def cluster_vectorspace(self, vectors, trace=False):
assert len(vectors) > 0
# set the parameters to initial values
dimensions = len(vectors[0])
means = self._means
priors = self._priors
if not priors:
priors = self._priors = numarray.ones(self._num_clusters,
numarray.Float64) / self._num_clusters
covariances = self._covariance_matrices
if not covariances:
covariances = self._covariance_matrices = \
[ numarray.identity(dimensions, numarray.Float64)
for i in range(self._num_clusters) ]
# do the E and M steps until the likelihood plateaus
lastl = self._loglikelihood(vectors, priors, means, covariances)
converged = False
while not converged:
if trace: print 'iteration; loglikelihood', lastl
# E-step, calculate hidden variables, h[i,j]
h = numarray.zeros((len(vectors), self._num_clusters),
numarray.Float64)
for i in range(len(vectors)):
for j in range(self._num_clusters):
h[i,j] = priors[j] * self._gaussian(means[j],
covariances[j], vectors[i])
h[i,:] /= sum(h[i,:])
# M-step, update parameters - cvm, p, mean
for j in range(self._num_clusters):
covariance_before = covariances[j]
new_covariance = numarray.zeros((dimensions, dimensions),
numarray.Float64)
new_mean = numarray.zeros(dimensions, numarray.Float64)
sum_hj = 0.0
for i in range(len(vectors)):
delta = vectors[i] - means[j]
new_covariance += h[i,j] * \
numarray.multiply.outer(delta, delta)
sum_hj += h[i,j]
new_mean += h[i,j] * vectors[i]
covariances[j] = new_covariance / sum_hj
means[j] = new_mean / sum_hj
priors[j] = sum_hj / len(vectors)
# bias term to stop covariance matrix being singular
covariances[j] += self._bias * \
numarray.identity(dimensions, numarray.Float64)
# calculate likelihood - FIXME: may be broken
l = self._loglikelihood(vectors, priors, means, covariances)
# check for convergence
if abs(lastl - l) < self._conv_threshold:
converged = True
lastl = l
def powell(F,x,h=0.1,tol=1.0e-6):
def f(s): return F(x + s*v) # F in direction of v
n = len(x) # Number of design variables
df = zeros((n),type=Float64) # Decreases of F stored here
u = identity(n)*1.0 # Vectors v stored here by rows
for j in range(30): # Allow for 30 cycles:
xOld = x.copy() # Save starting point
fOld = F(xOld)
# First n line searches record decreases of F
for i in range(n):
v = u[i]
a,b = bracket(f,0.0,h)
s,fMin = search(f,a,b)
df[i] = fOld - fMin
fOld = fMin
x = x + s*v
# Last line search in the cycle
v = x - xOld
a,b = bracket(f,0.0,h)
s,fLast = search(f,a,b)
x = x + s*v
# Check for convergence
if sqrt(dot(x-xOld,x-xOld)/n) < tol: return x,j+1
# Identify biggest decrease & update search directions
iMax = int(argmax(df))
for i in range(iMax,n-1):
u[i] = u[i+1]
u[n-1] = v
print "Powell did not converge"
def cluster(self, tokens, assign_clusters=False, trace=False):
assert chktype(1, tokens, [Token])
assert chktype(2, assign_clusters, bool)
assert chktype(3, trace, bool)
assert len(tokens) > 0
vectors = map(lambda tk: tk['FEATURES'], tokens)
# normalise the vectors
if self._should_normalise:
vectors = map(self._normalise, vectors)
# use SVD to reduce the dimensionality
if self._svd_dimensions and self._svd_dimensions < len(vectors[0]):
[u, d, vt] = numarray.linear_algebra.singular_value_decomposition(
numarray.transpose(numarray.array(vectors)))
S = d[:self._svd_dimensions] * \
numarray.identity(self._svd_dimensions, numarray.Float64)
T = u[:,:self._svd_dimensions]
Dt = vt[:self._svd_dimensions,:]
vectors = numarray.transpose(numarray.matrixmultiply(S, Dt))
self._Tt = numarray.transpose(T)
# call abstract method to cluster the vectors
self.cluster_vectorspace(vectors, trace)
# assign the tokens to clusters
if assign_clusters:
for token in tokens:
self.classify(token)
def matInv(a):
n = len(a[0])
aInv = identity(n)*1.0
a,seq = LUdecomp(a)
for i in range(n):
aInv[:,i] = LUsolve(a,aInv[:,i],seq)
return aInv
class _Matrix(NumArray):
def _rc(self, a):
if len(shape(a)) == 0:
return a
else:
return Matrix(a)
def __mul__(self, other):
aother = asarray(other)
if len(aother.shape) == 0:
return self._rc(self * aother)
else:
return self._rc(_dot(self, aother))
def __rmul__(self, other):
aother = asarray(other)
if len(aother.shape) == 0:
return self._rc(aother * self)
else:
return self._rc(_dot(aother, self))
def __imul__(self, other):
aother = asarray(other)
self[:] = _dot(self, aother)
return self
def __pow__(self, other):
shape = self.shape
if len(shape) != 2 or shape[0] != shape[1]:
raise TypeError, "matrix is not square"
if type(other) not in (type(1), type(1L)):
raise TypeError, "exponent must be an integer"
if other == 0:
return Matrix(identity(shape[0]))
if other < 0:
result = Matrix(LinearAlgebra.inverse(self))
x = Matrix(result)
other = -other
else:
result = self
x = result
if other <= 3:
while (other > 1):
result = result * x
other = other - 1
return result
# binary decomposition to reduce the number of Matrix
# Multiplies for other > 3.
beta = _binary(other)
t = len(beta)
Z, q = x.copy(), 0
while beta[t - q - 1] == '0':
Z *= Z
q += 1
result = Z.copy()
for k in range(q + 1, t):
Z *= Z
if beta[t - k - 1] == '1':
result *= Z
return result
def powell(F, x, h=0.1, tol=1.0e-6):
def f(s):
return F(x + s * v) # F in direction of v
n = len(x) # Number of design variables
df = zeros((n), type=Float64) # Decreases of F stored here
u = identity(n) * 1.0 # Vectors v stored here by rows
for j in range(30): # Allow for 30 cycles:
xOld = x.copy() # Save starting point
fOld = F(xOld)
# First n line searches record decreases of F
for i in range(n):
v = u[i]
a, b = bracket(f, 0.0, h)
s, fMin = search(f, a, b)
df[i] = fOld - fMin
fOld = fMin
x = x + s * v
# Last line search in the cycle
v = x - xOld
a, b = bracket(f, 0.0, h)
s, fLast = search(f, a, b)
x = x + s * v
# Check for convergence
if sqrt(dot(x - xOld, x - xOld) / n) < tol: return x, j + 1
# Identify biggest decrease & update search directions
iMax = int(argmax(df))
for i in range(iMax, n - 1):
u[i] = u[i + 1]
u[n - 1] = v
print "Powell did not converge"
def __pow__(self, other):
shape = self.shape
if len(shape)!=2 or shape[0]!=shape[1]:
raise TypeError, "matrix is not square"
if type(other) not in (type(1), type(1L)):
raise TypeError, "exponent must be an integer"
if other==0:
return Matrix(identity(shape[0]))
if other<0:
result=Matrix(LinearAlgebra.inverse(self))
x=Matrix(result)
other=-other
else:
result=self
x=result
if other <= 3:
while(other>1):
result=result*x
other=other-1
return result
# binary decomposition to reduce the number of Matrix
# Multiplies for other > 3.
beta = _binary(other)
t = len(beta)
Z,q = x.copy(),0
while beta[t-q-1] == '0':
Z *= Z
q += 1
result = Z.copy()
for k in range(q+1,t):
Z *= Z
if beta[t-k-1] == '1':
result *= Z
return result
def inverse(a):
"""inverse(a) -> inverse matrix of a
*a* may be either rank-2 or rank-3. If it is rank-2, it must square.
>>> A = [[1,2,3], [3,5,5], [5,6,7]]
>>> Ainv = inverse(A)
>>> _isClose(na.dot(A, Ainv), na.identity(3))
1
If *a* is rank-3, it is treated as an array of rank-2 matrices and
must be square along the last 2 axes.
>>> A = [[[1, 3], [2j, 3j]],
... [[2, 4], [4j, 4j]],
... [[3, 5], [6j, 5j]]]
>>> Ainv = inverse(A)
>>> _isClose(map(na.dot, A, Ainv), [na.identity(2)]*3)
1
If *a* is not square along its last two axes, a LinAlgError is raised.
>>> inverse(na.asarray(A)[...,:1])
Traceback (most recent call last):
...
LinearAlgebraError: Array (or it submatrices) must be square
"""
a = na.asarray(a)
I = na.identity(a.shape[-2])
if len(a.shape) == 3:
I.shape = (1,) + I.shape
return solve_linear_equations(a, I)
def computeP(a):
n = len(a)
p = identity(n)*1.0
for k in range(n-2):
u = a[k+1:n,k]
h = dot(u,u)/2.0
v = matrixmultiply(p[1:n,k+1:n],u)/h
p[1:n,k+1:n] = p[1:n,k+1:n] - outerproduct(v,u)
return p
def computeP(a):
n = len(a)
p = identity(n) * 1.0
for k in range(n - 2):
u = a[k + 1:n, k]
h = dot(u, u) / 2.0
v = matrixmultiply(p[1:n, k + 1:n], u) / h
p[1:n, k + 1:n] = p[1:n, k + 1:n] - outerproduct(v, u)
return p
def jacobi(a, tol=1.0e-9): # Jacobi method
def maxElem(a): # Find largest off-diag. element a[k,l]
n = len(a)
aMax = 0.0
for i in range(n - 1):
for j in range(i + 1, n):
if abs(a[i, j]) >= aMax:
aMax = abs(a[i, j])
k = i
l = j
return aMax, k, l
def rotate(a, p, k, l): # Rotate to make a[k,l] = 0
n = len(a)
aDiff = a[l, l] - a[k, k]
if abs(a[k, l]) < abs(aDiff) * 1.0e-36: t = a[k, l] / aDiff
else:
phi = aDiff / (2.0 * a[k, l])
t = 1.0 / (abs(phi) + sqrt(phi**2 + 1.0))
if phi < 0.0: t = -t
c = 1.0 / sqrt(t**2 + 1.0)
s = t * c
tau = s / (1.0 + c)
temp = a[k, l]
a[k, l] = 0.0
a[k, k] = a[k, k] - t * temp
a[l, l] = a[l, l] + t * temp
for i in range(k): # Case of i < k
temp = a[i, k]
a[i, k] = temp - s * (a[i, l] + tau * temp)
a[i, l] = a[i, l] + s * (temp - tau * a[i, l])
for i in range(k + 1, l): # Case of k < i < l
temp = a[k, i]
a[k, i] = temp - s * (a[i, l] + tau * a[k, i])
a[i, l] = a[i, l] + s * (temp - tau * a[i, l])
for i in range(l + 1, n): # Case of i > l
temp = a[k, i]
a[k, i] = temp - s * (a[l, i] + tau * temp)
a[l, i] = a[l, i] + s * (temp - tau * a[l, i])
for i in range(n): # Update transformation matrix
temp = p[i, k]
p[i, k] = temp - s * (p[i, l] + tau * p[i, k])
p[i, l] = p[i, l] + s * (temp - tau * p[i, l])
n = len(a)
maxRot = 5 * (n**2) # Set limit on number of rotations
p = identity(n) * 1.0 # Initialize transformation matrix
for i in range(maxRot): # Jacobi rotation loop
aMax, k, l = maxElem(a)
if aMax < tol: return diagonal(a), p
rotate(a, p, k, l)
print 'Jacobi method did not converge'
def jacobi(a,tol = 1.0e-9): # Jacobi method
def maxElem(a): # Find largest off-diag. element a[k,l]
n = len(a)
aMax = 0.0
for i in range(n-1):
for j in range(i+1,n):
if abs(a[i,j]) >= aMax:
aMax = abs(a[i,j])
k = i; l = j
return aMax,k,l
def rotate(a,p,k,l): # Rotate to make a[k,l] = 0
n = len(a)
aDiff = a[l,l] - a[k,k]
if abs(a[k,l]) < abs(aDiff)*1.0e-36: t = a[k,l]/aDiff
else:
phi = aDiff/(2.0*a[k,l])
t = 1.0/(abs(phi) + sqrt(phi**2 + 1.0))
if phi < 0.0: t = -t
c = 1.0/sqrt(t**2 + 1.0); s = t*c
tau = s/(1.0 + c)
temp = a[k,l]
a[k,l] = 0.0
a[k,k] = a[k,k] - t*temp
a[l,l] = a[l,l] + t*temp
for i in range(k): # Case of i < k
temp = a[i,k]
a[i,k] = temp - s*(a[i,l] + tau*temp)
a[i,l] = a[i,l] + s*(temp - tau*a[i,l])
for i in range(k+1,l): # Case of k < i < l
temp = a[k,i]
a[k,i] = temp - s*(a[i,l] + tau*a[k,i])
a[i,l] = a[i,l] + s*(temp - tau*a[i,l])
for i in range(l+1,n): # Case of i > l
temp = a[k,i]
a[k,i] = temp - s*(a[l,i] + tau*temp)
a[l,i] = a[l,i] + s*(temp - tau*a[l,i])
for i in range(n): # Update transformation matrix
temp = p[i,k]
p[i,k] = temp - s*(p[i,l] + tau*p[i,k])
p[i,l] = p[i,l] + s*(temp - tau*p[i,l])
n = len(a)
maxRot = 5*(n**2) # Set limit on number of rotations
p = identity(n)*1.0 # Initialize transformation matrix
for i in range(maxRot): # Jacobi rotation loop
aMax,k,l = maxElem(a)
if aMax < tol: return diagonal(a),p
rotate(a,p,k,l)
print 'Jacobi method did not converge'
def setParameters(self, mu = None, sigma = None, wi = None, sigma_type = 'full', \
tied_sigma = False, isAdjustable = False):
#============================================================
# set the mean :
# self.mean[i] = the mean for dimension i
# self.mean.shape = (self.nvalues, q1,q2,...,qn)
# where qi is the size of discrete parent i
if mu == None:
# set all mu to zeros
mu = na.zeros(shape=([self.nvalues]+self.discrete_parents_shape), \
type='Float32')
try:
mu = na.array(shape=[self.nvalues]+self.discrete_parents_shape, \
type='Float32')
except:
raise 'Could not convert mu to numarray of shape : %s, discrete parents = %s' %(str(self.discrete_parents_shape),
str([dp.name for dp in self.discrete_parents]))
self.mean = mu
#============================================================
# set the covariance :
# self.sigma[i,j] = the covariance between dimension i and j
# self.sigma.shape = (nvalues,nvalues,q1,q2,...,qn)
# where qi is the size of discrete parent i
if sigma == None:
eye = na.identity(self.nvalues, type = 'Float32')[...,na.NewAxis]
if len(self.discrete_parents) > 0:
q = reduce(lambda a,b:a*b,self.discrete_parents_shape) # number of different configurations for the parents
sigma = na.concatenate([eye]*q, axis=2)
sigma = na.array(sigma,shape=[self.nvalues,self.nvalues]+self.discrete_parents_shape)
try:
sigma = na.array(sigma, shape=[self.nvalues,self.nvalues]+self.discrete_parents_shape, type='Float32')
except:
raise 'Not a valid covariance matrix'
self.sigma = sigma
#============================================================
# set the weights :
# self.weights[i,j] = the regression for dimension i and continuous parent j
# self.weights.shape = (nvalues,x1,x2,...,xn,q1,q2,...,qn)
# where xi is the size of continuous parent i)
# and qi is the size of discrete parent i
if wi == None:
wi = na.ones(shape=[self.nvalues]+self.parents_shape, type='Float32')
try:
wi = na.array(wi, shape=[self.nvalues]+self.parents_shape, type='Float32')
except:
raise 'Not a valid weight'
self.weights = wi
def __init__(self, v, mu = None, sigma = None, wi = None, \
sigma_type = 'full', tied_sigma = False, \
isAdjustable = True, ignoreFamily = False):
Distribution.__init__(self, v, isAdjustable=isAdjustable, \
ignoreFamily=ignoreFamily)
self.distribution_type = 'Gaussian'
# check that current node is continuous
if v.discrete:
raise 'Node must be continuous'
self.discrete_parents = [parent for parent in self.parents \
if parent.discrete]
self.continuous_parents = [parent for parent in self.parents \
if not parent.discrete]
self.discrete_parents_shape = [dp.nvalues for dp \
in self.discrete_parents]
self.parents_shape = [p.nvalues for p in self.parents]
if not self.parents_shape:
self.parents_shape = [0]
# set defaults
# set all mu to zeros
self.mean = na.zeros(shape=([self.nvalues] + \
self.discrete_parents_shape), type='Float32')
# set sigma to ones along the diagonal
eye = na.identity(self.nvalues, type = 'Float32')[..., na.NewAxis]
if len(self.discrete_parents) > 0:
q = reduce(lambda a, b:a * b, self.discrete_parents_shape) # number of different configurations for the parents
sigma = na.concatenate([eye] * q, axis=2)
self.sigma = na.array(sigma, shape=[self.nvalues, self.nvalues] + \
self.discrete_parents_shape)
# set weights to
self.weights = na.ones(shape=[self.nvalues] + self.parents_shape, type='Float32')
# set the parameters : mean, sigma, weights
self.setParameters(mu=mu, sigma=sigma, wi=wi, sigma_type=sigma_type, \
tied_sigma=tied_sigma, isAdjustable=isAdjustable)
def inversePower(a,s,tol=1.0e-6):
n = len(a)
aStar = a - identity(n)*s # Form [a*] = [a] - s[I]
aStar = LUdecomp(aStar) # Decompose [a*]
x = zeros((n),type=Float64)
for i in range(n): # Seed [x] with random numbers
x[i] = random()
xMag = sqrt(dot(x,x)) # Normalize [x]
x =x/xMag
for i in range(50): # Begin iterations
xOld = x.copy() # Save current [x]
x = LUsolve(aStar,x) # Solve [a*][x] = [xOld]
xMag = sqrt(dot(x,x)) # Normalize [x]
x = x/xMag
if dot(xOld,x) < 0.0: # Detect change in sign of [x]
sign = -1.0
x = -x
else: sign = 1.0
if sqrt(dot(xOld - x,xOld - x)) < tol:
return s + sign/xMag,x
print 'Inverse power method did not converge'
def inversePower(a, s, tol=1.0e-6):
n = len(a)
aStar = a - identity(n) * s # Form [a*] = [a] - s[I]
aStar = LUdecomp(aStar) # Decompose [a*]
x = zeros((n), type=Float64)
for i in range(n): # Seed [x] with random numbers
x[i] = random()
xMag = sqrt(dot(x, x)) # Normalize [x]
x = x / xMag
for i in range(50): # Begin iterations
xOld = x.copy() # Save current [x]
x = LUsolve(aStar, x) # Solve [a*][x] = [xOld]
xMag = sqrt(dot(x, x)) # Normalize [x]
x = x / xMag
if dot(xOld, x) < 0.0: # Detect change in sign of [x]
sign = -1.0
x = -x
else:
sign = 1.0
if sqrt(dot(xOld - x, xOld - x)) < tol:
return s + sign / xMag, x
print 'Inverse power method did not converge'
def run(self):
# material parameter
lam = self.prms.lameLambda
mu = self.prms.lameMu
rho = self.prms.density
nu = self.prms.viscosity
# set up boundary conditions
pres = self.prms.pressure
shear = self.prms.shearForce
getLogger().info("Reading mesh from " + self.getMeshFileName())
domain = ReadMesh(self.getMeshFileName())
impact_forces = escript.Vector(0, FunctionOnBoundary(domain))
impact_forces.expand()
getLogger().info("Shape = " + str(impact_forces.getShape()))
x = FunctionOnBoundary(domain).getX()
getLogger().info(
"Initialising pressure and shearing boundary conditions...")
snapDist = 0.001
external_forces = \
(abs(x[1]-sup(x[1]))-snapDist).whereNegative()*[-shear,-pres] \
+ \
(abs(x[1]-inf(x[1]))-snapDist).whereNegative()*[shear,pres]
getLogger().info("Setting up PDE...")
mypde = LinearPDE(domain)
mypde.setLumpingOn()
mypde.setValue(D=rho * identity(mypde.getDim()))
getLogger().info("Initialising solution at t=0...")
u = Vector(0, ContinuousFunction(domain))
u_last = Vector(0, ContinuousFunction(domain))
v = Vector(0, ContinuousFunction(domain))
v_last = Vector(0, ContinuousFunction(domain))
a = Vector(0, ContinuousFunction(domain))
a_last = Vector(0, ContinuousFunction(domain))
# initialise iteration prms
tend = self.prms.maxTime
dt = self.prms.timeStepSize
# dt=1./5.*sqrt(rho/(lam+2*mu))*Lsup(domain.getSize())
getLogger().info("time step size = " + str(dt))
n = 0
t = 0
getLogger().info("Beginning iteration...")
while (t < tend):
getLogger().info("Running LSM time step...")
self.lsm.runTimeStep()
getLogger().info("Updating impact forces from LSM...")
self.updateImpactStresses(impact_forces)
getLogger().info(
"(inf(impactForces), sup(impact_forces)) = (" + \
str(inf(impact_forces)) + ", " + str(sup(impact_forces)) + ")"
)
# ... update FEM ...
getLogger().info("Initialising PDE coefficients...")
g = grad(u)
stress = (lam * trace(g)) * identity(
mypde.getDim()) + mu * (g + transpose(g))
mypde.setValue(X=-(1.0 / (1.0 + (nu * dt / 2.0))) * stress,
Y=-nu * v_last - (nu * dt / 2.0) * a_last,
y=external_forces + impact_forces)
getLogger().info("Solving PDE...")
a = mypde.getSolution()
getLogger().info("Updating displacements...")
# u_new=2*u-u_last+dt**2*a
# u_last=u
# u=u_new
v_new = v_last + (dt / 2.0) * (a + a_last)
v_last = v
v = v_new
u_new = u_last + dt * v + ((dt**2) / 2.0) * a
u_last = u
u = u_new
a_last = a
getLogger().info("Updating LSM mesh node positions...")
displacement = u - u_last
self.updateLsmMeshPosn(displacement)
t += dt
n += 1
getLogger().info(str(n) + "-th time step, t=" + str(t))
getLogger().info("a=" + str(inf(a)) + ", " + str(sup(a)))
getLogger().info("u=" + str(inf(u)) + ", " + str(sup(u)))
getLogger().info("inf(u-u_last) = " + str(inf(displacement)))
getLogger().info("sup(u-u_last) = " + str(sup(displacement)))
# ... save current acceleration and displacement
if ((self.prms.saveDxIncr > 0)
and ((n % self.prms.saveDxIncr) == 0)):
u.saveDX(
os.path.join(
self.prms.outputDir,
"displ.{0:d}.dx".format(n // self.prms.saveDxIncr)))
def getMaps(pt, pttype, output=False, screen=False, cyclic=True):
jac0 = pt.labels['LC']['data'].jac0
jac1 = pt.labels['LC']['data'].jac1
flow = pt.labels[pttype]['flow']
n = jac0.shape[0]
# Compute jacobian times vec
J = linalg.solve(jac1, jac0)
if output:
print "Jacobian J*x"
print "------------\n"
print J
print "\n"
print "Check Jacobian"
print "--------------\n"
print " eigs = ", linalg.eig(J)[0]
print " eigs = ", pt.labels['LC']['data'].evals
# Compute composition of flow maps
print "Flow maps"
print "---------\n"
ntst = len(flow)/2
maps = []
if not cyclic:
for i in range(ntst):
I = identity(n)
for j in mod(arange(i, i + ntst), ntst):
j = int(j)
I = linalg.solve(flow[2*j+1], matrixmultiply(flow[2*j], I))
maps.append(I)
# Check eigs of maps
evals = []
levecs = []
revecs = []
for m in maps:
w, vl, vr = linalg.eig(m, left=1, right=1)
evals.append(w)
levecs.append(vl)
revecs.append(vr)
if output:
for i in range(ntst):
print evals[i]
# Get left evecs along curve associated with 1 evalue
evec1 = []
good = []
for i in range(ntst):
ind = argsort(abs(evals[i]-1.))[0]
if abs(evals[i][ind]-1) > 0.05:
print "Bad floquet multipliers!"
else:
good.append(i)
evec1.append(levecs[i][:,ind])
else:
# CYCLIC METHOD
print "Similarity method!\n"
I = identity(n)
for i in range(ntst):
I = linalg.solve(flow[2*i+1], matrixmultiply(flow[2*i], I))
evec1 = []
w, vl, vr = linalg.eig(I, left=1, right=1)
print w, vl
ind = argsort(abs(w-1.))[0]
if abs(w[ind]-1) > 0.05:
raise "Bad floquet multipliers!"
else:
v = vl[:,ind]
print v
for i in range(ntst):
v = matrixmultiply(transpose(flow[2*i]), linalg.solve(transpose(flow[2*i+1]), v))
evec1.append(v/linalg.norm(v))
# print "Cyclic method!\n"
# evals = []
# levecs = []
# revecs = []
# C0 = zeros((n*ntst, n*ntst), Float64)
# C1 = zeros((n*ntst, n*ntst), Float64)
# for i in range(ntst):
# C0[(n*i):(n*(i+1)), int(mod(n*(i+1), n*ntst)):int(mod(n*(i+2), n*ntst))] = flow[2*i]
# C1[(n*i):(n*(i+1)), (n*i):(n*(i+1))] = flow[2*i+1]
#
# w, vl, vr = linalg.eig(C0, C1, left=1, right=1)
# print w
# Same direction - if right eigenvector
cycle = pt.labels[pttype]['cycle']
coords = cycle.coordnames
#for i in range(ntst):
# if matrixmultiply(evec1[i], (cycle[4*i+1]-cycle[4*i]).toarray()) < 0:
# evec1[i] = -1*evec1[i]
if screen:
x = cycle[coords[0]]
y = cycle[coords[1]]
pylab.plot(x,y)
for i in good:
a = [x[4*i], x[4*i] + 10*evec1[i][0]]
b = [y[4*i], y[4*i] + 10*evec1[i][1]]
pylab.plot(a, b, 'r', linewidth=1)
pylab.plot([x[4*i]], [y[4*i]], 'gs')
#pylab.plot([x[4*i], x[4*i] + 0.5*vhd[0]], [y[4*i], y[4*i] + 0.5*vhd[1]], 'b')
#print evec1[i], vhd, "\n"
#print matrixmultiply(evec1[i], vhd)
if screen:
Je, JE = linalg.eig(jac0, jac1)
ind = argsort(abs(Je-1.))[0]
#if abs(Je[ind]-1) > 0.05:
if 0:
print "Jacobian: Bad floquet multipliers!"
else:
u, s, vh = linalg.svd(jac0-jac1, compute_uv=1)
evec2 = [transpose(vh)[:,-1]]
#evec2 = [JE[:,ind]]
for i in range(ntst):
evec2.append(linalg.solve(flow[2*i+1], matrixmultiply(flow[2*i], evec2[i])))
# Same direction
for i in range(ntst):
if matrixmultiply(evec2[i], (cycle[4*i+1]-cycle[4*i]).toarray()) < 0:
evec2[i] = -1*evec2[i]
for i in range(ntst):
a = [x[4*i], x[4*i] + 0.5*evec2[i][0]]
b = [y[4*i], y[4*i] + 0.5*evec2[i][1]]
#pylab.plot(a, b, 'r', linewidth=1)
#pylab.plot([x[4*i]], [y[4*i]], 'gs')
return J, maps, evec1
#!/usr/bin/env python
# Generate the NetCDF Test dataset
from Scientific.IO import NetCDF
import numarray
nc = NetCDF.NetCDFFile('testdata.nc', 'w')
nc.createDimension('x', 10)
nc.createDimension('y', 10)
def funcform(x,y):
return (x-5)**2 + (y-5)**2
h = nc.createVariable('h', 'd', ('x','y') )
h.assignValue( numarray.fromfunction( funcform, (10,10) ) )
u = nc.createVariable('u', 'd', ('x','y') )
u.assignValue( numarray.identity(10) * 10 )
v = nc.createVariable('v', 'd', ('x','y') )
v.assignValue( numarray.ones( (10,10) ) * 5 )
nc.close()
def fminPowell(func,
x0,
args=(),
shouldBreak=True,
xtol=1e-4,
ftol=1e-4,
xi=None,
maxiter=None,
maxfun=None,
fulloutput=0,
printmessg=1):
"""xopt,{fval,warnflag} = fmin(function, x0, args=(), xtol=1e-4, ftol=1e-4,
maxiter=200*len(x0), maxfun=200*len(x0), fulloutput=0, printmessg=0)
Uses a Powell's algorithm to find the minimum of function
of one or more variables. See Section 10.5 of Numerical Recipes in C.
"""
p = Num.asarray(x0)
assert (len(p.shape) == 1)
n = len(p)
if maxiter is None:
maxiter = 200
iter = 0
if maxfun is None:
maxfun = n * 50000
func_calls = 0
if xi is None:
xi = Num.identity(n, p.typecode())
steps = [5] * n
fret = apply(func, (p, ) + args)
func_calls += 1
pt = p.copy()
ptt = p.copy()
xit = xi[0].copy()
while True:
ibig = -1
delta = -1e20
iter += 1
if (iter > maxiter):
if printmessg:
print "Warning: Maximum number of iterations has been exceeded: ", iter
warnflag = True
break
if (func_calls > maxfun):
if printmessg:
print "Warning: Maximum number of function evaluations has been exceeded: ", func_calls
warnflag = True
break
# perform a line search in each direction
fptt = fp = fret
for i in range(0, n):
fptt = fret
alpha, fret, fc, steps[i] = line_search(func, p, xi[i], args, xtol,
ftol, 3, 0, steps[i])
print "alpa:", alpha, "a-b/2", steps[i]
#print __name__, "xi[i]", xi[i], "alpha: ", alpha
print __name__, "p:", p
p += alpha * xi[i]
print __name__, "p:", p
func_calls += fc
if ((fptt - fret) > delta):
delta = fptt - fret
ibig = i
if (delta < 0):
if printmessg:
print "Warning: All directions head up hill"
warnflag = True
break
# find the total movement - and extrapolate in that direction
#ptt = 2*p-pt
xit = p - pt
pt = p.copy()
steps[ibig] = steps[0]
xi[ibig] = xi[0]
steps[0] = 1
xi[0] = xit
if not shouldBreak:
print __name__, "Big Old Direction Vector:", xi[ibig]
print __name__, "New Direction Vector:", xit
storeDirectionVectors(xi)
if (shouldBreak
and 2.0 * abs(fp - fret) <= ftol * (abs(fp) + abs(fret))):
if printmessg:
print "Optimisation successful"
warnflag = False
break
if printmessg:
print " Current function value: %f" % fret
print " Iterations: %d" % iter
print " Function evaluations: %d" % func_calls
try:
apply(func, (None, )) #tell the function thread to stop
except TypeError:
print "Function not support None argument"
if fulloutput:
return p, fret, func_calls, warnflag
else:
return p
def getMaps(pt, pttype, output=False, screen=False, cyclic=True):
jac0 = pt.labels['LC']['data'].jac0
jac1 = pt.labels['LC']['data'].jac1
flow = pt.labels[pttype]['flow']
n = jac0.shape[0]
# Compute jacobian times vec
J = linalg.solve(jac1, jac0)
if output:
print "Jacobian J*x"
print "------------\n"
print J
print "\n"
print "Check Jacobian"
print "--------------\n"
print " eigs = ", linalg.eig(J)[0]
print " eigs = ", pt.labels['LC']['data'].evals
# Compute composition of flow maps
print "Flow maps"
print "---------\n"
ntst = len(flow) / 2
maps = []
if not cyclic:
for i in range(ntst):
I = identity(n)
for j in mod(arange(i, i + ntst), ntst):
j = int(j)
I = linalg.solve(flow[2 * j + 1],
matrixmultiply(flow[2 * j], I))
maps.append(I)
# Check eigs of maps
evals = []
levecs = []
revecs = []
for m in maps:
w, vl, vr = linalg.eig(m, left=1, right=1)
evals.append(w)
levecs.append(vl)
revecs.append(vr)
if output:
for i in range(ntst):
print evals[i]
# Get left evecs along curve associated with 1 evalue
evec1 = []
good = []
for i in range(ntst):
ind = argsort(abs(evals[i] - 1.))[0]
if abs(evals[i][ind] - 1) > 0.05:
print "Bad floquet multipliers!"
else:
good.append(i)
evec1.append(levecs[i][:, ind])
else:
# CYCLIC METHOD
print "Similarity method!\n"
I = identity(n)
for i in range(ntst):
I = linalg.solve(flow[2 * i + 1], matrixmultiply(flow[2 * i], I))
evec1 = []
w, vl, vr = linalg.eig(I, left=1, right=1)
print w, vl
ind = argsort(abs(w - 1.))[0]
if abs(w[ind] - 1) > 0.05:
raise "Bad floquet multipliers!"
else:
v = vl[:, ind]
print v
for i in range(ntst):
v = matrixmultiply(transpose(flow[2 * i]),
linalg.solve(transpose(flow[2 * i + 1]), v))
evec1.append(v / linalg.norm(v))
# print "Cyclic method!\n"
# evals = []
# levecs = []
# revecs = []
# C0 = zeros((n*ntst, n*ntst), Float64)
# C1 = zeros((n*ntst, n*ntst), Float64)
# for i in range(ntst):
# C0[(n*i):(n*(i+1)), int(mod(n*(i+1), n*ntst)):int(mod(n*(i+2), n*ntst))] = flow[2*i]
# C1[(n*i):(n*(i+1)), (n*i):(n*(i+1))] = flow[2*i+1]
#
# w, vl, vr = linalg.eig(C0, C1, left=1, right=1)
# print w
# Same direction - if right eigenvector
cycle = pt.labels[pttype]['cycle']
coords = cycle.coordnames
#for i in range(ntst):
# if matrixmultiply(evec1[i], (cycle[4*i+1]-cycle[4*i]).toarray()) < 0:
# evec1[i] = -1*evec1[i]
if screen:
x = cycle[coords[0]]
y = cycle[coords[1]]
pylab.plot(x, y)
for i in good:
a = [x[4 * i], x[4 * i] + 10 * evec1[i][0]]
b = [y[4 * i], y[4 * i] + 10 * evec1[i][1]]
pylab.plot(a, b, 'r', linewidth=1)
pylab.plot([x[4 * i]], [y[4 * i]], 'gs')
#pylab.plot([x[4*i], x[4*i] + 0.5*vhd[0]], [y[4*i], y[4*i] + 0.5*vhd[1]], 'b')
#print evec1[i], vhd, "\n"
#print matrixmultiply(evec1[i], vhd)
if screen:
Je, JE = linalg.eig(jac0, jac1)
ind = argsort(abs(Je - 1.))[0]
#if abs(Je[ind]-1) > 0.05:
if 0:
print "Jacobian: Bad floquet multipliers!"
else:
u, s, vh = linalg.svd(jac0 - jac1, compute_uv=1)
evec2 = [transpose(vh)[:, -1]]
#evec2 = [JE[:,ind]]
for i in range(ntst):
evec2.append(
linalg.solve(flow[2 * i + 1],
matrixmultiply(flow[2 * i], evec2[i])))
# Same direction
for i in range(ntst):
if matrixmultiply(
evec2[i],
(cycle[4 * i + 1] - cycle[4 * i]).toarray()) < 0:
evec2[i] = -1 * evec2[i]
for i in range(ntst):
a = [x[4 * i], x[4 * i] + 0.5 * evec2[i][0]]
b = [y[4 * i], y[4 * i] + 0.5 * evec2[i][1]]
#pylab.plot(a, b, 'r', linewidth=1)
#pylab.plot([x[4*i]], [y[4*i]], 'gs')
return J, maps, evec1
def fminBFGS(f,
x0,
fprime=None,
args=(),
avegtol=1e-5,
maxiter=None,
fulloutput=0,
printmessg=1):
"""xopt = fminBFGS(f, x0, fprime=None, args=(), avegtol=1e-5,
maxiter=None, fulloutput=0, printmessg=1)
Optimize the function, f, whose gradient is given by fprime using the
quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)
See Wright, and Nocedal 'Numerical Optimization', 1999, pg. 198.
"""
app_fprime = 0
if fprime is None:
app_fprime = 1
x0 = Num.asarray(x0)
if maxiter is None:
maxiter = len(x0) * 200
func_calls = 0
grad_calls = 0
k = 0
N = len(x0)
gtol = N * avegtol
I = Num.identity(N)
Hk = I
if app_fprime:
gfk = apply(approx_fprime, (x0, f) + args)
func_calls = func_calls + len(x0) + 1
else:
gfk = apply(fprime, (x0, ) + args)
grad_calls = grad_calls + 1
xk = x0
sk = [2 * gtol]
while (Num.add.reduce(abs(gfk)) > gtol) and (k < maxiter):
pk = -Num.dot(Hk, gfk)
alpha_k, fc, gc = line_search_BFGS(f, xk, pk, gfk, args)
func_calls = func_calls + fc
xkp1 = xk + alpha_k * pk
sk = xkp1 - xk
xk = xkp1
if app_fprime:
gfkp1 = apply(approx_fprime, (xkp1, f) + args)
func_calls = func_calls + gc + len(x0) + 1
else:
gfkp1 = apply(fprime, (xkp1, ) + args)
grad_calls = grad_calls + gc + 1
yk = gfkp1 - gfk
k = k + 1
rhok = 1 / Num.dot(yk, sk)
A1 = I - sk[:, Num.NewAxis] * yk[Num.NewAxis, :] * rhok
A2 = I - yk[:, Num.NewAxis] * sk[Num.NewAxis, :] * rhok
Hk = Num.dot(A1, Num.dot(
Hk, A2)) + rhok * sk[:, Num.NewAxis] * sk[Num.NewAxis, :]
gfk = gfkp1
if printmessg or fulloutput:
fval = apply(f, (xk, ) + args)
if k >= maxiter:
warnflag = 1
if printmessg:
print "Warning: Maximum number of iterations has been exceeded"
print " Current function value: %f" % fval
print " Iterations: %d" % k
print " Function evaluations: %d" % func_calls
print " Gradient evaluations: %d" % grad_calls
else:
warnflag = 0
if printmessg:
print "Optimization terminated successfully."
print " Current function value: %f" % fval
print " Iterations: %d" % k
print " Function evaluations: %d" % func_calls
print " Gradient evaluations: %d" % grad_calls
if fulloutput:
return xk, fval, func_calls, grad_calls, warnflag
else:
return xk
#!/usr/bin/python
## example2_14
from numarray import array,ones,identity,Float64
from LUdecomp3 import *
n = 6
d = ones((n))*2.0
e = ones((n-1))*(-1.0)
c = e.copy()
d[n-1] = 5.0
aInv = identity(n)*1.0
c,d,e = LUdecomp3(c,d,e)
for i in range(n):
aInv[:,i] = LUsolve3(c,d,e,aInv[:,i])
print "\nThe inverse matrix is:\n",aInv
raw_input("\nPress return to exit")
#!/usr/bin/python
## example2_14
from numarray import array, ones, identity, Float64
from LUdecomp3 import *
n = 6
d = ones((n)) * 2.0
e = ones((n - 1)) * (-1.0)
c = e.copy()
d[n - 1] = 5.0
aInv = identity(n) * 1.0
c, d, e = LUdecomp3(c, d, e)
for i in range(n):
aInv[:, i] = LUsolve3(c, d, e, aInv[:, i])
print "\nThe inverse matrix is:\n", aInv
raw_input("\nPress return to exit")