def solve(self, A, b, x0,
max_iter=None,
tol_rel=0.0001,
tol_abs=10*np.finfo(np.float64).eps,
subsolver=None,
warmstart=None,
gradient=None,
searchmethod=None,
profile=False):
# Human readable dictionary of exit messages
msg = {1 : 'preprocessing', # flag = 1
2 : 'iterating', # flag = 2
3 : 'relative', # flag = 3
4 : 'absolute', # flag = 4
5 : 'stagnation', # flag = 5
6 : 'local minima', # flag = 6
7 : 'nondescent', # flag = 7
8 : 'maxlimit', # flag = 8
}
if not A.shape[0] == A.shape[1]:
raise NonSquareException("A is not a square matrix it has shape: {}".format(repr(A.shape)))
if not A.shape[0] == x0.shape[0]:
raise InvalidShapeException("A does not have the same dimensions as x0. A.shape: {}, while x0.shape: {}".format(repr(A.shape), repr(x0.shape)))
if not b.shape[0] == x0.shape[0]:
raise InvalidShapeException("x0 does not have the same dimensions as b. x0.shape: {}, while b.shape: {}".format(repr(x0.shape), repr(b.shape)))
N = x0.shape[0]
if max_iter:
self.max_iter = max_iter
else:
self.max_iter = np.floor(N/2.0)
if subsolver:
self.subsolver = subsolver
else:
self.subsolver = SubsolverPerturbation()
if gradient:
self.use_gradient_steps = gradient
self.take_gradient_step = False
grad_alpha=0.9
else:
self.use_gradient_steps = gradient
if searchmethod:
self.searchmethod = searchmethod
else:
self.searchmethod = ArmijoLineSearch(merit_func=self.fischer)
if warmstart:
self.warm_start = True
x0 = warmstart(A,b,x0, max_warm_iter)
else:
self.warm_start = False
if profile:
self.stat = StatStructure(self.max_iter+1, self.use_gradient_steps, self.warm_start)
x = x0
err = 1e20
self.iterate = 1
while self.iterate < self.max_iter:
y = np.dot(A,x) + b
phi = self.fischer(x,y)
old_err = err
err = 0.5*np.dot(phi.T,phi)
if profile:
self.stat.convergence[self.iterate] = err
if self.use_gradient_steps:
self.take_gradient_step = False
##### Test the stopping criterias used #####
rel_err = np.abs(err-old_err)/np.abs(old_err)
if rel_err < tol_rel:
if self.use_gradient_steps:
self.take_gradient_step = True
else:
if profile:
self.stat.flag = 3
break
if err < tol_abs:
if profile:
self.stat.flag = 4
break
# Call the chosen subsolver
(J, dx) = subsolver.solve(A, x, y, phi)
nabla_phi = np.dot(phi.T, J)
# Make it a column vector
nabla_phi = nabla_phi.T
# Perform gradient descent step if take_grad_step is defined
if self.use_gradient_steps and self.take_gradient_step:
# self.searchmethod.alpha = gradient.grad_alpha
self.searchmethod.alpha = grad_alpha
dx = -nabla_phi
if profile:
self.stat.gradient_steps['gradient descent'] +=1
# Tests whether the search direction is below machine
# precision.
if np.max(np.abs(dx)) < self.eps:
# If enabled and the relative change in error is low, use
# a gradient descent step
if take_gradient_step:
dx = -nabla_phi
if profile:
self.stat.gradient_steps['search direction'] += 1
else:
if profile:
self.stat.flag = 5
break
# Test whether we are stuck in a local minima
if np.linalg.norm(nabla_phi) < tol_abs:
if profile:
self.stat.flag = 6
break
# Test whether our direction is a sufficient descent direction
if np.dot(nabla_phi.T,dx) > -self.rho*(np.dot(dx.T, dx)):
# Otherwise we should try gradient direction instead.
if self.use_grad_steps:
dx = nabla_phi
if profile:
self.stat.gradient_steps['non descent'] += 1
else:
if profile:
self.stat.flag = 7
break
# For testing sufficient decrease in function
f_0 = err
# Gradient of f
grad_f = self.beta*np.dot(nabla_phi.T,dx)
# Search method returns the new x^{(k+1)}, instead of
# returning tau^{(k)}, which is more flexible?
x[:] = self.searchmethod.findNextx(A, x, b, dx, f_0, grad_f)
# Reset search alpha to original
self.searchmethod.alpha = self.alpha
self.iterate += 1
if self.iterate >= self.max_iter:
self.iterate -= 1
if profile:
self.stat.flag = 8
if profile:
self.stat.msg = msg[self.stat.flag]
self.stat.iterate = self.iterate
# Might want to change this to returning a
# FischerNewtonSolution containing all the relevant stuff. Or
# might this be too much of an overhead.
if profile:
return (x, err, self.stat)
else:
return (x, err)
def solve(self,
A,
b,
x0,
max_iter=None,
tol_rel=0.0001,
tol_abs=10 * np.finfo(np.float64).eps,
subsolver=None,
warmstart=None,
gradient=None,
searchmethod=None,
profile=False):
# Human readable dictionary of exit messages
msg = {
1: 'preprocessing', # flag = 1
2: 'iterating', # flag = 2
3: 'relative', # flag = 3
4: 'absolute', # flag = 4
5: 'stagnation', # flag = 5
6: 'local minima', # flag = 6
7: 'nondescent', # flag = 7
8: 'maxlimit', # flag = 8
}
if not A.shape[0] == A.shape[1]:
raise NonSquareException(
"A is not a square matrix it has shape: {}".format(
repr(A.shape)))
if not A.shape[0] == x0.shape[0]:
raise InvalidShapeException(
"A does not have the same dimensions as x0. A.shape: {}, while x0.shape: {}"
.format(repr(A.shape), repr(x0.shape)))
if not b.shape[0] == x0.shape[0]:
raise InvalidShapeException(
"x0 does not have the same dimensions as b. x0.shape: {}, while b.shape: {}"
.format(repr(x0.shape), repr(b.shape)))
N = x0.shape[0]
if max_iter:
self.max_iter = max_iter
else:
self.max_iter = np.floor(N / 2.0)
if subsolver:
self.subsolver = subsolver
else:
self.subsolver = SubsolverPerturbation()
if gradient:
self.use_gradient_steps = gradient
self.take_gradient_step = False
grad_alpha = 0.9
else:
self.use_gradient_steps = gradient
if searchmethod:
self.searchmethod = searchmethod
else:
self.searchmethod = ArmijoLineSearch(merit_func=self.fischer)
if warmstart:
self.warm_start = True
x0 = warmstart(A, b, x0, max_warm_iter)
else:
self.warm_start = False
if profile:
self.stat = StatStructure(self.max_iter + 1,
self.use_gradient_steps, self.warm_start)
x = x0
err = 1e20
self.iterate = 1
while self.iterate < self.max_iter:
y = np.dot(A, x) + b
phi = self.fischer(x, y)
old_err = err
err = 0.5 * np.dot(phi.T, phi)
if profile:
self.stat.convergence[self.iterate] = err
if self.use_gradient_steps:
self.take_gradient_step = False
##### Test the stopping criterias used #####
rel_err = np.abs(err - old_err) / np.abs(old_err)
if rel_err < tol_rel:
if self.use_gradient_steps:
self.take_gradient_step = True
else:
if profile:
self.stat.flag = 3
break
if err < tol_abs:
if profile:
self.stat.flag = 4
break
# Call the chosen subsolver
(J, dx) = subsolver.solve(A, x, y, phi)
nabla_phi = np.dot(phi.T, J)
# Make it a column vector
nabla_phi = nabla_phi.T
# Perform gradient descent step if take_grad_step is defined
if self.use_gradient_steps and self.take_gradient_step:
# self.searchmethod.alpha = gradient.grad_alpha
self.searchmethod.alpha = grad_alpha
dx = -nabla_phi
if profile:
self.stat.gradient_steps['gradient descent'] += 1
# Tests whether the search direction is below machine
# precision.
if np.max(np.abs(dx)) < self.eps:
# If enabled and the relative change in error is low, use
# a gradient descent step
if take_gradient_step:
dx = -nabla_phi
if profile:
self.stat.gradient_steps['search direction'] += 1
else:
if profile:
self.stat.flag = 5
break
# Test whether we are stuck in a local minima
if np.linalg.norm(nabla_phi) < tol_abs:
if profile:
self.stat.flag = 6
break
# Test whether our direction is a sufficient descent direction
if np.dot(nabla_phi.T, dx) > -self.rho * (np.dot(dx.T, dx)):
# Otherwise we should try gradient direction instead.
if self.use_grad_steps:
dx = nabla_phi
if profile:
self.stat.gradient_steps['non descent'] += 1
else:
if profile:
self.stat.flag = 7
break
# For testing sufficient decrease in function
f_0 = err
# Gradient of f
grad_f = self.beta * np.dot(nabla_phi.T, dx)
# Search method returns the new x^{(k+1)}, instead of
# returning tau^{(k)}, which is more flexible?
x[:] = self.searchmethod.findNextx(A, x, b, dx, f_0, grad_f)
# Reset search alpha to original
self.searchmethod.alpha = self.alpha
self.iterate += 1
if self.iterate >= self.max_iter:
self.iterate -= 1
if profile:
self.stat.flag = 8
if profile:
self.stat.msg = msg[self.stat.flag]
self.stat.iterate = self.iterate
# Might want to change this to returning a
# FischerNewtonSolution containing all the relevant stuff. Or
# might this be too much of an overhead.
if profile:
return (x, err, self.stat)
else:
return (x, err)
class FischerNewtonSolver:
"""
Fischer Newton's method for solving Linear Complementarity problems
Usage:
fns = FischerNewtonSolver()
(x, err, iterate, flag, convergence, msg) = fns.solve(A, b, x0, max_iter=None, tol_rel=0.0001, tol_abs=10*np.finfo(np.float64).eps, subsolver=None, profile=True, warmstart=None, gradient=None, searchmethod=None)
@params
A - The LCP matrix
b - The LCP vector
x0 - Initial guess
max_iter - Maximum number of iterations used to solve the LCP,
default to floor(problem_size / 2.0).
tol_rel - Relative tolerence, break if the error changes less
than this between two iterations, otherwise it can
be configured to using gradient steps instead.
tol_abs - Absolute tolerence, break if the error drops below
this.
subsolver - Chooses which subsolver for solving the Newton
subsystem, these can be perturbation, zero, random
or approximation (not implemented).
profile - whether or not to profile while running, log
convergence and perhaps other properties, number
of gradient steps, information about warm start etc.
warmstart - Class defining how to warm start the Fischer Newton
method. Should implement the entire warm start
procedure returning a new (and better) x0.
gradient -
searchmethod - Defines the searchmethod used, the default Armijo
backtracking line search
"""
def __init__(self,
eps=np.finfo(np.float64).eps,
rho=np.finfo(np.float64).eps,
beta=0.001,
alpha=0.5):
self.eps = eps
self.rho = rho
self.beta = beta
self.alpha = alpha
def solve(self, A, b, x0,
max_iter=None,
tol_rel=0.0001,
tol_abs=10*np.finfo(np.float64).eps,
subsolver=None,
warmstart=None,
gradient=None,
searchmethod=None,
profile=False):
# Human readable dictionary of exit messages
msg = {1 : 'preprocessing', # flag = 1
2 : 'iterating', # flag = 2
3 : 'relative', # flag = 3
4 : 'absolute', # flag = 4
5 : 'stagnation', # flag = 5
6 : 'local minima', # flag = 6
7 : 'nondescent', # flag = 7
8 : 'maxlimit', # flag = 8
}
if not A.shape[0] == A.shape[1]:
raise NonSquareException("A is not a square matrix it has shape: {}".format(repr(A.shape)))
if not A.shape[0] == x0.shape[0]:
raise InvalidShapeException("A does not have the same dimensions as x0. A.shape: {}, while x0.shape: {}".format(repr(A.shape), repr(x0.shape)))
if not b.shape[0] == x0.shape[0]:
raise InvalidShapeException("x0 does not have the same dimensions as b. x0.shape: {}, while b.shape: {}".format(repr(x0.shape), repr(b.shape)))
N = x0.shape[0]
if max_iter:
self.max_iter = max_iter
else:
self.max_iter = np.floor(N/2.0)
if subsolver:
self.subsolver = subsolver
else:
self.subsolver = SubsolverPerturbation()
if gradient:
self.use_gradient_steps = gradient
self.take_gradient_step = False
grad_alpha=0.9
else:
self.use_gradient_steps = gradient
if searchmethod:
self.searchmethod = searchmethod
else:
self.searchmethod = ArmijoLineSearch(merit_func=self.fischer)
if warmstart:
self.warm_start = True
x0 = warmstart(A,b,x0, max_warm_iter)
else:
self.warm_start = False
if profile:
self.stat = StatStructure(self.max_iter+1, self.use_gradient_steps, self.warm_start)
x = x0
err = 1e20
self.iterate = 1
while self.iterate < self.max_iter:
y = np.dot(A,x) + b
phi = self.fischer(x,y)
old_err = err
err = 0.5*np.dot(phi.T,phi)
if profile:
self.stat.convergence[self.iterate] = err
if self.use_gradient_steps:
self.take_gradient_step = False
##### Test the stopping criterias used #####
rel_err = np.abs(err-old_err)/np.abs(old_err)
if rel_err < tol_rel:
if self.use_gradient_steps:
self.take_gradient_step = True
else:
if profile:
self.stat.flag = 3
break
if err < tol_abs:
if profile:
self.stat.flag = 4
break
# Call the chosen subsolver
(J, dx) = subsolver.solve(A, x, y, phi)
nabla_phi = np.dot(phi.T, J)
# Make it a column vector
nabla_phi = nabla_phi.T
# Perform gradient descent step if take_grad_step is defined
if self.use_gradient_steps and self.take_gradient_step:
# self.searchmethod.alpha = gradient.grad_alpha
self.searchmethod.alpha = grad_alpha
dx = -nabla_phi
if profile:
self.stat.gradient_steps['gradient descent'] +=1
# Tests whether the search direction is below machine
# precision.
if np.max(np.abs(dx)) < self.eps:
# If enabled and the relative change in error is low, use
# a gradient descent step
if take_gradient_step:
dx = -nabla_phi
if profile:
self.stat.gradient_steps['search direction'] += 1
else:
if profile:
self.stat.flag = 5
break
# Test whether we are stuck in a local minima
if np.linalg.norm(nabla_phi) < tol_abs:
if profile:
self.stat.flag = 6
break
# Test whether our direction is a sufficient descent direction
if np.dot(nabla_phi.T,dx) > -self.rho*(np.dot(dx.T, dx)):
# Otherwise we should try gradient direction instead.
if self.use_grad_steps:
dx = nabla_phi
if profile:
self.stat.gradient_steps['non descent'] += 1
else:
if profile:
self.stat.flag = 7
break
# For testing sufficient decrease in function
f_0 = err
# Gradient of f
grad_f = self.beta*np.dot(nabla_phi.T,dx)
# Search method returns the new x^{(k+1)}, instead of
# returning tau^{(k)}, which is more flexible?
x[:] = self.searchmethod.findNextx(A, x, b, dx, f_0, grad_f)
# Reset search alpha to original
self.searchmethod.alpha = self.alpha
self.iterate += 1
if self.iterate >= self.max_iter:
self.iterate -= 1
if profile:
self.stat.flag = 8
if profile:
self.stat.msg = msg[self.stat.flag]
self.stat.iterate = self.iterate
# Might want to change this to returning a
# FischerNewtonSolution containing all the relevant stuff. Or
# might this be too much of an overhead.
if profile:
return (x, err, self.stat)
else:
return (x, err)
def fischer(self,a,b):
# Fischer function
# Cite reference to it.
return np.sqrt(np.power(a,2) + np.power(b,2)) - (a + b)
# return np.sqrt(y**2+x**2) - y - x
# def fischerLinearOperator(A,x,dx,phi,h,idx):
# fischer(np.dot(A[idx[0],:][:,idx[0]],(x[idx]+dx[idx]*h)),
# x[idx]+dx*h)-phi[idx]
class SubsolverPerturbation:
def __init__(self, gamma = 1e-28, eps=np.finfo(np.float64).eps):
self.gamma = gamma
self.eps = eps
def solve(self, A, x, y, phi):
"""Solves the sub Newton system of the Fischer Newton method
in order to obtain the Newton direction dx."""
# Bitmask of True/False values where the conditions are
# satisfied, that is where phi and x are near
# singular/singular.
S = np.logical_and(np.abs(phi) < self.gamma,
np.abs(x) < self.gamma)
N = np.size(x)
px = np.zeros(x.shape)
px[:] = x # Copy x
assert px.shape == (N,1), 'px is not a column vector, it has shape: ' + \
repr(px.shape)
assert np.all( np.isreal(px) ), 'px is not real'
direction = np.zeros(x.shape)
direction[:] = np.sign(x)
direction[ direction == 0 ] = 1
px[S] = self.gamma * direction[S]
p = np.zeros(px.shape)
p = (px / (np.sqrt(np.power(y,2) + np.power(px,2))))-1
assert p.shape == (N,1), 'p is not a column vector, it has shape: ' + \
repr(p.shape)
assert np.all(np.isreal(p)), 'p is not real'
q = np.zeros(y.shape)
q = (y / (np.sqrt(np.power(y,2) + np.power(px,2)))) - 1
assert q.shape == (N,1), 'q is not a column vector, it has shape: ' + \
repr(q.shape)
assert np.all(np.isreal(q)), 'q is not real'
J = np.dot(np.diag(p[:,0]),np.identity(N)) + np.dot(np.diag(q[:,0]),A)
assert J.shape == (N,N), 'J is not a square matrix, it has shape: ' + \
repr(J.shape)
assert np.all(np.isreal(J)), 'J is not real'
# Reciprocal conditioning number of J
rcond = 1/np.linalg.cond(J)
if np.isnan(rcond):
rcond = np.nan_to_num(rcond)
# Check conditioning of J, if bad use pinv to solve the system.
if rcond < 2*self.eps:
# dx = np.dot(np.linalg.pinv(J), (-phi))
if np.any(np.isnan(J)):
print "J contains nan"
if np.any(np.isnan(phi)):
print "phi contains nan"
try:
dx = np.dot(spl.pinv(J), (-phi))
except Exception:
print repr(dir(Exception))
dx = np.dot(spl.pinv2(J), (-phi))
else:
# if J.issparse():
J = sps.csc_matrix(J)
dx = spsl.dsolve.spsolve(J, (-phi), use_umfpack=True)
J = J.toarray()
# else:
# dx = np.linalg.solve(J,(-phi))
dx = dx.reshape(dx.size,1)
assert dx.shape == (N,1), 'dx is not a column vector, it has shape: ' + \
repr(dx.shape)
assert np.all(np.isreal(dx)), 'dx is not real'
return (J, dx)
class SubsolverRandom:
pass
class SubsolverZero:
pass
class SubsolverApproximation:
# This has not yet been implemented.
pass
class FischerNewtonSolver:
"""
Fischer Newton's method for solving Linear Complementarity problems
Usage:
fns = FischerNewtonSolver()
(x, err, iterate, flag, convergence, msg) = fns.solve(A, b, x0, max_iter=None, tol_rel=0.0001, tol_abs=10*np.finfo(np.float64).eps, subsolver=None, profile=True, warmstart=None, gradient=None, searchmethod=None)
@params
A - The LCP matrix
b - The LCP vector
x0 - Initial guess
max_iter - Maximum number of iterations used to solve the LCP,
default to floor(problem_size / 2.0).
tol_rel - Relative tolerence, break if the error changes less
than this between two iterations, otherwise it can
be configured to using gradient steps instead.
tol_abs - Absolute tolerence, break if the error drops below
this.
subsolver - Chooses which subsolver for solving the Newton
subsystem, these can be perturbation, zero, random
or approximation (not implemented).
profile - whether or not to profile while running, log
convergence and perhaps other properties, number
of gradient steps, information about warm start etc.
warmstart - Class defining how to warm start the Fischer Newton
method. Should implement the entire warm start
procedure returning a new (and better) x0.
gradient -
searchmethod - Defines the searchmethod used, the default Armijo
backtracking line search
"""
def __init__(self,
eps=np.finfo(np.float64).eps,
rho=np.finfo(np.float64).eps,
beta=0.001,
alpha=0.5):
self.eps = eps
self.rho = rho
self.beta = beta
self.alpha = alpha
def solve(self,
A,
b,
x0,
max_iter=None,
tol_rel=0.0001,
tol_abs=10 * np.finfo(np.float64).eps,
subsolver=None,
warmstart=None,
gradient=None,
searchmethod=None,
profile=False):
# Human readable dictionary of exit messages
msg = {
1: 'preprocessing', # flag = 1
2: 'iterating', # flag = 2
3: 'relative', # flag = 3
4: 'absolute', # flag = 4
5: 'stagnation', # flag = 5
6: 'local minima', # flag = 6
7: 'nondescent', # flag = 7
8: 'maxlimit', # flag = 8
}
if not A.shape[0] == A.shape[1]:
raise NonSquareException(
"A is not a square matrix it has shape: {}".format(
repr(A.shape)))
if not A.shape[0] == x0.shape[0]:
raise InvalidShapeException(
"A does not have the same dimensions as x0. A.shape: {}, while x0.shape: {}"
.format(repr(A.shape), repr(x0.shape)))
if not b.shape[0] == x0.shape[0]:
raise InvalidShapeException(
"x0 does not have the same dimensions as b. x0.shape: {}, while b.shape: {}"
.format(repr(x0.shape), repr(b.shape)))
N = x0.shape[0]
if max_iter:
self.max_iter = max_iter
else:
self.max_iter = np.floor(N / 2.0)
if subsolver:
self.subsolver = subsolver
else:
self.subsolver = SubsolverPerturbation()
if gradient:
self.use_gradient_steps = gradient
self.take_gradient_step = False
grad_alpha = 0.9
else:
self.use_gradient_steps = gradient
if searchmethod:
self.searchmethod = searchmethod
else:
self.searchmethod = ArmijoLineSearch(merit_func=self.fischer)
if warmstart:
self.warm_start = True
x0 = warmstart(A, b, x0, max_warm_iter)
else:
self.warm_start = False
if profile:
self.stat = StatStructure(self.max_iter + 1,
self.use_gradient_steps, self.warm_start)
x = x0
err = 1e20
self.iterate = 1
while self.iterate < self.max_iter:
y = np.dot(A, x) + b
phi = self.fischer(x, y)
old_err = err
err = 0.5 * np.dot(phi.T, phi)
if profile:
self.stat.convergence[self.iterate] = err
if self.use_gradient_steps:
self.take_gradient_step = False
##### Test the stopping criterias used #####
rel_err = np.abs(err - old_err) / np.abs(old_err)
if rel_err < tol_rel:
if self.use_gradient_steps:
self.take_gradient_step = True
else:
if profile:
self.stat.flag = 3
break
if err < tol_abs:
if profile:
self.stat.flag = 4
break
# Call the chosen subsolver
(J, dx) = subsolver.solve(A, x, y, phi)
nabla_phi = np.dot(phi.T, J)
# Make it a column vector
nabla_phi = nabla_phi.T
# Perform gradient descent step if take_grad_step is defined
if self.use_gradient_steps and self.take_gradient_step:
# self.searchmethod.alpha = gradient.grad_alpha
self.searchmethod.alpha = grad_alpha
dx = -nabla_phi
if profile:
self.stat.gradient_steps['gradient descent'] += 1
# Tests whether the search direction is below machine
# precision.
if np.max(np.abs(dx)) < self.eps:
# If enabled and the relative change in error is low, use
# a gradient descent step
if take_gradient_step:
dx = -nabla_phi
if profile:
self.stat.gradient_steps['search direction'] += 1
else:
if profile:
self.stat.flag = 5
break
# Test whether we are stuck in a local minima
if np.linalg.norm(nabla_phi) < tol_abs:
if profile:
self.stat.flag = 6
break
# Test whether our direction is a sufficient descent direction
if np.dot(nabla_phi.T, dx) > -self.rho * (np.dot(dx.T, dx)):
# Otherwise we should try gradient direction instead.
if self.use_grad_steps:
dx = nabla_phi
if profile:
self.stat.gradient_steps['non descent'] += 1
else:
if profile:
self.stat.flag = 7
break
# For testing sufficient decrease in function
f_0 = err
# Gradient of f
grad_f = self.beta * np.dot(nabla_phi.T, dx)
# Search method returns the new x^{(k+1)}, instead of
# returning tau^{(k)}, which is more flexible?
x[:] = self.searchmethod.findNextx(A, x, b, dx, f_0, grad_f)
# Reset search alpha to original
self.searchmethod.alpha = self.alpha
self.iterate += 1
if self.iterate >= self.max_iter:
self.iterate -= 1
if profile:
self.stat.flag = 8
if profile:
self.stat.msg = msg[self.stat.flag]
self.stat.iterate = self.iterate
# Might want to change this to returning a
# FischerNewtonSolution containing all the relevant stuff. Or
# might this be too much of an overhead.
if profile:
return (x, err, self.stat)
else:
return (x, err)
def fischer(self, a, b):
# Fischer function
# Cite reference to it.
return np.sqrt(np.power(a, 2) + np.power(b, 2)) - (a + b)
# return np.sqrt(y**2+x**2) - y - x
# def fischerLinearOperator(A,x,dx,phi,h,idx):
# fischer(np.dot(A[idx[0],:][:,idx[0]],(x[idx]+dx[idx]*h)),
# x[idx]+dx*h)-phi[idx]
class SubsolverPerturbation:
def __init__(self, gamma=1e-28, eps=np.finfo(np.float64).eps):
self.gamma = gamma
self.eps = eps
def solve(self, A, x, y, phi):
"""Solves the sub Newton system of the Fischer Newton method
in order to obtain the Newton direction dx."""
# Bitmask of True/False values where the conditions are
# satisfied, that is where phi and x are near
# singular/singular.
S = np.logical_and(
np.abs(phi) < self.gamma,
np.abs(x) < self.gamma)
N = np.size(x)
px = np.zeros(x.shape)
px[:] = x # Copy x
assert px.shape == (N,1), 'px is not a column vector, it has shape: ' + \
repr(px.shape)
assert np.all(np.isreal(px)), 'px is not real'
direction = np.zeros(x.shape)
direction[:] = np.sign(x)
direction[direction == 0] = 1
px[S] = self.gamma * direction[S]
p = np.zeros(px.shape)
p = (px / (np.sqrt(np.power(y, 2) + np.power(px, 2)))) - 1
assert p.shape == (N,1), 'p is not a column vector, it has shape: ' + \
repr(p.shape)
assert np.all(np.isreal(p)), 'p is not real'
q = np.zeros(y.shape)
q = (y / (np.sqrt(np.power(y, 2) + np.power(px, 2)))) - 1
assert q.shape == (N,1), 'q is not a column vector, it has shape: ' + \
repr(q.shape)
assert np.all(np.isreal(q)), 'q is not real'
J = np.dot(np.diag(p[:, 0]), np.identity(N)) + np.dot(
np.diag(q[:, 0]), A)
assert J.shape == (N,N), 'J is not a square matrix, it has shape: ' + \
repr(J.shape)
assert np.all(np.isreal(J)), 'J is not real'
# Reciprocal conditioning number of J
rcond = 1 / np.linalg.cond(J)
if np.isnan(rcond):
rcond = np.nan_to_num(rcond)
# Check conditioning of J, if bad use pinv to solve the system.
if rcond < 2 * self.eps:
# dx = np.dot(np.linalg.pinv(J), (-phi))
if np.any(np.isnan(J)):
print "J contains nan"
if np.any(np.isnan(phi)):
print "phi contains nan"
try:
dx = np.dot(spl.pinv(J), (-phi))
except Exception:
print repr(dir(Exception))
dx = np.dot(spl.pinv2(J), (-phi))
else:
# if J.issparse():
J = sps.csc_matrix(J)
dx = spsl.dsolve.spsolve(J, (-phi), use_umfpack=True)
J = J.toarray()
# else:
# dx = np.linalg.solve(J,(-phi))
dx = dx.reshape(dx.size, 1)
assert dx.shape == (N,1), 'dx is not a column vector, it has shape: ' + \
repr(dx.shape)
assert np.all(np.isreal(dx)), 'dx is not real'
return (J, dx)
class SubsolverRandom:
pass
class SubsolverZero:
pass
class SubsolverApproximation:
# This has not yet been implemented.
pass