def test_interpolation(self):
norm = numpy.linalg.norm
import random
grid = interpolation.chebyshev_grid(13, 0.0, 1.0)
for x in grid:
self.assertTrue( 0.0 <= x and x <= 1.0 )
self.assertTrue( abs(grid[0]-0.0) < 1E-15 and abs(grid[-1]-1) < 1E-15 )
n_in = 12
rand = lambda a: random.uniform(-a,a)
input_X = interpolation.chebyshev_grid(n_in, rand(0.3)-1, rand(0.3)+1)
input_Y = interpolation.chebyshev_grid(n_in, rand(0.3)-1, rand(0.3)+1)
output_X = numpy.arange(rand(0.25)-0.25, rand(0.25)+0.25, 0.01)
output_Y = numpy.arange(rand(0.25)-0.25, rand(0.25)+0.25, 0.01)
T = interpolation.lagrange_tensor(input_X, output_X, input_Y, output_Y)
#print T.left_matrix
#print T.right_matrix
test_function = lambda x,y: numpy.cos(x*y*6)
input = numpy.zeros((n_in, n_in))
output = numpy.zeros((len(output_X), len(output_Y)))
for i,j in range2(n_in):
input[i,j] = test_function(input_X[i], input_Y[j])
for i,j in range2(len(output_X), len(output_Y)):
output[i,j] = test_function(output_X[i], output_Y[j])
self.assertTrue( norm( T(input)-output )/norm(output) < 1E-4 )
def test_child_tensor(self ):
norm = numpy.linalg.norm
n_cheby = 20
unit_cheby_grid = interpolation.chebyshev_grid(n_cheby, 0.0, 1.0)
child_tensor = {}
for c1,c2 in range2(2,2):
child_tensor[c1,c2] = interpolation.lagrange_tensor(unit_cheby_grid, (unit_cheby_grid+c1)*0.5, unit_cheby_grid, (unit_cheby_grid+c2)*0.5)
test_function = lambda x,y: numpy.cos(6.3*x)*numpy.sin(6.3*y)
for c1,c2 in range2(2):
source = numpy.zeros((n_cheby, n_cheby))
for t1,t2 in range2(n_cheby):
x = unit_cheby_grid[t1]
y = unit_cheby_grid[t2]
source[t1, t2] = test_function(x,y)
output = child_tensor[c1,c2](source)
exact = numpy.zeros((n_cheby, n_cheby))
for t1,t2 in range2(n_cheby):
x=(unit_cheby_grid[t1]+c1)*0.5
y=(unit_cheby_grid[t2]+c2)*0.5
exact[t1, t2] = test_function(x,y)
self.assertTrue( norm(exact-output)/norm(exact) <= 1E-14 )
def bfio_eval(N, input_sources, input_phase, n_cheby, tree_length):
# some pre-processing
phase = lambda x,p: CONST_PI_SQRT2_J*N*p[0]*input_phase(x,(numpy.cos(CONST_2PI*p[1]),numpy.sin(CONST_2PI*p[1])))
sources = [ (polar(N, (float(j1)-N/2, float(j2)-N/2)), input_sources[j1,j2]) for j1,j2 in range2(N) ]
boxed_sources = {}
for j1,j2 in range2(2**tree_length):
a1,a2,b1,b2 = k_spacing*j1, k_spacing*j2, k_spacing*(j1+1), k_spacing*(j2+1)
leaf=[]
for (source_point, source) in sources:
if a1 <= source_point[0] and source_point[0] < b1 and a2 <= source_point[1] and source_point[1] < b2:
leaf.append( (source_point, source) )
boxed_sources[j1,j2] = leaf
sources = boxed_sources
# preliminaries
unit_cheby_grid_1d = interpolation.chebyshev_grid(n_cheby )*0.5+0.5
unit_cheby_grid_2d = numpy.zeros((n_cheby,n_cheby))
for i,j in range2(n_cheby):
unit_cheby_grid_2d[i,j] = unit_cheby_grid1[i]*unit_cheby_grid2[j]
lagrange_tensor
# initialization
print "initialization"
k_level = tree_length
x_level = log2(N)-tree_length
next_coefficients = {}
for i1,i2 in range2(2**x_level):
next_coefficients[i1,i2] = {}
for j1,j2 in range2(2**k_level):
for i1,i2 in range2(2**x_level):
next_coefficients[i1,i2][j1,j2] = numpy.zeros((n_cheby,n_cheby))
interp_grid1 = (unit_cheby_grid_1d+j1)*2**-k_level
interp_grid2 = (unit_cheby_grid_1d+j2)*2**-k_level
if len(sources[j1,j2])>0:
for i1,i2 in range2(2**x_level):
x_center = ((i1+0.5)*x_spacing, (i2+0.5)*2**-x_level)
for t1,t2 in range2(n_cheby):
interp_point = (interp_grid1[t1], interp_grid2[t2])
value = 0.0j
for (source_point, source) in sources:
prod = interpolation.lagrange_prod(source_point[0], interp_grid1, t1)
prod *= interpolation.lagrange_prod(source_point[1], interp_grid2, t2)
prod *= numpy.exp( phase(x_center, interp_point)-phase_p(x_center, source_point) )
value += prod*source
next_coefficients[i1,i2][j1,j2][t1,t2] = value
coefficients = next_coefficients
k_level -= 1
x_level += 1
#recurrence, first half
print "recurrence first half"
while k_level >= log2(N)/2:
print k_level
next_coefficients = {}
for i1,i2 in range2(2**x_level):
next_coefficients[i1,i2] = {}
for j1,j2 in range2(2**k_level): # looping over index of frequency-boxes
print '\t', j1,j2
cheby_grid1 = (unit_cheby_grid+j1)*k_spacing
cheby_grid2 = (unit_cheby_grid+j2)*k_spacing
for i1,i2 in range2(2**x_level): # looping over index of space-boxes
x_center = ( (0.5+i1)*x_spacing, (0.5+i2)*x_spacing)
points = []
values = []
for c1, c2 in [ (j1*2,j2*2), (j1*2+1,j2*2), (j1*2,j2*2+1), (j1*2+1,j2*2+1) ]: # looping over index of child frequency-boxes
temp = expansion_coefficients_firsthalf(N, phase, x_center, tree[k_level+1][i1/2,i2/2][c1,c2], cheby_grid1, cheby_grid2)
for point, value in temp:
if point in points:
values[points.index(point)] += value
else:
points.append(point)
values.append(value)
tree[k_level][i1,i2][j1,j2] = [ (points[k], values[k]) for k in range(len(points)) ]
k_level -= 1
x_level += 1
print "at middle; switching representation"
switch_representation(N, phase, tree, k_level, n_cheby )
print "recursion second half"
while k_level >= 0:
print k_level
x_level = log2(N)-k_level
x_spacing = 2**-x_level
k_spacing = 2**-k_level
for i1,i2 in range2(2**x_level):
tree[k_level][i1,i2] = {}
for i1,i2 in range2(2**x_level): # looping over index of space-boxes
print '\t', i1,i2
cheby_grid1 = (unit_cheby_grid+i1)*x_spacing
cheby_grid2 = (unit_cheby_grid+i2)*x_spacing
for j1,j2 in range2(2**k_level): # looping over index of frequency-boxes
points = []
values = []
for c1, c2 in [ (j1*2,j2*2), (j1*2+1,j2*2), (j1*2,j2*2+1), (j1*2+1,j2*2+1) ]: # looping over index of child frequency-boxes
k_center = ( (0.5+j1)*k_spacing, (0.5+j2)*k_spacing)
temp = expansion_coefficients_secondhalf(N, phase, k_center, tree[k_level+1][i1/2,i2/2][c1,c2], cheby_grid1, cheby_grid2)
for point, value in temp:
if point in points:
values[points.index(point)] += value
else:
points.append(point)
values.append(value)
tree[k_level][i1,i2][j1,j2] = [ (points[k], values[k]) for k in range(len(points)) ]
k_level -= 1
print "termination"
for i1,i2 in range2(N):
def bfio_prototype(N, sources, phase, n_cheby, start_level, end_level):
# preliminaries
unit_cheby_grid = interpolation.chebyshev_grid(n_cheby, 0.0, 1.0)
child_tensor = {}
for c1,c2 in range2(2,2):
child_tensor[c1,c2] = interpolation.lagrange_tensor(unit_cheby_grid, (unit_cheby_grid+c1)*0.5, unit_cheby_grid, (unit_cheby_grid+c2)*0.5)
def new_interp_grid_zeros():
return numpy.zeros((n_cheby, n_cheby), dtype=numpy.complex128)
def expansion_coefficients(level):
coefficients = {}
for i1,i2 in range2(int(N*2**-level)):
coefficients[i1,i2] = {}
for j1,j2 in range2(2**level):
coefficients[i1,i2][j1,j2] = new_interp_grid_zeros()
return coefficients
middle_level = (start_level+end_level)/2
# initialization
print "initialization"
k_level = start_level
x_level = log2(N)-start_level
next_coefficients = expansion_coefficients(k_level)
for j1,j2 in range2(2**k_level):
if len(sources[j1,j2]) > 0:
for i1,i2 in range2(2**x_level):
print j1, i1
lagrange1=interpolation.lagrange_interpolator((unit_cheby_grid+i1)*2**-k_level)
lagrange2=interpolation.lagrange_interpolator((unit_cheby_grid+i2)*2**-k_level)
x_center = ((i1+0.5)*2**-x_level, (i2+0.5)*2**-x_level)
for t1,t2 in range2(n_cheby):
target_point = (lagrange1[t1], lagrange2[t2])
value = 0
for (source_point, source) in sources[j1,j2]:
prod = lagrange1.basis(source_point[0], t1)
prod *= lagrange2.basis(source_point[1], t2)
prod *= numpy.exp( +phase(x_center, source_point))
value += prod*source
next_coefficients[i1,i2][j1,j2][t1,t2] = value*numpy.exp( -phase(x_center, source_point))
return 0
coefficients = next_coefficients
k_level -= 1
x_level += 1
#recurrence, first half
print "recurrence first half"
while k_level >= middle_level:
print k_level
next_coefficients = expansion_coefficients(k_level)
kernel_child = new_interp_grid_zeros()
source_points = new_interp_grid_zeros()
for j1,j2 in range2(2**k_level): # looping over index of frequency-boxes
for i1,i2 in range2(2**x_level): # looping over index of space-boxes
x_center = ((0.5+i1)*2**-x_level, (0.5+i2)*2**x_level)
for c1,c2 in [ (0,0), (0,1), (1,0), (1,1) ]: # looping over children
jc1, jc2 = j1*2+c1, j2*2+c2 # index of child frequency-boxes, at level k_level+1
for t1,t2 in range2(n_cheby):
source_point = ((jc1+unit_cheby_grid[t1])*2**-(k_level+1), (jc2+unit_cheby_grid[t2])*2**-(k_level+1))
kernel_child[t1,t2] = numpy.exp( +phase(x_center, source_point))
next_coefficients[i1,i2][j1,j2] += child_tensor[c1,c2]( kernel_child*coefficients[i1/2,i2/2][jc1,jc2] ) # element-wise pruduct
for t1,t2 in range2(n_cheby):
target_point = ( (j1+unit_cheby_grid[t1])*2**-(k_level), (j2+unit_cheby_grid[t2])*2**-(k_level) )
next_coefficients[i1,i2][j1,j2] *= numpy.exp( -phase(x_center, target_point))
coefficients = next_coefficients
k_level -= 1
x_level += 1
#at middle; switching representation
print "at middle; switching representation"
k_level += 1
x_level -= 1
next_coefficients = expansion_coefficients(k_level)
for j1,j2 in range2(2**k_level): # looping over index of frequency-boxes
for i1,i2 in range2(2**x_level): # looping over index of space-boxes
for t1,t2 in range2(n_cheby):
target_point = ((i1+unit_cheby_grid[t1])*2**-(x_level), (i2+unit_cheby_grid[t2])*2**-(x_level))
value = 0.0j
for s1,s2 in range2(n_cheby):
source_point = ((j1+unit_cheby_grid[s1])*2**-(k_level), (j2+unit_cheby_grid[s2])*2**-(k_level))
value += coefficients[i1,i2][j1,j2][s1,s2]*numpy.exp( +phase(target_point, source_point))
next_coefficients[i1,i2][j1,j2][t1,t2] = value
coefficients = next_coefficients
k_level -= 1
x_level += 1
value = 0.0j
for s1,s2 in range2(n_cheby):
source_point = ((j1+unit_cheby_grid[s1])*2**-(k_level), (j2+unit_cheby_grid[s2])*2**-(k_level))
value += coefficients[i1,i2][j1,j2][s1,s2]*numpy.exp( +phase(target_point, source_point))
next_coefficients[i1,i2][j1,j2][t1,t2] = value
coefficients = next_coefficients
k_level -= 1
x_level += 1
=======
phase = lambda x,p: CONST_PI_SQRT2_J*N*p[0]*input_phase(x,(numpy.cos(CONST_2PI*(p[1]-0.5)),numpy.sin(CONST_2PI*(p[1]-0.5))))
sources = bfio_polar_boxed_sources(N, input_sources, tree_length)
unit_cheby_grid = interpolation.chebyshev_grid(n_cheby, 0, 1)
child_tensor = {}
for c1,c2 in range2(2,2):
child_tensor[c1,c2] = interpolation.lagrange_tensor((unit_cheby_grid+c1)*0.5, unit_cheby_grid, (unit_cheby_grid+c2)*0.5, unit_cheby_grid)
def expansion_coefficients(level):
coefficients = {}
for i1,i2 in range2(int(N*2**-level)):
coefficients[i1,i2] = {}
for j1,j2 in range2(2**level):
coefficients[i1,i2][j1,j2] = numpy.zeros((n_cheby,n_cheby))
return coefficients
# initialization
print "initialization"