw) / (res.p0 -
r.tail(1)['p'])
return sp.lsq_linear(np.array(A),
np.array(B),
bounds=[(0., Waqmin), (np.inf, np.inf)]), A, B
def test_np_matrix(self):
# gh-10711
with np.testing.suppress_warnings() as sup:
sup.filter(PendingDeprecationWarning)
A = np.matrix([[20, -4, 0, 2, 3], [10, -2, 1, 0, -1]])
k = np.array([20, 15])
s_t = lsq_linear(A, k)
def edit_fibermodel(Felist, Fiblist, header_tup, B, avgB, A, args):
nifu = len(Fiblist)
wave = np.arange(header_tup[0]) * header_tup[2] + header_tup[1]
for i in xrange(nifu):
F = FiberModel(Fiblist[i])
nfib, nw = Felist[i][0].data.shape
through = A[i] * B[i, :] / avgB
for j in xrange(nfib):
if args.debug:
t1 = time.time()
print("Working on Fibermodel, fiber: %i, %i" % (i, j))
mask = np.where(np.isfinite(through[j, :]))[0]
basis = np.vstack(
[F.amplitudes[j].get_basis(F._scal_w(w)) for w in wave])
x = through[j, mask]
x[-1] = x[-2]
y = basis[mask, :]
ax, ay = y.shape
lb = -1. * np.ones((ay, ))
lb[-1] = -10.
ub = 1. * np.ones((ay, ))
ub[-1] = 10.
sol = lsq_linear(np.array(basis[mask, :]),
np.array(x),
bounds=(lb, ub))
F.amplitudes[j].A = sol['x'][:-1]
F.amplitudes[j].mean = sol['x'][-1]
filename = Fiblist[i][:-6] + 'adjpy_' + Fiblist[i][-6] + '.fmod'
if args.debug:
print("Writing out %s" % filename)
F.writeto(op.join(args.outfolder, filename))
if args.debug:
t2 = time.time()
print("Time Taken to reset amplitudes for IFU %i: %0.2f" %
(i, t2 - t1))
def _rank_pass(self, teams, passN=0, prev_rank=[] ):
'''Returns list of rankings using linear chi2
For pass 0, set passN = 0, all weights are 1
For all others, set passN = n and send prev_rank '''
A = [] # g x t array (games x teams)
b = [] # g list of game results R_g
N_g = self._calc_Ng_list(teams) # list of games with home/away ids
# error of game measurement w = 1/sig^2
if passN == 0:
sig_g = np.ones(len(N_g)) # equal weight first iteration
else:
sig_g = self._calc_sig_g(prev_rank, N_g)
# Loop over games & home/away team ids
for g, h_a in enumerate(N_g):
row_g = [] # row for game g in A matrix
# loop over teams
for team in teams:
if team.teamId == h_a[0]:
r_tg = 1./sig_g[g] # home team
elif team.teamId == h_a[1]:
r_tg = -1./sig_g[g] #away
else:
r_tg = 0 #not in game
row_g.append(r_tg)
# add row to A matrix
R_g = self._calc_Rg(h_a[2], h_a[3]) # send( score_home, score_away)
b.append(R_g/sig_g[g])
A.append(row_g)
# Calculate the ranks
rank = lsq_linear(A, b, bounds=(30,130))#,lsq_solver='exact')
if rank.success == False:
print('WARNING:\t%s'%rank.message)
for i,r in enumerate(rank.x):
teams[i].rank.lsq = r
return rank.x
def _find_conservative_equilibrium(self,guess,tolerance=1e-9,max_iter=10):
y = guess
lb = np.array([-np.pi,-np.pi, -1* y[2], -1 * y[3]])
ub = np.array([np.pi,np.pi,np.inf,np.inf])
fun = self.H_flow
jac = self.H_flow_jac
f = fun(y)[:-1]
J = jac(y)[:-1,:-1]
it=0
# Newton method iteration
while np.linalg.norm(f)>tolerance and it < max_iter:
# Note-- using constrained least-squares
# to avoid setting actions to negative
# values.
# The lower bounds ensure that I1 and I2 are positive quantities
lb[2:] = -1* y[2:-1]
dy = lsq_linear(J,-f,bounds=(lb,ub)).x
y[:-1] = y[:-1] + dy
f = fun(y)[:-1]
J = jac(y)[:-1,:-1]
it+=1
if it==max_iter:
raise RuntimeError("Max iterations reached!")
return y
def solve_linear_system(emplproj_list, cost_list, skills_list, idx_selected, totals):
'''
'idx_selected' is a tuple with row and column index of selected (empl,proj).
'totals' is a tuple constaining two tuples, one for emplTotals and one for projTotals.
'''
# Check that employee has skill
if not skills_list[idx_selected[0]][idx_selected[1]]:
status_msg = "NO_SKILL"
return status_msg
# User-supplied data in array form (#empl x #proj)
emplproj_arr = np.array(emplproj_list, np.dtype('d'))
nEmpl, nProj = emplproj_arr.shape
N = nEmpl*nProj
cost_arr = np.array(cost_list, np.dtype('d'))
skills_mask = np.array(skills_list, np.dtype('?'))
# Flatten arrays into vectors, taking only allowed values based on skills_mask
emplproj_vec = emplproj_arr[skills_mask]
cost_vec = cost_arr[skills_mask]
skills_mask_ind = np.nonzero(skills_mask.flatten())[0]
# Adjust index of selected [empl,proj] after flattening and skill filtering
idx_mat = np.arange(N).reshape(nEmpl,nProj)
idx_selec_mat = idx_mat[idx_selected[0]][idx_selected[1]]
idx_vec = idx_mat[skills_mask]
idx_selec_flat = np.where(np.isin(idx_vec, idx_selec_mat))[0][0]
# Create design matrix for fixed marginal totals, excluding user-selected element
A = create_design_matrix(emplproj_arr, skills_mask, excluded_elem=[idx_selected])
m, n = A.shape
# Create right-hand-side vector.
# Must subtract the user-supplied value (constant) from the totals!
selec_value = emplproj_vec[idx_selec_flat]
tot_empl = list(totals[0])
tot_empl[idx_selected[0]] = totals[0][idx_selected[0]] - selec_value
tot_proj = list(totals[1])
tot_proj[idx_selected[1]] = totals[1][idx_selected[1]] - selec_value
b = tot_empl + tot_proj[:-1]
# Compute least-squares solution of linear matrix equation
print(A)
print(b)
lb = np.ones(n)
ub = np.full(n, np.inf)
x = optimize.lsq_linear(A, b, bounds=(lb, ub))['x']
# Check that solution satisfies equations
if not np.allclose(np.dot(A, x), b):
status_msg = "NO_SOLUTION"
return status_msg
# Add user-selected element back to x vector
x = np.insert(x, idx_selec_flat, selec_value)
# Put all values back in matrix
new_mat = put_solution(emplproj_arr, x, skills_mask_ind)
return new_mat.tolist()
def down_sample_SchWMA_model(rx, dcv, num_slow_samps):
ry = np.zeros(2 * num_slow_samps)
A = np.zeros((len(ry), len(ry)))
C = np.zeros(len(ry))
for L in range(1, len(ry) + 1):
vec = np.concatenate([[L], 2 * (L - np.arange(1, L))])
A[L - 1, :len(vec)] = vec
vec = np.concatenate([[L * dcv],
2 * (L * dcv - np.arange(1, L * dcv))])
vec = vec[:np.minimum(len(vec), len(rx))]
C[L - 1] = np.sum(vec * rx[:len(vec)])
ry = opt.lsq_linear(A, C)['x']
# x0 = np.zeros(len(ry))
# x0[0] = np.sqrt(ry[0]+2*np.sum(ry[1:]))
# out = opt.minimize(lambda x: np.sum((acv_from_b(x)-ry)**2),x0)
# return out['x']
Ry = np.real(np.fft.fft(np.concatenate([ry, ry[1::][::-1]])))
Ry[Ry <= 0] = np.min(Ry[Ry > 0]) / 10.
alpha = .5 * np.log(Ry)
phi = np.imag(si.hilbert(alpha))
Bmp = np.exp(alpha + 1j * phi)
bmp = np.fft.fftshift(np.fft.ifft(Bmp))
return np.abs(bmp)
def test_almost_singular(self):
A = np.array(
[[0.8854232310355122, 0.0365312146937765, 0.0365312146836789],
[0.3742460132129041, 0.0130523214078376, 0.0130523214077873],
[0.9680633871281361, 0.0319366128718639, 0.0319366128718388]])
b = np.array(
[0.0055029366538097, 0.0026677442422208, 0.0066612514782381])
result = lsq_linear(A, b, method=self.method)
assert_(result.cost < 1.1e-8)
def test_dense_rank_deficient(self):
A = np.array([[-0.307, -0.184]])
b = np.array([0.773])
lb = [-0.1, -0.1]
ub = [0.1, 0.1]
for lsq_solver in self.lsq_solvers:
res = lsq_linear(A, b, (lb, ub), method=self.method,
lsq_solver=lsq_solver)
assert_allclose(res.x, [-0.1, -0.1])
A = np.array([
[0.334, 0.668],
[-0.516, -1.032],
[0.192, 0.384],
])
b = np.array([-1.436, 0.135, 0.909])
lb = [0, -1]
ub = [1, -0.5]
for lsq_solver in self.lsq_solvers:
res = lsq_linear(A, b, (lb, ub), method=self.method,
lsq_solver=lsq_solver)
assert_allclose(res.optimality, 0, atol=1e-11)
def solve_design_matrix(self, dmatrix, order, solver='lsmr',
solver_dict={}):
b = order.data.compressed() / order.uncertainty.compressed()
if solver == 'lsmr':
result = sparse.linalg.lsmr(dmatrix.tobsr(), b, **solver_dict)
elif solver == 'lsq':
lsq_dict = dict(bounds=(0, np.inf), lsmr_tol='auto', verbose=1)
lsq_dict.update(solver_dict)
result = lsq_linear(dmatrix, b, **lsq_dict)
result = [result.x, result]
else:
raise NotImplementedError('Solver {0} is not implemented'.format(
solver))
return result
def test_full_result(self):
lb = np.array([0, -4])
ub = np.array([1, 0])
res = lsq_linear(A, b, (lb, ub), method=self.method)
assert_allclose(res.x, [0.005236663400791, -4])
r = A.dot(res.x) - b
assert_allclose(res.cost, 0.5 * np.dot(r, r))
assert_allclose(res.fun, r)
assert_allclose(res.optimality, 0.0, atol=1e-12)
assert_equal(res.active_mask, [0, -1])
assert_(res.nit < 15)
assert_(res.status == 1 or res.status == 3)
assert_(isinstance(res.message, str))
assert_(res.success)
def invert_STF(st_data, st_synth, method='bound_lsq', len_stf=None, eps=1e-3):
# print('Using %d stations for STF inversion' % len(st_data))
# for tr in st_data:
# fig = plt.figure()
# ax = fig.add_subplot(111)
# ax.plot(tr.data, label='data')
# ax.plot(st_synth.select(station=tr.stats.station,
# network=tr.stats.network,
# location=tr.stats.location)[0].data,
# label='synth')
# ax.legend()
# fig.savefig('%s.png' % tr.stats.station)
# plt.close(fig)
# Calculate number of samples for STF:
if len_stf:
npts_stf = int(len_stf * st_data[0].stats.delta)
else:
npts_stf = st_data[0].stats.npts
d, G = _create_matrix_STF_inversion(st_data, st_synth, npts_stf)
if method == 'bound_lsq':
m = lsq_linear(G, d, (-0.1, 1.1))
stf = np.r_[m.x[(len(m.x) - 1) / 2:], m.x[0:(len(m.x) - 1) / 2]]
elif method == 'lsq':
stf, residual, rank, s = np.linalg.lstsq(G, d)
elif method == 'dampened':
# J from eq.3 in Sigloch (2006) to dampen later part of STFs
GTG = np.matmul(G.T, G)
diagGmean = np.mean(np.diag(GTG))
J = np.diag(np.linspace(0, diagGmean, G.shape[1]))
Ginv = np.matmul(np.linalg.inv(GTG + eps * J), G.T)
stf = np.matmul(Ginv, d)
else:
raise ValueError('method %s unknown' % method)
return stf
def test_dense_bounds(self):
# Solutions for comparison are taken from MATLAB.
lb = np.array([-1, -10])
ub = np.array([1, 0])
for lsq_solver in self.lsq_solvers:
res = lsq_linear(A, b, (lb, ub), method=self.method,
lsq_solver=lsq_solver)
assert_allclose(res.x, lstsq(A, b)[0])
lb = np.array([0.0, -np.inf])
for lsq_solver in self.lsq_solvers:
res = lsq_linear(A, b, (lb, np.inf), method=self.method,
lsq_solver=lsq_solver)
assert_allclose(res.x, np.array([0.0, -4.084174437334673]),
atol=1e-6)
lb = np.array([-1, 0])
for lsq_solver in self.lsq_solvers:
res = lsq_linear(A, b, (lb, np.inf), method=self.method,
lsq_solver=lsq_solver)
assert_allclose(res.x, np.array([0.448427311733504, 0]),
atol=1e-15)
ub = np.array([np.inf, -5])
for lsq_solver in self.lsq_solvers:
res = lsq_linear(A, b, (-np.inf, ub), method=self.method,
lsq_solver=lsq_solver)
assert_allclose(res.x, np.array([-0.105560998682388, -5]))
ub = np.array([-1, np.inf])
for lsq_solver in self.lsq_solvers:
res = lsq_linear(A, b, (-np.inf, ub), method=self.method,
lsq_solver=lsq_solver)
assert_allclose(res.x, np.array([-1, -4.181102129483254]))
lb = np.array([0, -4])
ub = np.array([1, 0])
for lsq_solver in self.lsq_solvers:
res = lsq_linear(A, b, (lb, ub), method=self.method,
lsq_solver=lsq_solver)
assert_allclose(res.x, np.array([0.005236663400791, -4]))
def solve_SAT2(self,cnf,number_of_variables,number_of_clauses):
#Solves CNFSAT by a Polynomial Time Approximation scheme:
# - Encode each clause as a linear equation in n variables: missing variables and negated variables are 0, others are 1
# - Solve previous system of equations by least squares algorithm to fit a line
# - Variable value above 0.5 is set to 1 and less than 0.5 is set to 0
# - Rounded of assignment array satisfies the CNFSAT with high probability
#Returns: a tuple with set of satisfying assignments
satass=[]
x=[]
self.solve_SAT(cnf,number_of_variables,number_of_clauses)
for clause in self.cnfparsed:
equation=[]
for n in xrange(number_of_variables):
equation.append(0)
#print "clause:",clause
for literal in clause:
if literal[0] != "!":
equation[int(literal[1:])-1]=1
else:
equation[int(literal[2:])-1]=0
self.equationsA.append(equation)
for n in xrange(number_of_clauses):
self.equationsB.append(1)
a = np.array(self.equationsA)
b = np.array(self.equationsB)
init_guess = []
for n in xrange(number_of_variables):
init_guess.append(0.00000000001)
initial_guess = np.array(init_guess)
self.A=a
self.B=b
#print "a:",a
#print "b:",b
#print "a.shape:",a.shape
#print "b.shape:",b.shape
matrixa=matrix(a,tc='d')
matrixb=matrix(b,tc='d')
x=None
if number_of_variables == number_of_clauses:
if self.Algorithm=="lsqr()":
#x = np.dot(np.linalg.inv(a),b)
#x = gmres(a,b)
#x = lgmres(a,b)
#x = minres(a,b)
#x = bicg(a,b)
#x = cg(a,b)
#x = cgs(a,b)
#x = bicgstab(a,b)
x = lsqr(a,b,atol=0,btol=0,conlim=0,show=True)
if self.Algorithm=="lapack()":
try:
x = gesv(matrixa,matrixb)
#x = gels(matrixa,matrixb)
#x = sysv(matrixa,matrixb)
#x = getrs(matrixa,matrixb)
x = [matrixb]
except:
print "Exception:",sys.exc_info()
return None
if self.Algorithm=="l1regls()":
l1x = l1regls(matrixa,matrixb)
ass=[]
x=[]
for n in l1x:
ass.append(n)
x.append(ass)
if self.Algorithm=="lsmr()":
x = lsmr(a,b,atol=0,btol=0,conlim=0,show=True,x0=initial_guess)
else:
if self.Algorithm=="solve()":
x = solve(a,b)
if self.Algorithm=="lstsq()":
x = lstsq(a,b,lapack_driver='gelsy')
if self.Algorithm=="lsqr()":
x = lsqr(a,b,atol=0,btol=0,conlim=0,show=True)
if self.Algorithm=="lsmr()":
#x = lsmr(a,b,atol=0.1,btol=0.1,maxiter=5,conlim=10,show=True)
x = lsmr(a,b,atol=0,btol=0,conlim=0,show=True,x0=initial_guess)
if self.Algorithm=="spsolve()":
x = dsolve.spsolve(csc_matrix(a),b)
if self.Algorithm=="pinv2()":
x=[]
#pseudoinverse_a=pinv(a)
pseudoinverse_a=pinv2(a,check_finite=False)
x.append(matmul(pseudoinverse_a,b))
if self.Algorithm=="lsq_linear()":
x = lsq_linear(a,b,lsq_solver='exact')
if self.Algorithm=="lapack()":
try:
#x = gesv(matrixa,matrixb)
x = gels(matrixa,matrixb)
#x = sysv(matrixa,matrixb)
#x = getrs(matrixa,matrixb)
x = [matrixb]
except:
print "Exception:",sys.exc_info()
return None
if self.Algorithm=="l1regls()":
l1x = l1regls(matrixa,matrixb)
ass=[]
x=[]
for n in l1x:
ass.append(n)
x.append(ass)
print "solve_SAT2(): ",self.Algorithm,": x:",x
if x is None:
return None
cnt=0
binary_parity=0
real_parity=0.0
if rounding_threshold == "Randomized":
randomized_rounding_threshold=float(random.randint(1,100000))/100000.0
else:
min_assignment=min(x[0])
max_assignment=max(x[0])
randomized_rounding_threshold=(min_assignment + max_assignment)/2
print "randomized_rounding_threshold = ", randomized_rounding_threshold
print "approximate assignment :",x[0]
for e in x[0]:
if e > randomized_rounding_threshold:
satass.append(1)
binary_parity += 1
else:
satass.append(0)
binary_parity += 0
real_parity += e
cnt+=1
print "solve_SAT2(): real_parity = ",real_parity
print "solve_SAT2(): binary_parity = ",binary_parity
return (satass,real_parity,binary_parity,x[0])
def test_dense_no_bounds(self):
for lsq_solver in self.lsq_solvers:
res = lsq_linear(A, b, method=self.method, lsq_solver=lsq_solver)
assert_allclose(res.x, lstsq(A, b)[0])