def h0h1(k):
'''
USE LONGFLOATS
Compute the h0 & h1 two-scale coefficients
h0_ij = sqrt(2)*int(phi_i(x)*phi_j(2x),x=0..1/2)
h1_ij = sqrt(2)*int(phi_i(x)*phi_j(2x-1),x=1/2..1)
but
h1_ij = (-1)^(i+j) h0_ij from symmetry property of the phi
The h0/h1 are useful because
phi_i(x) = sqrt(2)*sum(j) h0_ij*phi_j(2x) + h1_ij*phi_j(2x-1)
and
phi_i(2x) = (1/sqrt(2))*sum(j) h0_ji*phi_j(x) + g0_ji*psi(x)
phi_i(2x-1) = (1/sqrt(2))*sum(j) h1_ji*phi_j(x) + g1_ji*psi(x)
'''
q = Quadrature(k, [longfloat(0), longfloat((-1, 1))], uselongfloat=1)
x = q.points()
w = q.weights()
npt = len(x)
twox = range(npt)
for i in range(npt):
twox[i] = x[i] + x[i]
h0 = zeromatrix(k, k)
h1 = zeromatrix(k, k)
root2 = longfloat(2).sqrt()
for t in range(npt):
p = philong(x[t], k)
p2 = philong(twox[t], k)
for i in range(k):
faci = p[i] * w[t] * root2
for j in range(k):
h0[i][j] = h0[i][j] + faci * p2[j]
for i in range(k):
for j in range(k):
h1[i][j] = ((-1)**(i + j)) * h0[i][j]
return (h0, h1)
def h0h1(k):
'''
USE LONGFLOATS
Compute the h0 & h1 two-scale coefficients
h0_ij = sqrt(2)*int(phi_i(x)*phi_j(2x),x=0..1/2)
h1_ij = sqrt(2)*int(phi_i(x)*phi_j(2x-1),x=1/2..1)
but
h1_ij = (-1)^(i+j) h0_ij from symmetry property of the phi
The h0/h1 are useful because
phi_i(x) = sqrt(2)*sum(j) h0_ij*phi_j(2x) + h1_ij*phi_j(2x-1)
and
phi_i(2x) = (1/sqrt(2))*sum(j) h0_ji*phi_j(x) + g0_ji*psi(x)
phi_i(2x-1) = (1/sqrt(2))*sum(j) h1_ji*phi_j(x) + g1_ji*psi(x)
'''
q = Quadrature(k,[longfloat(0),longfloat((-1,1))],uselongfloat=1)
x = q.points()
w = q.weights()
npt = len(x)
twox = range(npt)
for i in range(npt):
twox[i] = x[i] + x[i]
h0 = zeromatrix(k,k)
h1 = zeromatrix(k,k)
root2 = longfloat(2).sqrt()
for t in range(npt):
p = philong(x[t],k)
p2= philong(twox[t],k)
for i in range(k):
faci = p[i]*w[t]*root2
for j in range(k):
h0[i][j] = h0[i][j] + faci*p2[j]
for i in range(k):
for j in range(k):
h1[i][j] = ((-1)**(i+j))*h0[i][j]
return (h0, h1)
#QuadratureTest(uselongfloat=1)
for k in range(1, 31):
print '\n\n Testing multiwavelets k =', k, '\n'
(h0, h1, g0, g1) = twoscalecoeffs(k)
# Demonstrate that the scaling functions are ortho-normal
i = 0
j = 0
def f(x):
global k, i, j
p = phi(x, k)
return p[i] * p[j]
q = Quadrature(k, [0, 1.0])
err = 0.0
for i in range(k):
for j in range(i + 1):
overlap = q.integrate(f)
if i == j: overlap = overlap - 1
err = max(err, abs(overlap))
print ' Maximum errror in the overlap of scaling functions', err
# Demonstrate that the wavelets are ortho-normal
i = 0
j = 0
def f(x):
global k, i, j, g0, g1
p = psi(x, k, g0, g1)
print '\n\n Beginning new run \n'
#QuadratureTest(uselongfloat=1)
for k in range(1,31):
print '\n\n Testing multiwavelets k =', k, '\n'
(h0,h1,g0,g1) = twoscalecoeffs(k)
# Demonstrate that the scaling functions are ortho-normal
i = 0
j = 0
def f(x):
global k, i, j
p = phi(x,k)
return p[i]*p[j]
q = Quadrature(k,[0,1.0])
err = 0.0
for i in range(k):
for j in range(i+1):
overlap = q.integrate(f)
if i == j: overlap = overlap - 1
err = max(err,abs(overlap))
print ' Maximum errror in the overlap of scaling functions', err
# Demonstrate that the wavelets are ortho-normal
i = 0
j = 0
def f(x):
global k, i, j, g0, g1
p = psi(x,k,g0,g1)
return p[i]*p[j]