def update(self, dt):
self.mass_cond.append(linalg.cond(self._world._mass))
self.imp_cond.append(linalg.cond(self._world._impedance))
curr_rk = []
for b in self._world.iterbodies():
curr_rk.append( self.rank(b.jacobian) )
self.c_rank.append(curr_rk)
def compDklgaussian(mu1, C1, mu2, C2):
'''
mu1 and C1 specify a multidimensional gaussian. mu2 and C2 specify another
one (of the same dimension). Return is a float giving the KL divergence
between the two Gaussians.
Inputs have to be matrix instances, and the mu inputs should be in row
vector shape (mu.shape[0] = 1, mu.shape[1] > 1)
'''
n = mu1.size
b = mu2 - mu1
C2inv = la.inv(C2)
C1sqrt = np.mat(sqrtPSDm(C1))
Term1 = C1sqrt * C2inv * C1sqrt
Term2 = b.transpose() * C2inv * b
det1 = la.det(C1)
det2 = la.det(C2)
tol = 1e8
if (la.cond(C1) > tol) | (la.cond(C2) > tol):
print('Determinants not non-zero. Ignoring determinants.')
Term3 = 0
else:
Term3 = .5 * np.log(det2 / det1)
d = .5 * np.trace(Term1) + .5 * Term2 - .5 * n + Term3
return d[0, 0]
def main(argv):
A = numpy.matrix([[float(sys.argv[1]), float(sys.argv[2]), float(sys.argv[3]), float(sys.argv[4])], [float(sys.argv[5]), float(sys.argv[6]), float(sys.argv[7]), float(sys.argv[8])], [float(sys.argv[9]), float(sys.argv[10]), float(sys.argv[11]), float(sys.argv[12])]])
B = numpy.matrix([[float(sys.argv[1]), float(sys.argv[2]), float(sys.argv[3])], [float(sys.argv[5]), float(sys.argv[6]), float(sys.argv[7])], [float(sys.argv[9]), float(sys.argv[10]), float(sys.argv[11])]])
#print A
#print 'top left number is: '+ str(A[0,0])
#backSub(forwardElim(A))
print LA.cond(B, numpy.inf)
def _B2formula(Ac, t1, t2, B2):
if t1 == 0 and t2 == 0:
term = B2
return term
n = Ac.shape[0]
tmp = np.eye(n) - expm(-Ac)
if cond(tmp) < 1000000.0:
term = np.dot(((expm(-Ac * t1) - expm(-Ac * t2)) * inv(tmp)), B2)
return term
# Numerical trouble. Perturb slightly and check the result
ntry = 0
k = np.sqrt(eps)
Ac0 = Ac
while ntry < 2:
Ac = Ac0 + k * rand(n, n)
tmp = np.eye(n) - expm(-Ac)
if cond(tmp) < 1 / np.sqrt(eps):
ntry = ntry + 1
if ntry == 1:
term = np.dot(np.dot(expm(-Ac * t1) - expm(-Ac * t2), inv(tmp)), B2)
else:
term1 = np.dot(np.dot(expm(-Ac * t1) - expm(-Ac * t2), inv(tmp)), B2)
k = k * np.sqrt(2)
if norm(term1 - term) > 0.001:
warn("Inaccurate calculation in mapCtoD.")
return term
def evaluate_rotation_matrix(R, this_theta):
pi = numpy.pi
print
print '###################'
theta = this_theta * 180.0 / pi
condition_number = linalg.cond(R)
smallest_singular_value = linalg.cond(R, -2)
two_norm = linalg.cond(R, 2)
print 'theta = ', theta, '\n', R, '\ncondition number = ', condition_number
print 'largest_singluar_value = ', two_norm
print 'smallest_singular_value = ', smallest_singular_value
print '###################'
print
return
def EV(sasb_n, sasb_nm1, sa_n, sa_nm1,debug1=False,debug2=False):
try:
dim = len(sa_n)
sasb_n = sasb_n.reshape(dim , dim)
sasb_nm1=sasb_nm1.reshape(dim , dim)
dsdk_n =sasb_n -outer(sa_n,sa_n)
dsdk_nm1=sasb_nm1-outer(sa_nm1,sa_n)
T=dot(dsdk_nm1,inv(dsdk_n))
if debug2:
print(cond(T))
#evs=sort(abs(eigvals(T)))
evs=sort(eigvals(T))
if debug1:
val,vec=eig(T)
print(val)
print('-----------------')
for i in range(len(val)):
print(val[i],vec[:,i])
print('-----------------')
lambda_ =max(abs(evs))
lambda_test =evs[-1]
#assert(lambda_==lambda_test)
if(len(evs)>1):
lambda_2=evs[-2]
else:
lambda_2=0
lambda_=RawSanitize(lambda_)
return lambda_, lambda_2
except np.linalg.linalg.LinAlgError:
return 0,0
def unteraufgabe_d():
n = np.arange(2,21,2)
xs = [x02,x04,x06,x08,x10,x12,x14,x16,x18,x20]
f = lambda x: np.sin(10*x*np.cos(x))
residuals = np.zeros_like(n,dtype=np.floating)
condition = np.ones_like(n,dtype=np.floating)
for i, x in enumerate(xs):
b = f(x)
A = interp_monom(x)
alpha = solve(A,b)
residuals[i] = norm(np.dot(A,alpha) - b)
condition[i] = cond(A)
plt.figure()
plt.plot(n,residuals,"-o")
plt.grid(True)
plt.xlabel(r"$n$")
plt.ylabel(r"$\|A \alpha - b\|_2$")
plt.savefig("residuals.eps")
plt.figure()
plt.semilogy(n,condition,"-o")
plt.grid(True)
plt.xlabel(r"$n$")
plt.ylabel(r"$\log(\mathrm{cond}(A))$")
plt.savefig("condition.eps")
def MdcNE(A, b):
"""
Résolution du problème des moindres carrés linéaire :
Min_{alpha} || b - A*alpha ||^2
par factorisation de Cholesky du système des équations normales.
Parameters
----------
A : np.array ou np.matrix
b : np.array ou np.matrix
Returns
-------
alpha : np.array (dans tous les cas)
solution du problème des moindres carrés linéaire
"""
S = np.matrix(A).T * np.matrix(A)
# Vérification au préalable du conditionnement du système et de la stabilité
# numérique de la résolution qui va suivre
c = la.cond(S)
if c > 1e16:
print('Attention : le conditionnement de la matrice des équations')
print(' normales est très grand ---> %0.5g' % c)
# Factorisation suivi de la résolution
L = la.cholesky(S) # matrice triangulaire inférieure
m = b.size
bvect = np.matrix( b.reshape(m,1) )
y = A.T * bvect
z = la.solve(L, y)
alpha = np.array( la.solve(L.T, z) )
return alpha
def crunch(self):
'''Mainline to number crunch the calculation of
member foreces.'''
# echo data:
printline = self.printline
lineread, noNodes, noMembers, noAuxiliaries,\
noLoadCases, noLoadLines = self.prolog()
nodes = self.doNodes(lineread, noNodes, noAuxiliaries)
printline("nodes = ")
printdict(printline, nodes)
members = self.doMembers(lineread, noMembers, noAuxiliaries)
printline('members = ')
printdict(printline, members)
memprops = self.doMemprops( noMembers, noAuxiliaries, members, nodes)
printline('memprops =')
printdict(self.printline, memprops)
ndim = noNodes * 2
printline('Dimension of connection mat ndim =' + str(ndim))
connection = np.zeros((ndim, ndim), dtype = "float")
for i in range(noMembers + noAuxiliaries):
member = memprops[i + 1]
connection[member[0] - 1, i] += member[4]
connection[member[1] - 1, i] += member[5]
if i < noMembers:
connection[member[2] - 1, i] += -member[4]
connection[member[3] - 1, i] += -member[5]
printline(' ')
printline('Connection matrix')
printerm(printline, connection)
try:
conditionnumber = la.cond(connection)
inv = la.inv(connection)
except la.LinAlgError, e:
printline( '%s %s' % (type(e).__name__, e))
def fitmat(p, q):
"""
Compute the best transformation to map p onto q without translation.
The transformation is best according to the least square residuals criteria and correspond to the matrix M in:
p-p0 = M(q-q0) + T
where p0 and q0 are the barycentre of the list of points p and q.
:Parameters:
p : array(M,2)
List of points p, one point per line, first column for x, second for y
q : array(M,2)
List of points q, one point per line, first column for x, second for y
"""
pc = p.sum(0) / p.shape[0]
qc = q.sum(0) / q.shape[0]
p = asmatrix(p - pc)
q = asmatrix(q - qc)
A = p.T * p
if cond(A) > 1e15:
return
V = p.T * q
M = (A.I * V).A
return M.T
def computeRegressorLinDepsSVD(self):
"""get base regressor and identifiable basis matrix with iDynTree (SVD)"""
with helpers.Timer() as t:
# get subspace basis (for projection to base regressor/parameters)
Yrand = self.getRandomRegressor(5000)
#A = iDynTree.MatrixDynSize(self.num_params, self.num_params)
#self.generator.generate_random_regressors(A, False, True, 2000)
#Yrand = A.toNumPy()
U, s, Vh = la.svd(Yrand, full_matrices=False)
r = np.sum(s>self.opt['min_tol'])
self.B = -Vh.T[:, 0:r]
self.num_base_params = r
print("tau: {}".format(self.tau.shape), end=' ')
print("YStd: {}".format(self.YStd.shape), end=' ')
# project regressor to base regressor, Y_base = Y_std*B
self.YBase = np.dot(self.YStd, self.B)
if self.opt['verbose']:
print("YBase: {}, cond: {}".format(self.YBase.shape, la.cond(self.YBase)))
self.num_base_params = self.YBase.shape[1]
if self.showTiming:
print("Getting the base regressor (iDynTree) took %.03f sec." % t.interval)
def check_input_data(a, b, x=None):
"""Does some sanity checks of the input data. Arguments:
a - input matrix
b - input vector
x - partial solution for the Jacobi Method
Returns:
args - list of input arguments, if valid - otherwise raises
InputDataException
"""
from sys import float_info
from time import sleep
if x is None:
args = [a, b]
else:
args = [a, b, x]
for arg in args:
if type(arg) != np.ndarray:
raise InputDataError(repr(np.ndarray) + ' expected, got ' +
repr(type(arg)) + ' instead.')
if (a.ndim, b.ndim, len(a)) != (2, 2, len(b)):
raise InputDataError('Input arrays have different sizes/dimensions')
if x is not None:
if (x.ndim, x.size) != (2, b.size):
raise InputDataError('Input arrays have different sizes/dimensions')
if a.shape[0] != a.shape[1]:
raise InputDataError('Only square matrices are supported')
if la.cond(a) > 1 / float_info.epsilon:
raise InputDataError('Singular matrix')
return args
def _energy(self, face, i):
"""Returns energy value and polygon area for a provided polygon."""
TV1 = array([self.Mesh.vertices[face[0]], self.Mesh.vertices[face[1]], self.Mesh.vertices[face[2]]])
TV2 = array([self.vnormal[face[0]],self.vnormal[face[1]],self.vnormal[face[2]]])
if array_equal(TV1[0], TV1[1]) or array_equal(TV1[0], TV1[2]) or array_equal(TV1[1], TV1[2]):
print "Warning: Duplicate vertices in polygon %s." % i
print "Ignoring this polygon for energy calculation, but editing surface to remove duplicate vertices prior to DNE calculation is encouraged."
return [0,1]
b1 = TV1[1] - TV1[0]
b2 = TV1[2] - TV1[0]
g = array(([dot(b1,b1), dot(b1,b2)],[dot(b2,b1), dot(b2,b2)]))
if self.docondition:
if cond(g) > 10**5:
self.high_condition_faces.append([i, cond(g)])
return [0,1]
c1 = TV2[1] - TV2[0]
c2 = TV2[2] - TV2[0]
fstarh = array(([dot(c1,c1), dot(c1,c2)], [dot(c2,c1), dot(c2,c2)]))
gm = mat(g)
try:
gminv = gm.I
except LinAlgError as err:
condition = cond(g)
if condition > 10**5:
err.args += ('G matrix for polygon %s is singular and an inverse cannot be determined. Condition number is %s, turning condition number checking on will cause this polygon to be ignored for energy calculation.' % (i, cond(g)),)
raise
else:
err.args += ('G matrix for polygon %s is singular and an inverse cannot be determined. Condition number is %s, turning condition number checking on will not cause this polygon to be ignored for energy calculation. Further mesh processing is advised.' % (i, cond(g)),)
raise
e = trace((gminv*fstarh))
facearea = 0.5 * sqrt(g[0,0]*g[1,1]-g[0,1]*g[1,0])
if isnan(e):
self.nan_faces.append(i)
return [e,facearea]
def test_inverse(self):
for n in xrange(1, 10):
a = hilbert(n)
b = invhilbert(n)
# The Hilbert matrix is increasingly badly conditioned,
# so take that into account in the test
c = cond(a)
assert_allclose(a.dot(b), eye(n), atol=1e-15*c, rtol=1e-15*c)
def const_r():
for i in range(0, 50):
size = 200
mat = np.zeros((size,size))
r = rdft.generate_r(size) # use constant r
for j in range(0,size):
for k in range(0,size):
mat[j,k] = random.uniform(-100,100)
f = rdft.generate_f(size)
fr = f.dot(r)
fra = fr.dot(mat)
(a_maxcond,_,_) = rdft.get_leading_maxcond(mat)
(fra_maxcond,_,_) = rdft.get_leading_maxcond(fra)
a_cond = linalg.cond(mat)
fra_cond = linalg.cond(fra)
print("A: ", a_maxcond/a_cond)
print("FRA:", fra_maxcond/fra_cond)
def diagonalize(correlator_pannel, t0, td, generalized=False):
length = correlator_pannel.shape[0]
n = int(np.sqrt(length))
assert t0 is not None
# Here we access the pannel major_xs gives time(n), mean incase it
# was a multi correlator should have no effect on an already averaged one
A = np.matrix(np.reshape(correlator_pannel.major_xs(td).mean().values, (n, n)))
B = np.matrix(np.reshape(correlator_pannel.major_xs(t0).mean().values, (n, n)))
# Require A and B to be hermition for our generalized eigen value
# problem method to work. Here we force the matricies to be
# hermtion. This is justified since it is just useing the other
# measurement of the same value and averaging them.
A = hermitionize(A)
B = hermitionize(B)
logging.debug("A = {} \n B = {}".format(A, B))
if generalized:
logging.info("running generalized eigensolver")
evals, evecs = LA.eigh(A, b=B) #gerenalized eig problem, eigh works only if hermitian
evecs = np.matrix(evecs)
V = evecs
else:
# Instead of using generalized eigen problem, we could solve a
# regular eigen problem involving Binvqrt
Binvsqrt = LA.inv(LA.sqrtm(B))
logging.info("condition number: {}".format(cond(Binvsqrt*A*Binvsqrt)))
evals, evecs = LA.eigh(Binvsqrt*A*Binvsqrt)
evecs = np.matrix(evecs)
V = np.matrix(Binvsqrt)*evecs
if min(evals) < 0.05:
logging.warn("Warning, low eigenvalue detected. Eval={}".format(min(evals)))
else:
logging.info("lowest eigenvalue={}".format(min(evals)))
logging.debug("eigen values are {}".format(evals))
logging.debug("eigen vectors are {}".format(evecs))
n = len(evecs)
logging.debug("Matrix size {N}x{N}".format(N=n))
def rotate(x):
M = np.matrix(np.resize(x, (n, n)))
M = hermitionize(M)
D = V.H * M * V
R = np.array(D).flatten()
P = pd.Series(R)
return P
diag = correlator_pannel.apply(rotate, "items")
diag.items = ["{}{}".format(i,j) for i in reversed(range(n)) for j in reversed(range(n))]
# This method simultaniously diagaonlizes at t0 and td. Should be
# identity at t0 and the eigenvalues at td
assert compare_matrix(np.reshape(diag.major_xs(t0).mean().values, (n, n)),
np.identity(n)), "Rotation error: is not ~identity at t0"
assert compare_matrix(np.reshape(diag.major_xs(td).mean().values, (n, n)),
np.diag(evals)), "Rotation error: Is not ~Lambda at td"
return diag
def diagnostics(basename):
dagfilename = join(DATADIR, basename+'.dag')
prob = read_problem(dagfilename, plot_dag=False, show_sparsity=False)
partial_code = prepare_evaluation_code(prob)
rev_ad = import_code(partial_code)
con, jac_ad = rev_ad.evaluate(prob.refsols[0], prob.ncons, prob.nvars, prob.nzeros)
print('Constraint infinity norm: ', norm(con, np.inf))
print('Condition number estimate:', cond(jac_ad.todense()))
tests.JacobianTest.dump_code(partial_code, dagfilename)
def getTransformationMat(A): #A = matrix([[1,5,2,7],[1,1,3,31],[1,3,4,17],[1,1,1,1]]) #B = matrix([[77,75,74,61],[35,39,36,41],[9.2,9.2,7.2,-20.8],[1,1,1,1]]) #A = matrix([[-7.68,-7.855,-18,-6.24],[5.46,8.912,10.35,5.3816],[110.15,121.22,124.47,129.4126],[1,1,1,1]]) #Kinect Frame #B = matrix([[-10,0,0,10],[25,21.22,36,21],[8.7,12.86,8.7,15],[1,1,1,1]]) #Arm Frame B = matrix([[-10,25,8.7,1],[0,21.22,12.86,1],[0,36,8.7,1],[10,21,15,1],[-15,31,20,1],[10,31,20,1],[10,29,15,1],[15,29,15,1],[-15,20,15,1],[-10,20,15,1],[3,18,5,1],[5,20,25,1],[-5,25,25,1],[-5,30,6,1],[15,20,6,1]]) #Arm Frame B = B.transpose() # C = (inv((A.transpose())*A))*(A.transpose())*B if pts rows C = B * (A.transpose()) * inv(A * (A.transpose())) print cond(C) #test = matrix([[-5.56606237302],[11.4300619608],[111.661235336],[1]]) #point = C*test #GoToPos(point[0],point[1],point[2],'close') #print (C*test)[0] return C
def get_K_cond(K):
start = time.time()
K_cond = linalg.cond(K)
end = time.time()
if end - start > timer_thresh:
print 'get_K_cond:', end - start, 'sec'
return K_cond
def const_a(sample, size, rand_range):
result = []
mat = np.zeros((size,size))
for j in range(0,size):
for k in range(0,size):
mat[j,k] = random.uniform(-rand_range,rand_range)
for i in range(0, sample):
f = rdft.generate_f(size)
r = rdft.generate_r(size)
fr = f.dot(r)
fra = fr.dot(mat)
(a_maxcond,_,a_subcond) = rdft.get_leading_maxcond(mat)
(fra_maxcond,_,fra_subcond) = rdft.get_leading_maxcond(fra)
a_cond = linalg.cond(mat)
fra_cond = linalg.cond(fra)
result.append([mat, a_maxcond/a_cond, fra_maxcond/fra_cond, fra, a_subcond, fra_subcond])
#print("A: ", a_maxcond/a_cond)
#print("FRA:", fra_maxcond/fra_cond)
return result
def random_well_condition(size, val_range):
import random
r = np.zeros((size,size), dtype=np.complex128)
while True:
for i in range(0,size):
for j in range(0,size):
r[i,j] = random.uniform(-val_range,val_range)
if linalg.cond(r) < 10000:
break
return r
def crunch(self):
'''Mainline to number crunch the calculation of
member foreces.'''
# echo data:
printline = self.printline
lineread, noNodes, noMembers, noAuxiliaries,\
noLoadCases, noLoadLines = self.prolog()
nodes = self.doNodes(lineread, noNodes, noAuxiliaries)
printline("nodes = ")
printdict(printline, nodes)
members = self.doMembers(lineread, noMembers, noAuxiliaries)
printline('members = ')
printdict(printline, members)
memprops = self.doMemprops( noMembers, noAuxiliaries, members, nodes)
printline('memprops =')
printdict(self.printline, memprops)
ndim = noNodes * 2
printline('Dimension of connection mat ndim =' + str(ndim))
connection = np.zeros((ndim, ndim), dtype = "float")
for i in range(noMembers + noAuxiliaries):
member = memprops[i + 1]
connection[member[0] - 1, i] += member[4]
connection[member[1] - 1, i] += member[5]
if i < noMembers:
connection[member[2] - 1, i] += -member[4]
connection[member[3] - 1, i] += -member[5]
printline(' ')
printline('Connection matrix')
printerm(printline, connection)
try:
conditionnumber = la.cond(connection)
inv = la.inv(connection)
except la.LinAlgError as e:
printline( '%s %s' % (type(e).__name__, e))
else:
i = np.dot(inv, connection)
loads = np.zeros((ndim, 1), dtype = "float")
line = lineread()
assert line.strip() == 'loads', 'Flag for loads is loads'
for i in range(noLoadLines):
lst = lineread().split()
nodeNo = int(lst[0])
node = nodes[nodeNo]
loads[node[3] - 1] = -float(lst[1])
loads[node[4] - 1] = -float(lst[2])
printline('')
printline('Load matrix = ')
printerm(printline, loads)
memberActions = np.dot(inv, loads)
printline(' ')
printline("Member Actions, [Ni | Ri] matrix.")
printerm(printline, memberActions)
printline(' ')
printline('matrix condition number = ' + str(conditionnumber))
printline(' ')
def solve_getinfo(a,b,x,sample):
mat = np.array(a)
(size,_) = a.shape
result = []
a_cond = linalg.cond(mat)
(a_maxcond,_,a_subcond) = rdft.get_leading_maxcond(mat)
for i in range(0, sample):
# RDFT solve
(x1, fra, fr, ra, x1_orig) = solve_rdft(a,b,x)
(fra_maxcond,_,fra_subcond) = rdft.get_leading_maxcond(fra)
fra_cond = linalg.cond(fra)
err_rdft_iter = linalg.norm(x1-x)
err_rdft = linalg.norm(x1_orig-x)
# RDFT improved solve
(x1_imp, fra_imp, fr_imp, ra_imp, x1_imp_orig) = solve_rdft_improved(a,b,x)
(fra_maxcond_imp,_,fra_subcond_imp) = rdft.get_leading_maxcond(fra)
fra_cond_imp = linalg.cond(fra_imp)
err_rdft_imp_iter = linalg.norm(x1_imp-x)
err_rdft_imp = linalg.norm(x1_imp_orig-x)
# RDFT givens solve
(x5_imp, x5_imp_orig) = solve_rdft_givens(a,b,x)
err_rdft_givens_iter = linalg.norm(x5_imp-x)
err_rdft_givens = linalg.norm(x5_imp_orig-x)
# RDFT givens solve
(x7_imp, x7_imp_orig) = solve_rdft_givens_both(a,b,x)
err_rdft_givens_both_iter = linalg.norm(x7_imp-x)
err_rdft_givens_both = linalg.norm(x7_imp_orig-x)
# GAUSS solve
(x2, ga, _, x2_orig) = solve_gauss(a,b,x)
(ga_maxcond,_,ga_subcond) = rdft.get_leading_maxcond(fra)
ga_cond = linalg.cond(ga)
err_gauss_iter = linalg.norm(x2-x)
err_gauss = linalg.norm(x2_orig-x)
# Partial Pivot
a_float = np.array(a,dtype=np.float64)
b_float = np.array(b,dtype=np.float64)
(x3, pl, pu, swapped_a, swapped_b) = pp.solve(a_float,b_float)
err_pp = linalg.norm(x3-x)
##x3_i = iteration.iteration(swapped_a, pl, pu, swapped_b, x3, linalg.cond(a_float))
#result make
result.append(([a_maxcond/a_cond, fra_maxcond/fra_cond, fra, a_subcond, fra_subcond, fr, ra, err_rdft_iter, err_rdft,err_rdft_imp_iter,err_rdft_imp],[err_rdft_givens_iter,err_rdft_givens],[err_rdft_givens_both_iter,err_rdft_givens_both],[ga_maxcond/ga_cond, ga, ga_subcond, err_gauss_iter, err_gauss],[err_pp]))
return result
def solve_rdft_improved(a,b,x): (size,_) = a.shape f = rdft.generate_f(size) r = rdft.generate_random_r(size) fr = f.dot(r) ra = r.dot(a) (x1, l, u, fra, frb, _) = rdft.rdft_lu_solver_with_lu(a,b,r) x2 = np.array(x1) x2 = iteration.iteration_another(a, l, u, fr, b, x2, linalg.cond(fra)) x2 = iteration.remove_imag(x2) return (x2, fra, fr, ra, x1)
def MatrixConditionCheck(A, MaxConditionNumber = 10.0):
"""Check the condition number of a matrix.
Only write output to screen if the condition number is too high.
Should return something, really."""
ConditionNumber = la.cond(A)
if ConditionNumber > MaxConditionNumber:
print("The condition number of the matrix\n{0}\n"\
"is too large (i.e., it is {1:.4} which is larger"\
" than {2:.4}).\n".\
format(A, ConditionNumber, MaxConditionNumber))
def get_cov_by_clazz(self, clazz):
if clazz not in self.inv_cov:
cov_clazz = self.__cov_by_clazz(clazz)
self.cov[clazz] = cov_clazz
if linalg.cond(cov_clazz) < 1 / sys.float_info.epsilon:
self.inv_cov[clazz] = linalg.inv(cov_clazz)
self.det_cov[clazz] = linalg.det(cov_clazz)
else: # computing pseudo inverse/determinant to a singular covariance matrix
self.inv_cov[clazz] = linalg.pinv(cov_clazz)
eig_values, _ = linalg.eig(cov_clazz)
self.det_cov[clazz] = np.product(eig_values[(eig_values > 1e-12)])
return self.cov[clazz], self.inv_cov[clazz], self.det_cov[clazz]
def synthesis_operator(LF_selected_values, HF_selected_values, L, LF_new):
# LF_selected_values is $u^L(\gamma)$
# HF_selected_values is $u^H(\gamma)$
# L is the Cholesky product from the node selection algorithm
# LF_new is $u^L(\mathbf{z})$
G = dot(L, L.T)
G_inv = inv(G)
g = dot(LF_selected_values.T, LF_new)
c = dot(G_inv, g)
# MF_new approximates $u^H(\mathbf{z})$
MF_new = dot(HF_selected_values, c).squeeze()
return MF_new, cond(G)
def solve_gauss(a,b,x): (size,_) = a.shape g = rdft.generate_g(size) ga = g.dot(a) ga_save = np.array(ga) gb = g.dot(b) (l,u) = lu.lu(ga) y = lu.l_step(l,gb) x1 = lu.u_step(u,y) x2 = np.array(x1) x2 = iteration.iteration_another(a, l, u, g, b, x2, linalg.cond(ga)) return (x2, ga, a, x1)
def getSubregressorsConditionNumbers(self):
# get condition number for each of the links
linkConds = list()
for i in range(0, self.N_LINKS):
#get columns of base regressor that are dependent on std parameters of link i
#TODO: check if this is a sound approach
#TODO: try going further down to e.g. condition number of link mass, com, inertial
#get parts of base regressor with only independent columns (identifiable space)
#get all independent std columns for link i
base_columns = [j for j in range(0, self.num_base_params) \
if self.independent_cols[j] in range(i*10, i*10+9)]
#add dependent columns (not really correct, see getting self.B)
#for j in range(0, self.num_base_params):
# for dep in np.where(np.abs(self.linear_deps[j, :])>self.opt['min_tol'])[0]:
# if dep in range(i*10, i*10+9):
# base_columns.append(j)
if not len(base_columns):
linkConds.append(10e15)
else:
linkConds.append(la.cond(self.YBase[:, base_columns]))
#use base column dependencies to get parts of base regressor with influence on each each link
'''
base_columns = list()
for k in range(i*10, i*10+9):
for j in range(0, self.num_base_params):
if self.param_syms[k] in self.base_deps[j].free_symbols:
if not j in base_columns:
base_columns.append(j)
continue
if not len(base_columns):
linkConds.append(10e15)
else:
linkConds.append(la.cond(self.YBase[:, base_columns]))
'''
#use std regressor directly
'''
c = la.cond(self.YStd[:, i*10:i*10+9])
if np.isfinite(c):
linkConds.append(c)
else:
linkConds.append(10e15)
'''
print("Condition numbers of link sub-regressors: [{}]".format(dict(enumerate(linkConds))))
return linkConds
def test_randsvd():
"""Simple test of randsvd THIS WILL FAIL ~10% of the time"""
n = 18
k = 1e+20
a = rogues.randsvd(n, kappa=k)
c = nl.cond(a) / k
# The actual condition number should be within an order of magnitude
# of what we asked for but, sometimes even a order of magnitude doesn't
# quite cut it so we loosen up even more (even this doesn't always pass
# so we must think of a better check (looks like this fails a little
# less than 10% of the time.
assert(c > 0.05 and c < 50.)
def srrqr(M, k, f=1., verbose=False):
m, n = M.shape
minmn = np.min([m, n])
assert k <= minmn, "k must be less than min{m,n} k = %d,m = %d, n = %d" % (
k, m, n)
#QR with column pivoting
Q, R, p = qr(M, mode='economic', pivoting=True)
if k == n: return Q, R, p
increase_found = True
counter_perm = 0
iterc = 0
while (increase_found) and iterc <= 100:
iterc += 1
A = R[:k, :k]
AinvB = np.linalg.solve(A, R[:k, k:]) #Form A^{-1}B
C = R[k:minmn, k:]
#Compute column norms of C
if k < m: gamma = np.apply_along_axis(norm, 0, C[:, :n - k])
#Find row norms of A^{-1}
omega = np.apply_along_axis(norm, 0, inv(A).T)
F = AinvB**2.
if k < m: F += (np.outer(omega, gamma))**2.
ind = np.argwhere(F > f**2.)
if ind.size == 0: #finished
increase_found = False
else: #we can increase |det(A)|
i, j = ind[0, :]
counter_perm += 1
#permute columns i and j
R[:, [i, j + k]] = R[:, [j + k, i]]
p[[i, j + k]] = p[[j + k, i]]
#retriangularize R
q, R = qr(R, mode='economic')
Q = np.dot(Q, q)
#print p
Rkk = R[:k, :k]
inv_norm = norm(inv(Rkk), 2)
res_norm = norm(R[k:minmn, k:], 2) if k < minmn else 0.
if verbose:
print "Norm of inverse is %g" % (inv_norm)
print "Norm of residual is %g" % (res_norm)
sgn, det = slogdet(Rkk)
print "Log-determinant of selected columns is %g with sign %g" %\
(det, sgn)
print "Conditioning of selected columns is %g" % (cond(Rkk))
p = p[:k]
return Q, R, p
def do(self, a, b):
c = asarray(a) # a might be a matrix
s = linalg.svd(c, compute_uv=False)
old_assert_almost_equal(s[0] / s[-1], linalg.cond(a, 2), decimal=5)
def lncond(G):
return np.log(linalg.cond(np.dot(G.T, G)))
def assemble_operator(self, term, current_stage="online"):
if term == "projection_truth_snapshots":
assert current_stage in (
"online", "offline_rectification_postprocessing")
if current_stage == "online": # load from file
self.operator["projection_truth_snapshots"].load(
self.folder["reduced_operators"],
"projection_truth_snapshots")
return self.operator["projection_truth_snapshots"]
elif current_stage == "offline_rectification_postprocessing":
# the affine expansion storage contains only the inner product matrix
assert len(self.truth_problem.inner_product) == 1
inner_product = self.truth_problem.inner_product[0]
for n in range(1, self.N + 1):
# the affine expansion storage contains only the inner product matrix
assert len(self.inner_product) == 1
inner_product_n = self.inner_product[:n, :n][0]
basis_functions_n = self.basis_functions[:n]
projection_truth_snapshots_expansion = OnlineAffineExpansionStorage(
1)
projection_truth_snapshots = OnlineMatrix(n, n)
for (i, snapshot_i) in enumerate(self.snapshots[:n]):
projected_truth_snapshot_i = OnlineFunction(n)
solver = LinearSolver(
inner_product_n, projected_truth_snapshot_i,
transpose(basis_functions_n) * inner_product *
snapshot_i)
solver.set_parameters(
self._linear_solver_parameters)
solver.solve()
for j in range(n):
projection_truth_snapshots[
j,
i] = projected_truth_snapshot_i.vector()[j]
projection_truth_snapshots_expansion[
0] = projection_truth_snapshots
print("\tcondition number for n = " + str(n) + ": " +
str(cond(projection_truth_snapshots)))
self.operator[
"projection_truth_snapshots"][:n, :
n] = projection_truth_snapshots_expansion
# Save
self.operator["projection_truth_snapshots"].save(
self.folder["reduced_operators"],
"projection_truth_snapshots")
return self.operator["projection_truth_snapshots"]
else:
raise ValueError("Invalid stage in assemble_operator().")
elif term == "projection_reduced_snapshots":
assert current_stage in (
"online", "offline_rectification_postprocessing")
if current_stage == "online": # load from file
self.operator["projection_reduced_snapshots"].load(
self.folder["reduced_operators"],
"projection_reduced_snapshots")
return self.operator["projection_reduced_snapshots"]
elif current_stage == "offline_rectification_postprocessing":
# Backup mu
bak_mu = self.mu
# Prepare rectification for all possible online solve arguments
for n in range(1, self.N + 1):
print("\tcondition number for n = " + str(n))
projection_reduced_snapshots_expansion = OnlineAffineExpansionStorage(
len(self.online_solve_kwargs_without_rectification)
)
for (q, online_solve_kwargs) in enumerate(
self.online_solve_kwargs_without_rectification
):
projection_reduced_snapshots = OnlineMatrix(n, n)
for (i, mu_i) in enumerate(self.snapshots_mu[:n]):
self.set_mu(mu_i)
projected_reduced_snapshot_i = self.solve(
n, **online_solve_kwargs)
for j in range(n):
projection_reduced_snapshots[
j,
i] = projected_reduced_snapshot_i.vector(
)[j]
projection_reduced_snapshots_expansion[
q] = projection_reduced_snapshots
print("\t\tonline solve options " + str(
dict(self.
online_solve_kwargs_with_rectification[q])
) + ": " + str(cond(projection_reduced_snapshots)))
self.operator[
"projection_reduced_snapshots"][:n, :
n] = projection_reduced_snapshots_expansion
# Save and restore previous mu
self.set_mu(bak_mu)
self.operator["projection_reduced_snapshots"].save(
self.folder["reduced_operators"],
"projection_reduced_snapshots")
return self.operator["projection_reduced_snapshots"]
else:
raise ValueError("Invalid stage in assemble_operator().")
else:
return EllipticCoerciveReducedProblem_DerivedClass.assemble_operator(
self, term, current_stage)
def printStats(self, summary_only=False):
idf = self.idf
if idf.opt['selectBlocksFromMeasurements']:
if len(idf.data.usedBlocks):
print("used {} of {} blocks: {}".format(
len(idf.data.usedBlocks),
len(idf.data.usedBlocks) + len(idf.data.unusedBlocks),
[b for (b, bs, cond, linkConds) in idf.data.usedBlocks]))
else:
print("\ncurrent block: {}".format(idf.data.block_pos))
#print "unused blocks: {}".format(idf.unusedBlocks)
print("condition number: {}".format(la.cond(idf.model.YBase)))
if idf.opt['identifyGravityParamsOnly']:
fric = idf.model.num_dofs * idf.opt['identifyFriction']
sum_id = np.sum(idf.model.xStd[0:idf.model.num_identified_params -
fric:4])
else:
sum_id = np.sum(idf.model.xStd[0:idf.model.num_model_params:10])
print(Style.BRIGHT + "Parameters" + Style.RESET_ALL)
sum_apriori = np.sum(
idf.model.xStdModel[0:idf.model.num_model_params:10])
print("Estimated overall mass: {} kg vs. a priori {} kg".format(
sum_id, sum_apriori),
end="")
if idf.urdf_file_real:
print(" vs. real {} kg".format(
np.sum(self.xStdReal[0:idf.model.num_model_params:10])))
else:
print()
if idf.opt['showStandardParams']:
if idf.opt['showTriangleConsistency']:
cons_apriori = idf.paramHelpers.checkPhysicalConsistency(
idf.model.xStdModel, full=True)
cons_ident = idf.paramHelpers.checkPhysicalConsistency(
idf.model.xStd)
print("Consistency (including triangle inequality):")
else:
cons_apriori = idf.paramHelpers.checkPhysicalConsistencyNoTriangle(
idf.model.xStdModel, full=True)
cons_ident = idf.paramHelpers.checkPhysicalConsistencyNoTriangle(
idf.model.xStd)
if False in list(cons_apriori.values()):
print(Fore.RED +
"A priori parameters are not physical consistent!" +
Fore.RESET)
print("Per-link physical consistency (a priori): {}".format(
cons_apriori))
else:
print("A priori parameters are physical consistent")
if False in list(cons_ident.values()):
print("Identified parameters are not physical consistent,")
print("per-link physical consistency (identified): {}".format(
cons_ident))
else:
print("Identified parameters are physical consistent")
if idf.opt['identifyGravityParamsOnly']:
p_idf = idf.model.identified_params
else:
p_idf = idf.model.identifiable
if idf.urdf_file_real:
if idf.opt['showStandardParams']:
#if idf.opt['useEssentialParams']:
# print("Mean relative error of essential std params: {}%".\
# format(sum_diff_r_pc_ess / len(idf.stdEssentialIdx)))
#print("Mean relative error of all std params: {}%".format(sum_diff_r_pc_all/len(idf.model.xStd)))
#if idf.opt['useEssentialParams']:
# print("Mean error delta (a priori error vs approx error) of essential std params: {}%".\
# format(sum_pc_delta_ess/len(idf.stdEssentialIdx)))
#print("Mean error delta (a priori error vs approx error) of all std params: {}%".\
# format(sum_pc_delta_all/len(idf.model.xStd)))
sq_error_apriori = np.square(
la.norm(self.xStdReal[p_idf] - idf.model.xStdModel[p_idf]))
if idf.opt['identifyGravityParamsOnly']:
xStd_full = idf.model.xStdModel.copy()
xStd_full[p_idf] = idf.model.xStd
sq_error_idf = np.square(
la.norm(self.xStdReal[p_idf] - xStd_full[p_idf]))
else:
sq_error_idf = np.square(
la.norm(self.xStdReal[p_idf] - idf.model.xStd[p_idf]))
print("Squared distance of identifiable std parameter vectors (identified, a priori) to real: {} vs. {}".\
format(sq_error_idf, sq_error_apriori))
#sq_error_apriori = np.square(la.norm(xStdReal - idf.model.xStdModel))
#sq_error_idf = np.square(la.norm(xStdReal - idf.model.xStd))
#print( "Squared distance of std parameter vectors (identified, a priori) to real: {} vs. {}".\
# format(sq_error_idf, sq_error_apriori))
if idf.opt['showBaseParams'] and not summary_only and idf.opt[
'estimateWith'] not in ['urdf', 'std_direct']:
#print("Mean error (a priori - approx) of all base params: {:.5f}".\
# format(sum_error_all_base/len(idf.model.xBase)))
sq_error_apriori = np.square(
la.norm(self.xBaseReal - idf.model.xBaseModel))
sq_error_idf = np.square(
la.norm(self.xBaseReal - idf.model.xBase))
print("Squared distance of base parameter vectors (identified, a priori) to real: {} vs. {}".\
format(sq_error_idf, sq_error_apriori))
else:
if idf.opt['showStandardParams'] and not summary_only:
if idf.opt['identifyGravityParamsOnly']:
xStd_full = idf.model.xStdModel.copy()
xStd_full[p_idf] = idf.model.xStd
sq_error_apriori = np.square(
la.norm(xStd_full[p_idf] - idf.model.xStdModel[p_idf]))
else:
sq_error_apriori = np.square(
la.norm(self.xStd[p_idf] - idf.model.xStdModel[p_idf]))
print("Squared distance of identifiable std parameter vectors to a priori: {}".\
format(sq_error_apriori))
if idf.opt['showBaseParams'] and not summary_only and idf.opt[
'estimateWith'] not in ['urdf', 'std_direct']:
sq_error_apriori = np.square(
la.norm(idf.model.xBase - idf.model.xBaseModel))
print("Squared distance of base parameter vectors (identified vs. a priori): {}".\
format(sq_error_apriori))
print(Style.BRIGHT + "\nTorque prediction errors" + Style.RESET_ALL)
# get percentual error (i.e. how big is the error relative to the measured magnitudes)
idf.estimateRegressorTorques(
estimateWith='urdf') #estimate torques with CAD params
idf.estimateRegressorTorques(
) #estimate torques again with identified parameters
idf.apriori_error = sla.norm(idf.tauAPriori -
idf.model.tauMeasured) * 100 / sla.norm(
idf.model.tauMeasured)
idf.res_error = sla.norm(idf.tauEstimated -
idf.model.tauMeasured) * 100 / sla.norm(
idf.model.tauMeasured)
print("Relative mean residual error: {}% vs. A priori: {}%".\
format(idf.res_error, idf.apriori_error))
idf.abs_apriori_error = np.mean(
sla.norm(idf.tauAPriori - idf.model.tauMeasured, axis=1))
idf.abs_res_error = idf.base_error #np.mean(sla.norm(idf.tauEstimated-idf.model.tauMeasured, axis=1))
print("Absolute mean residual error: {} vs. A priori: {}".format(
idf.abs_res_error, idf.abs_apriori_error))
torque_limits = []
for joint in idf.model.jointNames:
torque_limits.append(idf.model.limits[joint]['torque'])
idf.abs_apriori_error = helpers.getNRMSE(idf.model.tauMeasured,
idf.tauAPriori,
limits=torque_limits)
idf.abs_res_error = helpers.getNRMSE(idf.model.tauMeasured,
idf.tauEstimated,
limits=torque_limits)
print("NRMS of residual error: {}% vs. A priori: {}%".format(
idf.abs_res_error, idf.abs_apriori_error))
y = []
z = 0
norma = 0
k = len(A)
for i in range (0, k):
kolumna = A[:, [i]]
for j in range (0, len(kolumna)):
z += kolumna[j]
z = float(z)
y.append(z)
z = 0
return max(y)
print("Norma macierzy Hilberta n = ", n1)
print(norma(m1))
print("Norma macierzy Hilberta n = ", n2)
print(norma(m2))
print("Norma macierzy Hilberta n = ", n3)
print(norma(m3))
print("Współczynnik uwarunkowania macierzy Hilberta n = ", n1)
print(cond(m1))
print("Współczynnik uwarunkowania macierzy Hilberta n = ", n2)
print(cond(m2))
print("Współczynnik uwarunkowania macierzy Hilberta n = ", n3)
print(cond(m3))
def functional(eps, dist_mat, rbf, target_cond):
return np.log(la.cond(phi(dist_mat, eps)) / target_cond)**2
def solve_continuous_are(a, b, q, r, e=None, s=None, balanced=True):
r"""
Solves the continuous-time algebraic Riccati equation (CARE).
The CARE is defined as
.. math::
X A + A^H X - X B R^{-1} B^H X + Q = 0
The limitations for a solution to exist are :
* All eigenvalues of :math:`A` on the right half plane, should be
controllable.
* The associated hamiltonian pencil (See Notes), should have
eigenvalues sufficiently away from the imaginary axis.
Moreover, if ``e`` or ``s`` is not precisely ``None``, then the
generalized version of CARE
.. math::
E^HXA + A^HXE - (E^HXB + S) R^{-1} (B^HXE + S^H) + Q = 0
is solved. When omitted, ``e`` is assumed to be the identity and ``s``
is assumed to be the zero matrix with sizes compatible with ``a`` and
``b``, respectively.
Parameters
----------
a : (M, M) array_like
Square matrix
b : (M, N) array_like
Input
q : (M, M) array_like
Input
r : (N, N) array_like
Nonsingular square matrix
e : (M, M) array_like, optional
Nonsingular square matrix
s : (M, N) array_like, optional
Input
balanced : bool, optional
The boolean that indicates whether a balancing step is performed
on the data. The default is set to True.
Returns
-------
x : (M, M) ndarray
Solution to the continuous-time algebraic Riccati equation.
Raises
------
LinAlgError
For cases where the stable subspace of the pencil could not be
isolated. See Notes section and the references for details.
See Also
--------
solve_discrete_are : Solves the discrete-time algebraic Riccati equation
Notes
-----
The equation is solved by forming the extended hamiltonian matrix pencil,
as described in [1]_, :math:`H - \lambda J` given by the block matrices ::
[ A 0 B ] [ E 0 0 ]
[-Q -A^H -S ] - \lambda * [ 0 E^H 0 ]
[ S^H B^H R ] [ 0 0 0 ]
and using a QZ decomposition method.
In this algorithm, the fail conditions are linked to the symmetry
of the product :math:`U_2 U_1^{-1}` and condition number of
:math:`U_1`. Here, :math:`U` is the 2m-by-m matrix that holds the
eigenvectors spanning the stable subspace with 2-m rows and partitioned
into two m-row matrices. See [1]_ and [2]_ for more details.
In order to improve the QZ decomposition accuracy, the pencil goes
through a balancing step where the sum of absolute values of
:math:`H` and :math:`J` entries (after removing the diagonal entries of
the sum) is balanced following the recipe given in [3]_.
.. versionadded:: 0.11.0
References
----------
.. [1] P. van Dooren , "A Generalized Eigenvalue Approach For Solving
Riccati Equations.", SIAM Journal on Scientific and Statistical
Computing, Vol.2(2), :doi:`10.1137/0902010`
.. [2] A.J. Laub, "A Schur Method for Solving Algebraic Riccati
Equations.", Massachusetts Institute of Technology. Laboratory for
Information and Decision Systems. LIDS-R ; 859. Available online :
http://hdl.handle.net/1721.1/1301
.. [3] P. Benner, "Symplectic Balancing of Hamiltonian Matrices", 2001,
SIAM J. Sci. Comput., 2001, Vol.22(5), :doi:`10.1137/S1064827500367993`
Examples
--------
Given `a`, `b`, `q`, and `r` solve for `x`:
>>> from scipy import linalg
>>> a = np.array([[4, 3], [-4.5, -3.5]])
>>> b = np.array([[1], [-1]])
>>> q = np.array([[9, 6], [6, 4.]])
>>> r = 1
>>> x = linalg.solve_continuous_are(a, b, q, r)
>>> x
array([[ 21.72792206, 14.48528137],
[ 14.48528137, 9.65685425]])
>>> np.allclose(a.T.dot(x) + x.dot(a)-x.dot(b).dot(b.T).dot(x), -q)
True
"""
# Validate input arguments
a, b, q, r, e, s, m, n, r_or_c, gen_are = _are_validate_args(
a, b, q, r, e, s, 'care')
H = np.empty((2 * m + n, 2 * m + n), dtype=r_or_c)
H[:m, :m] = a
H[:m, m:2 * m] = 0.
H[:m, 2 * m:] = b
H[m:2 * m, :m] = -q
H[m:2 * m, m:2 * m] = -a.conj().T
H[m:2 * m, 2 * m:] = 0. if s is None else -s
H[2 * m:, :m] = 0. if s is None else s.conj().T
H[2 * m:, m:2 * m] = b.conj().T
H[2 * m:, 2 * m:] = r
if gen_are and e is not None:
J = block_diag(e, e.conj().T, np.zeros_like(r, dtype=r_or_c))
else:
J = block_diag(np.eye(2 * m), np.zeros_like(r, dtype=r_or_c))
if balanced:
# xGEBAL does not remove the diagonals before scaling. Also
# to avoid destroying the Symplectic structure, we follow Ref.3
M = np.abs(H) + np.abs(J)
M[np.diag_indices_from(M)] = 0.
_, (sca, _) = matrix_balance(M, separate=1, permute=0)
# do we need to bother?
if not np.allclose(sca, np.ones_like(sca)):
# Now impose diag(D,inv(D)) from Benner where D is
# square root of s_i/s_(n+i) for i=0,....
sca = np.log2(sca)
# NOTE: Py3 uses "Bankers Rounding: round to the nearest even" !!
s = np.round((sca[m:2 * m] - sca[:m]) / 2)
sca = 2**np.r_[s, -s, sca[2 * m:]]
# Elementwise multiplication via broadcasting.
elwisescale = sca[:, None] * np.reciprocal(sca)
H *= elwisescale
J *= elwisescale
# Deflate the pencil to 2m x 2m ala Ref.1, eq.(55)
q, r = qr(H[:, -n:])
H = q[:, n:].conj().T.dot(H[:, :2 * m])
J = q[:2 * m, n:].conj().T.dot(J[:2 * m, :2 * m])
# Decide on which output type is needed for QZ
out_str = 'real' if r_or_c == float else 'complex'
_, _, _, _, _, u = ordqz(H,
J,
sort='lhp',
overwrite_a=True,
overwrite_b=True,
check_finite=False,
output=out_str)
# Get the relevant parts of the stable subspace basis
if e is not None:
u, _ = qr(np.vstack((e.dot(u[:m, :m]), u[m:, :m])))
u00 = u[:m, :m]
u10 = u[m:, :m]
# Solve via back-substituion after checking the condition of u00
up, ul, uu = lu(u00)
if 1 / cond(uu) < np.spacing(1.):
raise LinAlgError('Failed to find a finite solution.')
# Exploit the triangular structure
x = solve_triangular(
ul.conj().T,
solve_triangular(uu.conj().T, u10.conj().T, lower=True),
unit_diagonal=True,
).conj().T.dot(up.conj().T)
if balanced:
x *= sca[:m, None] * sca[:m]
# Check the deviation from symmetry for lack of success
# See proof of Thm.5 item 3 in [2]
u_sym = u00.conj().T.dot(u10)
n_u_sym = norm(u_sym, 1)
u_sym = u_sym - u_sym.conj().T
sym_threshold = np.max([np.spacing(1000.), 0.1 * n_u_sym])
if norm(u_sym, 1) > sym_threshold:
raise LinAlgError('The associated Hamiltonian pencil has eigenvalues '
'too close to the imaginary axis')
return (x + x.conj().T) / 2
def test(self):
A = array([[1., 0, 0], [0, -2., 0], [0, 0, 3.]])
assert_almost_equal(linalg.cond(A, inf), 3.)
