288 lines
6.9 KiB
Python
288 lines
6.9 KiB
Python
import sys
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import numpy as np
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import itertools
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from interp.tools import log
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def get_phis(X, R):
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"""
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The get_phis function is used to get barycentric coordonites for a point on
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a triangle or tetrahedron:
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in 2D:
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X -- the destination point (2D)
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X = [0,0]
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r -- the three points that make up the containing triangular simplex (2D)
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r = [[-1, -1], [0, 2], [1, -1]]
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this will return [0.333, 0.333, 0.333]
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in 3D:
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X -- the destination point (3D)
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X = [0,0,0]
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R -- the four points that make up the containing simplex, tetrahedron (3D)
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R = [
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[0.0, 0.0, 1.0],
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[0.94280904333606508, 0.0, -0.3333333283722672],
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[-0.47140452166803232, 0.81649658244673617, -0.3333333283722672],
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[-0.47140452166803298, -0.81649658244673584, -0.3333333283722672],
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]
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this will return [0.25, 0.25, 0.25, 0.25]
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"""
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# baker: eq 7
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# TODO: perhaps also test R .. ?
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if len(X) == 2:
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A = np.array([
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[ 1, 1, 1],
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[R[0][0], R[1][0], R[2][0]],
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[R[0][1], R[1][1], R[2][1]],
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])
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b = np.array([ 1,
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X[0],
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X[1]
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])
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elif len(X) == 3:
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A = np.array([
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[ 1, 1, 1, 1 ],
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[R[0][0], R[1][0], R[2][0], R[3][0]],
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[R[0][1], R[1][1], R[2][1], R[3][1]],
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[R[0][2], R[1][2], R[2][2], R[3][2]],
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])
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b = np.array([ 1,
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X[0],
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X[1],
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X[2]
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])
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else:
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raise Exception("inapropriate demension on X")
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try:
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phi = np.linalg.solve(A,b)
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except np.linalg.LinAlgError as e:
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msg = "calculation of phis yielded a linearly dependant system (%s)" % e
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log.error(msg)
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raise Exception(msg)
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phi = np.dot(np.linalg.pinv(A), b)
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return phi
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def qlinear(X, R):
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"""
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this calculates the linear portion of q from R to X
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also, this is baker eq 3
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X = destination point
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R = simplex points
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q = quantity of interest at the simplex points
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"""
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phis = get_phis(X, R.verts)
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qlin = np.sum([q_i * phi_i for q_i, phi_i in zip(R.q, phis)])
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return phis, qlin
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def get_error(phi, R, S, order = 2):
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log.debug("len(phi): %d"% len(phi))
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B = [] # baker eq 9
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w = [] # baker eq 11
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p = pattern(order, len(phi), offset = -1)
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log.debug("pattern: %s" % p)
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for (s,q) in zip(S.verts, S.q):
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cur_phi, cur_qlin = qlinear(s, R)
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l = []
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for i in p:
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cur_sum = cur_phi[i[0]]
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for j in i[1:]:
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cur_sum *= cur_phi[j]
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l.append(cur_sum)
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B.append(l)
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w.append(q - cur_qlin)
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log.debug("B: %s" % B)
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log.debug("w: %s" % w)
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B = np.array(B)
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w = np.array(w)
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A = np.dot(B.T, B)
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b = np.dot(B.T, w)
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# baker solve eq 10
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try:
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abc = np.linalg.solve(A,b)
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except np.linalg.LinAlgError as e:
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log.error("linear calculation went bad, resorting to np.linalg.pinv: %s" % e)
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abc = np.dot(np.linalg.pinv(A), b)
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log.debug(len(abc) == len(p))
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error_term = 0.0
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for (a, i) in zip(abc, p):
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cur_sum = a
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for j in i:
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cur_sum *= phi[j]
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error_term += cur_sum
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log.debug("error_term smb: %s" % error_term)
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return error_term, abc
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def run_baker(X, R, S, order=2):
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"""
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This is the main function to call to get an interpolation to X from the input meshes
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X -- the destination point (2D)
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X = [0,0]
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R = Simplex
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S = extra points
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"""
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log.debug("order = %d" % order)
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log.debug("extra points = %d" % len(S.verts))
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answer = {
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'qlin': None,
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'error': None,
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'final': None,
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}
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# calculate values only for the simplex triangle
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phi, qlin = qlinear(X, R)
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if order == 1:
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answer['qlin'] = qlin
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return answer
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elif order in (2,3,4,5,6):
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error_term, abc = get_error(phi, R, S, order)
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else:
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raise Exception('unsupported order for baker method')
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q_final = qlin + error_term
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answer['qlin' ] = qlin
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answer['error'] = error_term
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answer['final'] = q_final
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answer['abc' ] = abc
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return answer
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def run_baker_3D(X, R, S):
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"""
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This is the main function to call to get an interpolation to X from the input meshes
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X -- the destination point (3D)
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X = [0,0,0]
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R = Simplex (4 points, contains X)
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S = extra points (surrounding, in some manner, R and X, but not in R)
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"""
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# calculate values only for the triangle
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phi, qlin = qlinear_3D(X, R)
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if len(S.verts) == 0:
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answer = {
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'a': None,
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'b': None,
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'c': None,
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'd': None,
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'e': None,
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'f': None,
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'qlin': qlin,
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'error': None,
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'final': None,
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}
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return answer
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B = [] # baker eq 9
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w = [] # baker eq 11
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for (s, q) in zip(S.verts, S.q):
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cur_phi, cur_qlin = qlinear_3D(s, R)
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(phi1, phi2, phi3, phi4) = cur_phi
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B.append(
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[
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phi1 * phi2,
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phi1 * phi3,
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phi1 * phi4,
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phi2 * phi3,
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phi2 * phi4,
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phi3 * phi4,
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]
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)
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w.append(q - cur_qlin)
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B = np.array(B)
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w = np.array(w)
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A = np.dot(B.T, B)
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b = np.dot(B.T, w)
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# baker solve eq 10
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try:
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(a, b, c, d, e, f) = np.linalg.solve(A,b)
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except np.linalg.LinAlgError as e:
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log.error("linear calculation went bad, resorting to np.linalg.pinv: %s", e)
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(a, b, c, d, e, f) = np.dot(np.linalg.pinv(A), b)
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error_term = a * phi[0] * phi[1]\
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+ b * phi[0] * phi[2]\
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+ c * phi[0] * phi[3]\
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+ d * phi[1] * phi[2]\
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+ e * phi[1] * phi[3]\
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+ f * phi[2] * phi[3]
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q_final = qlin + error_term
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answer = {
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'a': a,
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'b': b,
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'c': c,
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'd': d,
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'e': e,
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'f': f,
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'qlin': qlin,
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'error': error_term,
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'final': q_final,
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}
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return answer
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def _boxings(n, k):
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"""\
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source for this function:
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http://old.nabble.com/Simple-combinatorics-with-Numpy-td20086915.html
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http://old.nabble.com/Re:-Simple-combinatorics-with-Numpy-p20099736.html
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"""
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seq, i = [n] * k + [0], k
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while i:
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yield tuple(seq[i] - seq[i+1] for i in xrange(k))
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i = seq.index(0) - 1
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seq[i:k] = [seq[i] - 1] * (k-i)
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def _samples_ur(items, k, offset = 0):
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"""Returns k unordered samples (with replacement) from items."""
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n = len(items)
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for sample in _boxings(k, n):
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selections = [[items[i]]*count for i,count in enumerate(sample)]
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yield tuple([x + offset for sel in selections for x in sel])
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def pattern(power, phicount, offset = 0):
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log.debug("(power = %s, phicount = %s)" % (power, phicount))
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r = []
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for i in _samples_ur(range(1, phicount + 1), power, offset):
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if not len(set(i)) == 1:
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r.append(i)
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return r
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