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polynomial.py
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import numpy
import quadratures
class polynomial:
eMod = 'Modal'
eNod = 'Nodal'
eChe = 'Chebyshev'
eLeg = 'Legendre'
eLag = 'Lagrange'
eMon = 'Monomial'
def __init__(self, dim, degree, type, ptype, z, w, quadrature, f):
self.basisType = type
self.dim = dim
self.n = degree + 1 # polynomial order
self.node = numpy.zeros([self.n, dim])
self.T = numpy.zeros(self.n)
self.polyType = ptype
if self.polyType == self.eMon: self.basisType = self.eNod
if self.basisType == self.eMod:
self.l2_Pro(f, z, w, quadrature)
else:
self.interpolate(f, z)
aux = self.evaluate(z, len(z))
self.value = aux[:,0]
self.tangent = aux[:,1]
self.hessian = aux[:,2]
def getType(self):
return (self.basisType + '-'+ self.polyType)
def buildBasis(self, order, z, dimZ):
poly = numpy.zeros([dimZ, order])
dpoly = numpy.zeros([dimZ, order])
d2poly = numpy.zeros([dimZ, order])
for j in range(dimZ):
if dimZ == 1: t = z
else: t = z[j]
if self.basisType == self.eMod:
if (self.polyType == self.eLeg):
poly[j,0] = 1.0
if (order > 1):
poly [j,1] = t
dpoly[j,1] = 1.0
for d in range (1, order - 1):
aux = (2.0 * d + 1.0) / (d + 1.0)
auxx = float(d) / (d + 1.0)
poly [j,d + 1] = aux * t * poly[j,d] - auxx * poly [j,d - 1]
dpoly [j,d + 1] = aux * ( poly[j,d] + t * dpoly [j,d]) - auxx * dpoly [j,d - 1]
d2poly[j,d + 1] = aux * (2.0 * dpoly[j,d] + t * d2poly[j,d]) - auxx * d2poly[j,d - 1]
else: # Chebyshev !! NOT DEFINED FOR DERIVATIVES !!
poly [j,0] = 1.0 # 1st kind: we will use those
if (order > 1):
poly [j,1] = t
#poly [j,1] = 2.0 * t
for d in range (1, order - 1):
poly [j,d + 1] = 2.0 * t * poly [j,d] - poly [j, d - 1]
elif self.basisType == self.eLag:
for d in range (order):
poly [j,d] = self.basis (t,d)
dpoly [j,d] = self.basisD (t,d)
d2poly[j,d] = self.basisD2(t,d)
else:
for d in range (order):
poly[j,d] = pow(t, d)
if d == 0: continue
dpoly[j,d] = pow(t, j - 1)
if j == 1: continue
d2poly[j,d] = pow(t, j - 2)
return poly, dpoly, d2poly
def getBasis(self, order, z, dimZ):
basis, der_basis, der2_basis = self.buildBasis(order, z, len(z))
return basis
def evaluate(self, z, dimZ):
basis, der_basis, der2_basis = self.buildBasis(self.n, z, len(z))
res = numpy.zeros([dimZ, 3, self.dim]) # value, derivative, 2nd derivative,
for j in range(dimZ):
for i in range(self.n):
for k in range(self.dim):
res[j,0,k] += self.node[i][k] * basis[j,i]
res[j,1,k] += self.node[i][k] * der_basis[j,i]
res[j,2,k] += self.node[i][k] * der2_basis[j,i]
return res
def interpolate(self, X, T):
if (self.polyType == self.eMon):
# Vandermonde matrix
n = len(T)
V = numpy.zeros([n, n])
for i in range(n):
for j in range(n):
if (j == 0): V[i,j] = 1.0
else: V[i,j] = pow(T[i], j)
f = numpy.zeros(n)
if (self.dim == 1):
a = numpy.linalg.solve(V, X)
for i in range(n): self.node[i,0] = a[i]
else:
for d in range (self.dim):
for i in range(n): f[i] = X[i,d]
a = numpy.linalg.solve(V, f)
for i in range(n): self.node[i,d] = a[i]
else:
self.T = T
self.node = X
def getPolyWeights(self, n):
w = numpy.zeros(n)
if self.polyType == self.eLeg:
for i in range(self.n): w[i] = 2.0 / (2.0 * i + 1.0)
elif self.polyType == self.eChe:
w[:] = numpy.pi * 0.5
w[0] = numpy.pi
else:
print(' cant set weihght for ', self.polyType)
quit()
return w
def l2_Pro (self, f, z, w, qtype):
phi = numpy.zeros([self.n, self.dim])
Q = len(z)
uvals = numpy.zeros([Q, self.dim])
poly, dpoly, d2poly = self.buildBasis(self.n, z, len(z))
polyWeights = self.getPolyWeights(self.n)
for i in range(self.n):
basis_sum = 0.0
cheat = 0.0
for j in range(Q):
wf = 1.0
if qtype == quadratures.eGC and self.polyType == self.eLeg:
wf = numpy.sqrt(1.0 - z[j] * z[j])
elif qtype != quadratures.eGC and self.polyType == self.eChe:
wf = 1.0 / numpy.sqrt(1.0 - z[j] * z[j])
phi[i] += f[j] * poly[j,i] * wf * w[j]
basis_sum += wf * w[j]
cheat += -0.5 * numpy.pi * w[j]
for i in range(self.n):
for k in range(self.dim):
self.node[i,k] = phi[i,k] / polyWeights[i] # account for polynomial inner product
for j in range(Q):
for i in range(self.n):
for k in range(self.dim):
uvals[j,k] += self.node[i,k] * poly[j,i]
return uvals
def basis(self, x, j):
b = [(x - self.T[m]) / (self.T[j] - self.T[m])
for m in range(self.n) if m != j]
return numpy.prod(b, axis=0)
def basis1(self, x, j, k):
b = [(x - self.T[m]) / (self.T[j] - self.T[m])
for m in range(self.n) if m != j and m != k]
if len(b) == 0: return 1.0
return numpy.prod(b, axis=0)
def basis2(self, x, j, k, i):
b = [(x - self.T[m]) / (self.T[j] - self.T[m])
for m in range(self.n) if m != j and m != k and m != i]
if len(b) == 0: return 1.0
return numpy.prod(b, axis=0)
def basisD(self, x, j):
b = [(self.basis1(x, j, m) / (self.T[j] - self.T[m]))
for m in range(self.n) if m != j ]
return numpy.sum(b, axis=0)
def basisD2(self, x, j):
sum = 0.0
for i in range(self.n):
if j == i: continue
b = [(self.basis2(x, j, i, m) / (self.T[j] - self.T[m]))
for m in range(self.n) if m != i and m != j]
sum += numpy.sum(b, axis=0) / (self.T[j] - self.T[i])
return sum