def sparse_grid(func, order, dim, skew=None):
X, W = [], []
bindex = ber.bindex(order - dim + 1, order, dim)
if skew is None:
skew = np.zeros(dim, dtype=int)
else:
skew = np.array(skew, dtype=int)
assert len(skew) == dim
for i in xrange(ber.terms(order, dim) - ber.terms(order - dim, dim)):
I = bindex[i]
x, w = func(skew + I)
w *= (-1)**(order - sum(I)) * comb(dim - 1, order - sum(I))
X.append(x)
W.append(w)
X = np.concatenate(X, 1)
W = np.concatenate(W, 0)
order = np.lexsort(X)
X = X.T[order].T
W = W[order]
# identify non-unique terms
diff = np.diff(X.T, axis=0)
ui = np.ones(len(X.T), bool)
ui[1:] = (diff != 0).any(axis=1)
# merge duplicate nodes
N = len(W)
i = 1
while i < N:
while i < N and ui[i]:
i += 1
j = i + 1
while j < N and not ui[j]:
j += 1
if j - i > 1:
W[i - 1] = np.sum(W[i - 1:j])
i = j + 1
X = X[:, ui]
W = W[ui]
return X, W
def sparse_grid(func, order, dim, skew=None):
X, W = [], []
bindex = ber.bindex(order-dim+1, order, dim)
if skew is None:
skew = np.zeros(dim, dtype=int)
else:
skew = np.array(skew, dtype=int)
assert len(skew)==dim
for i in xrange(ber.terms(order, dim)-ber.terms(order-dim, dim)):
I = bindex[i]
x,w = func(skew+I)
w *= (-1)**(order-sum(I))*comb(dim-1,order-sum(I))
X.append(x)
W.append(w)
X = np.concatenate(X, 1)
W = np.concatenate(W, 0)
X = np.around(X, 15)
order = np.lexsort(tuple(X))
X = X.T[order].T
W = W[order]
# identify non-unique terms
diff = np.diff(X.T, axis=0)
ui = np.ones(len(X.T), bool)
ui[1:] = (diff!=0).any(axis=1)
# merge duplicate nodes
N = len(W)
i = 1
while i<N:
while i<N and ui[i]: i+=1
j = i+1
while j<N and not ui[j]: j+=1
if j-i>1:
W[i-1] = np.sum(W[i-1:j])
i = j+1
X = X[:,ui]
W = W[ui]
return X, W
def lagrange_polynomial(X, sort="GR"):
"""
Lagrange Polynomials
X : array_like
Sample points where the lagrange polynomials shall be
"""
X = np.asfarray(X)
if len(X.shape) == 1:
X = X.reshape(1, X.size)
dim, size = X.shape
order = 1
while ber.terms(order, dim) <= size:
order += 1
indices = np.array(ber.bindex(1, order, dim, sort)[:size])
s, t = np.mgrid[:size, :size]
M = np.prod(X.T[s]**indices[t], -1)
det = np.linalg.det(M)
if det == 0:
raise np.linalg.LinAlgError, "invertable matrix"
v = po.basis(1, order, dim, sort)[:size]
coeffs = np.zeros((size, size))
if size == 2:
coeffs = np.linalg.inv(M)
else:
for i in xrange(size):
for j in xrange(size):
coeffs[i, j] += np.linalg.det(M[1:, 1:])
M = np.roll(M, -1, axis=0)
M = np.roll(M, -1, axis=1)
coeffs /= det
return po.sum(v * (coeffs.T), 1)
def lagrange_polynomial(X, sort="GR"):
"""
Lagrange Polynomials
X : array_like
Sample points where the lagrange polynomials shall be
"""
X = np.asfarray(X)
if len(X.shape)==1:
X = X.reshape(1,X.size)
dim,size = X.shape
order = 1
while ber.terms(order, dim)<=size: order += 1
indices = np.array(ber.bindex(1, order, dim, sort)[:size])
s,t = np.mgrid[:size, :size]
M = np.prod(X.T[s]**indices[t], -1)
det = np.linalg.det(M)
if det==0:
raise np.linalg.LinAlgError, "invertable matrix"
v = po.basis(1, order, dim, sort)[:size]
coeffs = np.zeros((size, size))
if size==2:
coeffs = np.linalg.inv(M)
else:
for i in xrange(size):
for j in xrange(size):
coeffs[i,j] += np.linalg.det(M[1:,1:])
M = np.roll(M, -1, axis=0)
M = np.roll(M, -1, axis=1)
coeffs /= det
return po.sum(v*(coeffs.T), 1)
def orth_bert(N, dist, normed=False, sort="GR"):
"""
# Stabilized process for generating orthogonal
polynomials in a weighted function space.
Add a comment to this line
Parameters
----------
N : int
The upper polynomial order.
dist : Dist
Weighting distribution(s) defining orthogonality.
normed
Returns
-------
P : Poly
The orthogonal polynomial expansion.
Examples
--------
>>> Z = cp.MvNormal([0,0], [[1,.5],[.5,1]])
>>> P = orth_bert(2, Z)
>>> print P
[1.0, q0, q1-0.5q0, q0^2-1.0, -0.5q0^2+q0q1, 0.25q0^2-0.75+q1^2-q0q1]
"""
dim = len(dist)
sort = sort.upper()
# Start orthogonalization
x = po.basis(1,1,dim)
if not ("R" in sort):
x = x[::-1]
foo = ber.Fourier_recursive(dist)
# Create order=0
pool = [po.Poly(1, dim=dim, shape=())]
# start loop
M = ber.terms(N,dim)
for i in xrange(1, M):
par, ax0 = ber.parent(i, dim)
gpar, ax1 = ber.parent(par, dim)
oneup = ber.child(0, dim, ax0)
# calculate rank to cut some terms
rank = ber.multi_index(i, dim)
while rank[-1]==0: rank = rank[:-1]
rank = dim - len(rank)
candi = x[ax0]*pool[par]
for j in xrange(gpar, i):
# cut irrelevant term
if rank and np.any(ber.multi_index(j, dim)[-rank:]):
continue
A = foo(oneup, par, j)
P = pool[j]
candi = candi - P*A
if normed:
candi = candi/np.sqrt(foo(i, i, 0))
pool.append(candi)
if "I" in sort:
pool = pool[::-1]
P = po.Poly([_.A for _ in pool], dim, (ber.terms(N, dim),))
return P
def orth_bert(N, dist, normed=False, sort="GR"):
"""
# Stabilized process for generating orthogonal
polynomials in a weighted function space.
Add a comment to this line
Parameters
----------
N : int
The upper polynomial order.
dist : Dist
Weighting distribution(s) defining orthogonality.
normed
Returns
-------
P : Poly
The orthogonal polynomial expansion.
Examples
--------
>>> Z = cp.MvNormal([0,0], [[1,.5],[.5,1]])
>>> P = orth_bert(2, Z)
>>> print P
[1.0, q0, q1-0.5q0, q0^2-1.0, -0.5q0^2+q0q1, 0.25q0^2-0.75+q1^2-q0q1]
"""
dim = len(dist)
sort = sort.upper()
# Start orthogonalization
x = po.basis(1, 1, dim)
if not ("R" in sort):
x = x[::-1]
foo = ber.Fourier_recursive(dist)
# Create order=0
pool = [po.Poly(1, dim=dim, shape=())]
# start loop
M = ber.terms(N, dim)
for i in xrange(1, M):
par, ax0 = ber.parent(i, dim)
gpar, ax1 = ber.parent(par, dim)
oneup = ber.child(0, dim, ax0)
# calculate rank to cut some terms
rank = ber.multi_index(i, dim)
while rank[-1] == 0:
rank = rank[:-1]
rank = dim - len(rank)
candi = x[ax0] * pool[par]
for j in xrange(gpar, i):
# cut irrelevant term
if rank and np.any(ber.multi_index(j, dim)[-rank:]):
continue
A = foo(oneup, par, j)
P = pool[j]
candi = candi - P * A
if normed:
candi = candi / np.sqrt(foo(i, i, 0))
pool.append(candi)
if "I" in sort:
pool = pool[::-1]
P = po.Poly([_.A for _ in pool], dim, (ber.terms(N, dim), ))
return P