def to_function(self):
if not self.position.is_diag:
raise ValueError("Position must be diagonal!")
def rec_a(i):
return float(1 / np.sqrt(i + 1))
def rec_b(i):
return -float(np.sqrt(i) / np.sqrt(i + 1))
dirs = self.position.dirs
hermite_dirs = []
for i, d in enumerate(dirs):
var = func.Function.xyz[d]
h0 = func.Function('1', dirs=dirs)
μ, σ = self.position.mean[i], self.position.cov[i][i]
f1 = np.sqrt(σ) * (var - μ)
h1 = func.Function(f1, dirs=dirs)
hermite_dirs.append([h0, h1])
for j in range(1, self.degree):
hermite_dirs[i].append(hermite_dirs[i][-1] *
hermite_dirs[i][1] * rec_a(j) +
hermite_dirs[i][-2] * rec_b(j))
result = func.Function('0', dirs=dirs)
for im, m in enumerate(self.multi_indices()):
result_m = func.Function(self.coeffs[im], dirs=dirs)
for i in range(0, self.position.dim):
result_m *= hermite_dirs[i][m[i]]
result += result_m
return result
def __call__(self, arg):
if not isinstance(arg, (func.Function, Operator)):
try:
arg = Operator(arg, dirs=self.dirs)
except TypeError:
arg = func.Function(arg, dirs=self.dirs)
variables = [func.Function.x_sub[d] for d in self.dirs]
sym = self.sym.subs(self.f(*variables), arg.sym).doit()
return Operator(sym, dirs=self.dirs) if isinstance(arg, Operator) \
else func.Function(sym, dirs=self.dirs)
def test_construct_from_string(self):
function_str1 = 'x*cos(y) + exp(z)**2'
function_str2 = 'x0*cos(x1) + exp(x2)**2'
function_sym1 = x * sym.cos(y) + sym.exp(z)**2
function_sym2 = v[0] * sym.cos(v[1]) + sym.exp(v[2])**2
function1 = func.Function(function_str1)
function2 = func.Function(function_str2)
function3 = func.Function(function_sym1)
function4 = func.Function(function_sym2)
self.assertTrue(function1 == function2)
self.assertTrue(function2 == function3)
self.assertTrue(function3 == function4)
def split(self):
MAX_ORDER = 5
variables = [func.Function.x_sub[d] for d in self.dirs]
unknown, dim = self.f(*variables), len(variables)
result, rem = {}, self.sym.expand()
mult = list(
m for m in itertools.product(range(MAX_ORDER + 1), repeat=dim)
if sum(m) <= MAX_ORDER)
for m in mult:
if rem == 0:
break
test, der = 1, unknown
for i, v in zip(m, variables):
test *= v**i / math.factorial(i)
der = sympy.diff(der, v, i)
remargs = rem.args if isinstance(rem, sympy.Add) else [rem]
term, rem = 0, 0
for arg in remargs: # Convoluted to avoid rounding errors
termarg = arg.subs(unknown, test).doit()
if termarg == 0:
rem += arg
else:
term += termarg
if isinstance(term, sympy.Basic):
term = sympy.simplify(term)
if term != 0:
result[tuple(m)] = func.Function(term, dirs=self.dirs)
if rem != 0:
raise ValueError("Nonzero remainder")
return result
def discretize(self, f):
if not isinstance(f, symfunc.Function):
f = symfunc.Function(f)
f = f.as_string(format='array', toC=True)
function = core.discretize(f, self.nodes, self.position.mean,
self.position.factor)
return function
def tensorize(args, sparse=None):
for a in args:
if not isinstance(a, Varf):
raise ValueError("Invalid argument(s)")
if len(args) == 1:
return args[0]
sparse = rc.settings['sparse'] if sparse is None else sparse
index_set, degree = args[0].index_set, args[0].degree
mats = {}
factor = 1
for a in args:
if not isinstance(a, Varf) or \
not a.index_set == index_set or \
not a.degree == degree:
raise ValueError("Invalid argument(s)")
key = frozenset(a.position.dirs)
mats[key] = a.matrix
factor *= a.factor.sym
matrix = core.tensorize(mats, sparse=sparse, index_set=index_set)
position = pos.Position.tensorize([a.position for a in args])
factor = func.Function(factor, dirs=position.dirs)
return Varf(matrix, position, factor=factor, index_set=index_set)
def __init__(self,
coeffs,
position,
factor=1,
index_set="triangle",
significant=0):
self.coeffs = coeffs
self.position = position
self.index_set = index_set
dim, npolys = self.position.dim, len(self.coeffs)
if position.dirs == []:
self.degree = 0
else:
self.degree = core.iterator_get_degree(dim,
npolys,
index_set=index_set)
if not len(self.multi_indices()) == len(self.coeffs):
raise ValueError("Invalid arguments: length of self.coeffs \
does not match number of multi-indices!")
if significant != 0:
for i, c in enumerate(self.coeffs):
self.coeffs[i] = round(c, significant)
self.factor = func.Function(factor, dirs=self.position.dirs)
if not self.coeffs.flags['C_CONTIGUOUS']:
self.coeffs = self.coeffs.copy(order='C')
def map(self, factor):
if not isinstance(factor, func.Function):
factor = func.Function(factor, dirs=self.dirs)
if not factor.dirs == self.dirs:
raise ValueError("Directions differ")
variables = [func.Function.x_sub[d] for d in self.dirs]
unknown = self.f(*variables)
sym = self.sym.subs(unknown, (unknown * factor).sym)
sym = (sym.doit() / factor.sym).expand()
return Operator(sym, dirs=self.dirs)
def __init__(self, matrix, position, factor=1, index_set="triangle"):
self.is_sparse = isinstance(matrix, ss.csr_matrix)
self.matrix = matrix
self.position = position
self.index_set = index_set
dim, npolys = self.position.dim, self.matrix.shape[0]
self.degree = core.iterator_get_degree(dim, npolys,
index_set=index_set)
assert len(self.multi_indices()) == self.matrix.shape[0]
self.factor = func.Function(factor, dirs=self.position.dirs)
def wrapper(*args, **kwargs):
do_tensorize = rc.settings['tensorize']
if 'tensorize' in kwargs:
do_tensorize = kwargs['tensorize']
del kwargs['tensorize']
if not do_tensorize:
return func(*args, **kwargs)
quad, function = args[0], args[arg_num]
if not quad.position.is_diag or \
isinstance(function, np.ndarray) or \
quad.position.dim is 1:
return func(*args, **kwargs)
results = []
if not isinstance(function, symfunc.Function):
function = symfunc.Function(function,
dim=quad.position.dim)
for add in function.split():
if len(add) == 2:
new_args = list(args).copy()
new_args[arg_num] = add[0]
results.append(
func(*new_args, **kwargs) * float(add[1]))
continue
func_dirs = []
for d in range(quad.position.dim):
new_args = list(args).copy()
new_args[0] = quad.project(d)
new_args[arg_num] = add[d]
func_dir = func(*new_args, **kwargs)
func_dirs.append(func_dir)
if rc.settings['debug']:
print("Tensorizing results")
kwargs_func = {'sparse': kwargs['sparse']} \
if 'sparse' in kwargs else {}
t = type(func_dirs[0])
tens_fun = t.tensorize if t is hv.Varf or t is hs.Series \
else core.tensorize
tensorized = tens_fun(func_dirs, **kwargs_func)
# pdb.set_trace()
results.append(tensorized * float(add[-1]))
return sum(results[1:], results[0])
def test_simple_split(self):
newf = func.Function
f = func.Function('x*y**2 + 5*exp(z)*y + 3*log(x*z)')
split = f.split()
self.assertTrue(len(f.dirs) == 3)
self.assertTrue(len(split) == 3)
for i in range(3):
if split[i][-1] == 1:
self.assertTrue(len(split[i]) == 4)
self.assertTrue(split[i][0] == newf('x', dirs=[0]))
self.assertTrue(split[i][1] == newf('y*y', dirs=[1]))
if split[i][-1] == 5:
self.assertTrue(len(split[i]) == 4)
self.assertTrue(split[i][0] == newf('1', dirs=[0]))
self.assertTrue(split[i][1] == newf('y', dirs=[1]))
self.assertTrue(split[i][2] == newf('exp(z)', dirs=[2]))
if split[i][-1] == 3:
self.assertTrue(len(split[i]) == 2)
self.assertTrue(split[i][0] == newf('log(x*z)', dim=3))
def _tensorize(s1, s2):
if not isinstance(s1, Series) or not isinstance(s2, Series):
raise ValueError("Arguments must be Series!")
if s1.position.dim == 0:
return Series(s1.coeffs[0] * s2.coeffs,
s2.position,
factor=s2.factor,
index_set=s2.index_set)
if s2.position.dim == 0:
return Series(s2.coeffs[0] * s1.coeffs,
s1.position,
factor=s1.factor,
index_set=s1.index_set)
# Inner product only makes sense in weighted space
# Product in appropriate weighted space
common = set(s1.position.dirs).intersection(s2.position.dirs)
f1 = s1.factor.project(list(common))
f2 = s2.factor.project(list(common))
if not f1 == f2:
raise ValueError("Factors should be match!")
if not s1.degree == s2.degree:
raise ValueError("s1 and s2 should have the same degree")
if not s1.index_set == s2.index_set:
raise ValueError("s1 and s2 should have the same index_set")
f1, f2 = s1.factor.sym / f1.sym, s2.factor.sym / f2.sym
d1, d2 = s1.position.dirs, s2.position.dirs
c1, c2 = s1.coeffs, s2.coeffs
result = core.inner(c1, c2, d1, d2, index_set=s1.index_set)
position = pos.Position.tensorize([s1.position, s2.position])
factor = func.Function(f1 * f2, dirs=position.dirs)
return Series(result,
position,
factor=factor,
index_set=s1.index_set)
def plot(self, series, factor=None, ax=None):
assert self.position.is_diag
assert self.position == series.position
if factor is None:
factor = self.position.weight()
if not isinstance(factor, np.ndarray):
# ipdb.set_trace()
factor = symfunc.Function(factor, dirs=self.position.dirs)
factor = self.discretize(factor)
n_nodes = []
r_nodes = []
for i in range(self.position.dim):
n_nodes.append(len(self.nodes[i]))
r_nodes.append(
self.project(i).discretize('x')) # Problematic line?
solution = self.eval(series) * factor
solution = solution.reshape(*n_nodes).T
if self.position.dim == 1:
return ax.plot(*r_nodes, solution)
elif self.position.dim == 2:
return ax.contourf(*r_nodes, solution, 100)
def __init__(self, expr, dirs=None):
if isinstance(expr, Operator):
self.sym = expr.sym
self.dirs = expr.dirs
return
aux = func.Function(expr)
if aux.sym == 0:
if dirs is None:
raise ValueError("Invalid argument!")
self.sym, self.dirs = aux.sym, dirs
return
functions = list(aux.sym.atoms(sympy.core.function.AppliedUndef))
if len(functions) != 1:
raise TypeError("There should be exactly 1 functional argument!")
self.sym = aux.sym.subs(functions[0].func, self.f)
variables = list(self.sym.atoms(
sympy.core.function.AppliedUndef))[0].args
self.dirs = [func.Function.x_sub.index(v) for v in variables]
assert len(self.dirs) == len(functions[0].args)
def test_as_xyz(self):
function_str1 = 'x0*cos(x1) + exp(2*x2)'
function_str2 = 'x*cos(y) + exp(2*z)'
function = func.Function(function_str1)
self.assertTrue(str(function.sym) == function_str1)
self.assertTrue(str(function.as_xyz()) == function_str2)
return symbolic
for i in range(nterms):
print("Solving for power " + str(i))
unk[i] = sym.Function('u{}'.format(i))(x)
u[i] = unk[i]
rhs = sym.Integer(0)
if i > 0:
rhs += -L1.subs(f, u[i - 1])
if i > 1:
rhs += -L2.subs(f, u[i - 2])
split = func.Function(rhs.doit().expand(), dim=2).split()
for term in split:
x_part = term[-1] * term[0].as_xyz()
y_part = term[1].as_xyz()
t_rhs = quad_num.transform(y_part, degree)
centered[i] += round(t_rhs.coeffs[0], 10) * x_part
u[i] += x_part * iL0(y_part)
u[i] = func.Function.sanitize(u[i])
centered[i] = func.Function.sanitize(centered[i])
# }}}
# Some manual calculations {{{
# Operator
fx = sym.Function('fx')(x)
LFP = ((1 / β) * sym.exp(-β * Vp) * (fx * sym.exp(β * Vp)).diff(x)).diff(x)
def __mul__(self, other):
if not isinstance(other, func.Function):
other = func.Function(other, dirs=self.dirs)
if not self.dirs == other.dirs:
raise ValueError("Invalid argument!")
return Operator(self.sym * other.sym, dirs=self.dirs)