First submit of an algorithm

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import math
import numpy as np
import mpmath as mp
import matplotlib.pyplot as plt
from matplotlib import cm
from mpmath import mpf, workdps
from mpl_toolkits.mplot3d import Axes3D
def ethalon_function(x, y):
return np.exp(-x**2)*x**2 + np.exp(-y**2)*y**2
def fdx(x, y):
return 2*x*np.exp(-x**2)*(-x**2 + 1)
def fdy(x, y):
return 2*y*np.exp(-y**2)*(-y**2 + 1)
L = 0.6
left = -2.0
right = 2.0
discr = 100
x_grid = np.linspace(left, right, discr, endpoint=False)
y_grid = np.linspace(left, right, discr, endpoint=False)
x_mesh, y_mesh = np.meshgrid(x_grid, y_grid)
values_mesh = ethalon_function(x_mesh, y_mesh)
#plt.contourf(x_mesh, y_mesh, values_mesh)
#plt.show()
# returns smallest power of 2 that is greater or equal than v
def power_2_bound(v):
v -= 1
v |= v >> 1
v |= v >> 2
v |= v >> 4
v |= v >> 8
v |= v >> 16
return v + 1
# Only for N=2 function!
def nth(n):
return 2**(n+1) - 3
def eps_generator(gamma):
coeff = 1.0 * (gamma - 3) / (gamma - 1)
def eps(k):
return coeff * gamma ** (-k)
return eps
epsilon = eps_generator(8.0)
def betas_generate(n):
return [0] + [nth(i) for i in xrange(1, n)]
# find_minimum_k
def find_minimum_k(delta_f, gamma = mpf(8.0), k_last=None):
# delta func
delta_f = lambda k: (gamma - 2) / (gamma - 1) * gamma**(-k)
if k_last == None:
return 1
assert k_last > 0
return max(mp.ceil(-mp.log(delta_f(k_last)/C, 2)), \
k_last + 1)
def search_optimal_k(inequality, low, high):
while True:
if high == low:
return low
if high == low + 1:
if inequality(epsilon(low) * L, low):
return low
else:
return high
middle = mp.floor((high + low) / 2)
modulo = epsilon(middle) * L
if inequality(modulo, middle):
high = middle
#return search_optimal_k(inequality, modulo, middle)
else:
low = middle + 1
#return search_optimal_k(inequality, middle + 1, high)
def find_optimal_k(inequality, k_last=None):
k_next = find_minimum_k(k_last)
valid = k_next
i = 1
while True:
modulo = epsilon(k_next) * L
if inequality(modulo, k_next):
print 'k_next =', k_next
k_returned = search_optimal_k(inequality, valid, k_next)
print 'k_returned =', k_returned
return epsilon(k_returned) * L, k_returned
print 'reached_k:', k_next
k_next *= 2**i
i += 1
def k_finder(values, betas, L, C=2, N=2, amount_of_k=3, dps=4096):
beta_map = {}
max_size = amount_of_k + 1
with workdps(dps):
# find proper gamma for N
power_of_two = power_2_bound(2*N + 3)
gamma_pow = int(math.log(power_of_two, 2))
gamma = mpf(power_of_two)
# define a_value function
A_CONST = mpf(C) * (2*N + 1) / (N + 1)
a_value_empty = lambda r: A_CONST * gamma**beta_map[kseq[r-1]] * mpf(2)
a_value = lambda n, r: A_CONST * gamma**beta_map[kseq[r-1]] * mpf(2)**(1 - kseq[n-1])
estimated_norm = mpf(np.max(np.abs(values)))
bseq = [estimated_norm]
# First iteration - very simple
inequality = lambda modulo, k: modulo <= estimated_norm / mpf(2*N + 2)
b_one, k_one = find_optimal_k(inequality)
kseq = [k_one]
beta_map[k_one] = nth(k_one)
bseq.append(b_one)
# Now, make a_nr and c_nj sequences
c_nj = [[0]*max_size for _ in xrange(max_size)]
a_nr = [[0]*max_size for _ in xrange(max_size)]
for i in xrange(max_size):
a_nr[i][i] = mpf(N) / mpf(N + 1)
a_nr[i][0] = mpf('1')
c_nj[i][i] = mpf('1')
for j in xrange(i + 1, max_size):
a_nr[i][j] = None
a_nr[1][0] = a_value(1, 0)
c_nj[1][0] = a_nr[1][1] * c_nj[0][0]
partial_sums = []
# Next iterations
for t in xrange(2, max_size):
j_max = t - 1
for r in xrange(t):
a_nr[t][r] = a_value_empty(r)
ssum = sum([c_nj[j_max - 1][k] * bseq[k] for k in xrange(j_max)])
partial_sums.append(a_nr[t][j_max] * ssum)
second_part = sum([c_nj[j_max][j] * bseq[j] for j in xrange(t)]) / (2*N + 2)
#print 'SP:', second_part
inequality = lambda modulo, k: (modulo + sum(partial_sums) / (2**(k))) <= second_part
b_next, k_next = find_optimal_k(inequality, kseq[-1])
kseq.append(k_next)
bseq.append(b_next)
beta_map[k_next] = nth(k_next)
print b_next, k_next
for r in xrange(t):
a_nr[t][r] /= 2**(k_next)
for j in xrange(t - 1, -1, -1):
c_nj[t][j] = sum([a_nr[t][k] * c_nj[k-1][j] for k in xrange(j+1, t + 1)])
return kseq, bseq, a_nr, c_nj